standing in line experiment

Solomon Asch Conformity Line Experiment Study

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Solomon Asch experimented with investigating the extent to which social pressure from a majority group could affect a person to conform .

He believed the main problem with Sherif’s (1935) conformity experiment was that there was no correct answer to the ambiguous autokinetic experiment.  How could we be sure that a person conformed when there was no correct answer?

Asch (1951) devised what is now regarded as a classic experiment in social psychology, whereby there was an obvious answer to a line judgment task.

If the participant gave an incorrect answer, it would be clear that this was due to group pressure.

Experimental Procedure

Asch used a lab experiment to study conformity, whereby 50 male students from Swarthmore College in the USA participated in a ‘vision test.’

Using a line judgment task, Asch put a naive participant in a room with seven confederates/stooges. The confederates had agreed in advance what their responses would be when presented with the line task.

The real participant did not know this and was led to believe that the other seven confederates/stooges were also real participants like themselves.

Each person in the room had to state aloud which comparison line (A, B or C) was most like the target line. The answer was always obvious.  The real participant sat at the end of the row and gave his or her answer last.

At the start, all participants (including the confederates) gave the correct answers. However, after a few rounds, the confederates started to provide unanimously incorrect answers.

There were 18 trials in total, and the confederates gave the wrong answer on 12 trials (called the critical trials).  Asch was interested to see if the real participant would conform to the majority view.

Asch’s experiment also had a control condition where there were no confederates, only a “real participant.”

Asch measured the number of times each participant conformed to the majority view. On average, about one third (32%) of the participants who were placed in this situation went along and conformed with the clearly incorrect majority on the critical trials.

Over the 12 critical trials, about 75% of participants conformed at least once, and 25% of participants never conformed.

In the control group , with no pressure to conform to confederates, less than 1% of participants gave the wrong answer.

Why did the participants conform so readily?  When they were interviewed after the experiment, most of them said that they did not really believe their conforming answers, but had gone along with the group for fear of being ridiculed or thought “peculiar.

A few of them said that they did believe the group’s answers were correct.

Apparently, people conform for two main reasons: because they want to fit in with the group ( normative influence ) and because they believe the group is better informed than they are ( informational influence ).

Critical Evaluation

One limitation of the study is that is used a biased sample. All the participants were male students who all belonged to the same age group. This means that the study lacks population validity and that the results cannot be generalized to females or older groups of people.

Another problem is that the experiment used an artificial task to measure conformity – judging line lengths. How often are we faced with making a judgment like the one Asch used, where the answer is plain to see?

This means that the study has low ecological validity and the results cannot be generalized to other real-life situations of conformity. Asch replied that he wanted to investigate a situation where the participants could be in no doubt what the correct answer was. In so doing he could explore the true limits of social influence.

Some critics thought the high levels of conformity found by Asch were a reflection of American, 1950’s culture and told us more about the historical and cultural climate of the USA in the 1950s than then they did about the phenomena of conformity.

In the 1950s America was very conservative, involved in an anti-communist witch-hunt (which became known as McCarthyism) against anyone who was thought to hold sympathetic left-wing views.

Perrin and Spencer

Conformity to American values was expected. Support for this comes from studies in the 1970s and 1980s that show lower conformity rates (e.g., Perrin & Spencer, 1980).

Perrin and Spencer (1980) suggested that the Asch effect was a “child of its time.” They carried out an exact replication of the original Asch experiment using engineering, mathematics, and chemistry students as subjects. They found that in only one out of 396 trials did an observer join the erroneous majority.

Perrin and Spencer argue that a cultural change has taken place in the value placed on conformity and obedience and in the position of students.

In America in the 1950s, students were unobtrusive members of society, whereas now, they occupy a free questioning role.

However, one problem in comparing this study with Asch is that very different types of participants are used. Perrin and Spencer used science and engineering students who might be expected to be more independent by training when it came to making perceptual judgments.

Finally, there are ethical issues : participants were not protected from psychological stress which may occur if they disagreed with the majority.

Evidence that participants in Asch-type situations are highly emotional was obtained by Back et al. (1963) who found that participants in the Asch situation had greatly increased levels of autonomic arousal.

This finding also suggests that they were in a conflict situation, finding it hard to decide whether to report what they saw or to conform to the opinion of others.

Asch also deceived the student volunteers claiming they were taking part in a “vision” test; the real purpose was to see how the “naive” participant would react to the behavior of the confederates. However, deception was necessary to produce valid results.

The clip below is not from the original experiment in 1951, but an acted version for television from the 1970s.

Factors Affecting Conformity

In further trials, Asch (1952, 1956) changed the procedure (i.e., independent variables) to investigate which situational factors influenced the level of conformity (dependent variable).

His results and conclusions are given below:

Asch (1956) found that group size influenced whether subjects conformed. The bigger the majority group (no of confederates), the more people conformed, but only up to a certain point.

With one other person (i.e., confederate) in the group conformity was 3%, with two others it increased to 13%, and with three or more it was 32% (or 1/3).

Optimum conformity effects (32%) were found with a majority of 3. Increasing the size of the majority beyond three did not increase the levels of conformity found. Brown and Byrne (1997) suggest that people might suspect collusion if the majority rises beyond three or four.

According to Hogg & Vaughan (1995), the most robust finding is that conformity reaches its full extent with 3-5 person majority, with additional members having little effect.

Lack of Group Unanimity / Presence of an Ally

The study also found that when any one individual differed from the majority, the power of conformity significantly decreased.

This showed that even a small dissent can reduce the power of a larger group, providing an important insight into how individuals can resist social pressure.

As conformity drops off with five members or more, it may be that it’s the unanimity of the group (the confederates all agree with each other) which is more important than the size of the group.

In another variation of the original experiment, Asch broke up the unanimity (total agreement) of the group by introducing a dissenting confederate.

Asch (1956) found that even the presence of just one confederate that goes against the majority choice can reduce conformity by as much as 80%.

For example, in the original experiment, 32% of participants conformed on the critical trials, whereas when one confederate gave the correct answer on all the critical trials conformity dropped to 5%.

This was supported in a study by Allen and Levine (1968). In their version of the experiment, they introduced a dissenting (disagreeing) confederate wearing thick-rimmed glasses – thus suggesting he was slightly visually impaired.

Even with this seemingly incompetent dissenter, conformity dropped from 97% to 64%. Clearly, the presence of an ally decreases conformity.

The absence of group unanimity lowers overall conformity as participants feel less need for social approval of the group (re: normative conformity).

Difficulty of Task

When the (comparison) lines (e.g., A, B, C) were made more similar in length it was harder to judge the correct answer and conformity increased.

When we are uncertain, it seems we look to others for confirmation. The more difficult the task, the greater the conformity.

Answer in Private

When participants were allowed to answer in private (so the rest of the group does not know their response), conformity decreased.

This is because there are fewer group pressures and normative influence is not as powerful, as there is no fear of rejection from the group.

Frequently Asked Questions

How has the asch conformity line experiment influenced our understanding of conformity.

The Asch conformity line experiment has shown that people are susceptible to conforming to group norms even when those norms are clearly incorrect. This experiment has significantly impacted our understanding of social influence and conformity, highlighting the powerful influence of group pressure on individual behavior.

It has helped researchers to understand the importance of social norms and group dynamics in shaping our beliefs and behaviors and has had a significant impact on the study of social psychology.

What are some real-world examples of conformity?

Examples of conformity in everyday life include following fashion trends, conforming to workplace norms, and adopting the beliefs and values of a particular social group. Other examples include conforming to peer pressure, following cultural traditions and customs, and conforming to societal expectations regarding gender roles and behavior.

Conformity can have both positive and negative effects on individuals and society, depending on the behavior’s context and consequences.

What are some of the negative effects of conformity?

Conformity can have negative effects on individuals and society. It can limit creativity and independent thinking, promote harmful social norms and practices, and prevent personal growth and self-expression.

Conforming to a group can also lead to “groupthink,” where the group prioritizes conformity over critical thinking and decision-making, which can result in poor choices.

Moreover, conformity can spread false information and harmful behavior within a group, as individuals may be afraid to challenge the group’s beliefs or actions.

How does conformity differ from obedience?

Conformity involves adjusting one’s behavior or beliefs to align with the norms of a group, even if those beliefs or behaviors are not consistent with one’s personal views. Obedience , on the other hand, involves following the orders or commands of an authority figure, often without question or critical thinking.

While conformity and obedience involve social influence, obedience is usually a response to an explicit request or demand from an authority figure, whereas conformity is a response to implicit social pressure from a group.

What is the Asch effect?

The Asch Effect is a term coined from the Asch Conformity Experiments conducted by Solomon Asch. It refers to the influence of a group majority on an individual’s judgment or behavior, such that the individual may conform to perceived group norms even when those norms are obviously incorrect or counter to the individual’s initial judgment.

This effect underscores the power of social pressure and the strong human tendency towards conformity in group settings.

What is Solomon Asch’s contribution to psychology?

Solomon Asch significantly contributed to psychology through his studies on social pressure and conformity.

His famous conformity experiments in the 1950s demonstrated how individuals often conform to the majority view, even when clearly incorrect.

His work has been fundamental to understanding social influence and group dynamics’ power in shaping individual behaviors and perceptions.

Allen, V. L., & Levine, J. M. (1968). Social support, dissent and conformity. Sociometry , 138-149.

Asch, S. E. (1951). Effects of group pressure upon the modification and distortion of judgment. In H. Guetzkow (ed.) Groups, leadership and men . Pittsburgh, PA: Carnegie Press.

Asch, S. E. (1952). Group forces in the modification and distortion of judgments.

Asch, S. E. (1956). Studies of independence and conformity: I. A minority of one against a unanimous majority. Psychological monographs: General and applied, 70(9) , 1-70.

Back, K. W., Bogdonoff, M. D., Shaw, D. M., & Klein, R. F. (1963). An interpretation of experimental conformity through physiological measures. Behavioral Science, 8(1) , 34.

Bond, R., & Smith, P. B. (1996). Culture and conformity : A meta-analysis of studies using Asch’s (1952b, 1956) line judgment task.  Psychological bulletin ,  119 (1), 111.

Longman, W., Vaughan, G., & Hogg, M. (1995). Introduction to social psychology .

Perrin, S., & Spencer, C. (1980). The Asch effect: a child of its time? Bulletin of the British Psychological Society, 32, 405-406.

Sherif, M., & Sherif, C. W. (1953). Groups in harmony and tension . New York: Harper & Row.

The Mechanics of Conformity: Inside Asch’s Line and Length Experiments

Table of Contents

Have you ever found yourself agreeing with a group even though you had doubts? It’s a common experience that can make us wonder about the nature of our own decisions. This curiosity about human behavior is precisely what led social psychologist conformity experiments">Solomon Asch to explore the phenomenon of conformity in the 1950s. His line judgment experiments, known as the Asch Paradigm , uncovered the subtle yet powerful influence of group pressure on individual judgments. Let’s dive into the mechanics of conformity through the lens of Asch’s groundbreaking research.

The Asch Conformity Experiments

Imagine sitting in a room with a group of people you believe are participants like yourself. You’re shown a card with a line on it, followed by another card with three lines of varying lengths. Your task seems simple: identify which of the three lines is the same length as the one on the first card. But there’s a catch—everyone else in the group, who are actually confederates in the experiment, gives the wrong answer on purpose. Would you stick to what you see, or would you go along with the group?

Setting the Stage

Asch’s experiment placed one real participant in a room with several confederates. The real participant was unaware that the others were in on the experiment. This setup allowed Asch to observe the influence of the majority on the individual.

The Power of the Majority

Asch found that a significant number of participants conformed to the majority’s incorrect answer. Even with clear evidence that the majority was wrong, individuals often chose to deny their own perceptions to fit in with the group.

Understanding Why We Conform

Conformity isn’t just about choosing the same line length—it’s a reflection of how social influence shapes our everyday lives. From fashion trends to political opinions, conformity is a powerful social force.

Social and Normative Influences

Asch’s experiments highlighted two main types of conformity: normative and informational. Normative influence occurs when individuals conform to be liked or accepted by the group. Informational influence happens when people assume the group is better informed than they are.

Cultural Context and Conformity

Cultural context plays a significant role in how and why we conform. In cultures that value social harmony and collective well-being over individuality, conformity rates may be higher.

The Lasting Impact of Asch’s Findings

The implications of Asch’s findings extend beyond a psychology lab. They help us understand phenomena like peer pressure , corporate culture , and even the spread of fake news .

Conformity in the Digital Age

In today’s digital world, social media platforms can create an echo chamber effect , reinforcing conformity. Understanding the Asch Paradigm can help us navigate these virtual spaces more critically.

Challenging Conformity

Asch’s work reminds us that it’s okay to stand against the majority when we believe we’re right. Encouraging critical thinking and fostering environments where dissent ing opinions are valued can help reduce undue conformity.

Key Takeaways from Asch’s Experiments

Asch’s line and length experiments were more than a study of visual perception; they were a window into the human psyche. They revealed our tendencies, the reasons behind them, and the implications for our society.

The Role of Dissent

One crucial finding was that the presence of just one dissenter can significantly reduce conformity. This highlights the importance of allies and the courage to be the first to stand up against the majority.

Conformity Across Time

While Asch’s experiments took place over half a century ago, the core insights remain relevant. The human inclination to conform persists, but each generation faces its own unique set of pressures.

Asch’s line and length experiments shed light on the intricate dance between individuality and social influence. They challenge us to reflect on our own decisions and the factors that shape them. As we navigate a world that’s more connected than ever, understanding the mechanics of conformity is key to maintaining our sense of self amidst the crowd.

What do you think? Would you have conformed in Asch’s experiment? How do you see conformity influencing your daily life?

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Social Psychology

1 Definition, Concept and Research Methods in Social Psychology

• Definition and Concept of Social Psychology
• Research Methods in Social Psychology
• Experimental Methods
• Non-Experimental Methods
• Other Research Methods
• Research Ethics

2 Historical Perspective of Social Psychology, Social Psychology and Other Related Disciplines

• Historical Perspective
• Landmarks in the History of Social Psychology
• Social Psychology and Other Related Disciplines
• Significance of Social Psychology Today

3 Social and Person Perception – Definition, Description and Functional Factors

• Social Cognition – Description and Nature
• Social Perception – Definition
• Understanding Temporary States
• Understanding of the Most Permanent or Lasting Characteristics – Attributions
• Impression Formation
• Implicit Personality Theory
• Person Perception
• Social Categorisation

4 Cognitive Basis and Dynamics of Social Perception and Person Perception

• Cognitive and Motivational Basis of Social and Person Perception
• Bias in Attribution
• Role of Emotions and Motivation in Information Processing
• Motivated Person Perception
• Effect of Cognitive and Emotional States

5 Definition, Concept, Description, Characteristic of Attitude

• Defining Attitudes
• Attitudes, Values, and Beliefs
• Formation of Attitudes
• Functions of Attitudes

6 Components of Attitude

• ABCs of Attitudes
• Properties of Attitudes

7 Predicting Behaviour from Attitude

• Relationship between Attitude and Behaviour
• Attitudes Predict Behaviour
• Attitudes Determine Behaviour?
• Behaviour Determine Attitudes

8 Effecting Attitudinal Change and Cognitive Dissonance Theory, Compliance of Self-perception Theory, Self-affirmation

• Self Presentation
• Cognitive Dissonance
• Self Perception
• Self Affirmation

9 Introduction to Groups- Definition, Characteristics and Types of Groups

• Groups-Definition Meaning and Concepts
• Characteristics Features of Group
• Types of Group
• The Role of Groups

10 Group Process- Social Facilitation, Social Loafing, Group Interaction, Group Polarization and Group Mind

• Social Facilitation
• Social Loafing
• Group Interaction
• Group Polarization

11 Group Behaviour- Influence of Norms, Status and Roles; Introduction to Crowd Behavioural Theory, Crowd Psychology (Classical and Convergence Theories)

• Human Behaviour in Groups
• Influence of Norms Status and Roles
• Crowd Behavioural Theory
• Crowd Psychology

12 Crowd Psychology- Collective Consciousness and Collective Hysteria

• Crowd: Definition and Characteristics
• Crowd Psychology: Definition and Characteristics
• Collective Behaviour
• Collective Hysteria

13 Definition of Norms, Social Norms, Need and Characteristics Features of Norms

• Meaning of Norms
• Types of Norms
• Violation of Social Norms
• Need and Importance of Social Norms
• Characteristic Features of Social Norms

14 Norm Formation, Factors Influencing Norms, Enforcement of Norms, Norm Formation and Social Conformity

• Norm Formation
• Factors Influencing Norm Formation
• Enforcement of Norms
• Social Conformity

15 Autokinetic Experiment in Norm Formation

• Autokinetic Effect
• Sherif’s Experiment
• Salient Features of Sherif’s Autokinetic Experiments
• Critical Appraisal
• Related Latest Research on Norm Formation

16 Norms and Conformity- Asch’s Line of Length Experiments

• Solomon E. Asch – A Leading Social Psychologist
• Line and Length Experiments
• Alternatives Available with Probable Consequences
• Explanation of the Yielding Behaviour
• Variants in Asch’s Experiments
• Salient Features
• Related Research on Asch’s Findings

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The Asch Conformity Experiments

What These Experiments Say About Group Behavior

What Is Conformity?

Factors that influence conformity.

The Asch conformity experiments were a series of psychological experiments conducted by Solomon Asch in the 1950s. The experiments revealed the degree to which a person's own opinions are influenced by those of a group . Asch found that people were willing to ignore reality and give an incorrect answer in order to conform to the rest of the group.

At a Glance

The Asch conformity experiments are among the most famous in psychology's history and have inspired a wealth of additional research on conformity and group behavior. This research has provided important insight into how, why, and when people conform and the effects of social pressure on behavior.

Do you think of yourself as a conformist or a non-conformist? Most people believe that they are non-conformist enough to stand up to a group when they know they are right, but conformist enough to blend in with the rest of their peers.

Research suggests that people are often much more prone to conform than they believe they might be.

Imagine yourself in this situation: You've signed up to participate in a psychology experiment in which you are asked to complete a vision test.

Seated in a room with the other participants, you are shown a line segment and then asked to choose the matching line from a group of three segments of different lengths.

The experimenter asks each participant individually to select the matching line segment. On some occasions, everyone in the group chooses the correct line, but occasionally, the other participants unanimously declare that a different line is actually the correct match.

So what do you do when the experimenter asks you which line is the right match? Do you go with your initial response, or do you choose to conform to the rest of the group?

Conformity in Psychology

In psychological terms, conformity refers to an individual's tendency to follow the unspoken rules or behaviors of the social group to which they belong. Researchers have long been been curious about the degree to which people follow or rebel against social norms.

Asch was interested in looking at how pressure from a group could lead people to conform, even when they knew that the rest of the group was wrong. The purpose of the Asch conformity experiment was to demonstrate the power of conformity in groups.

Methodology of Asch's Experiments

Asch's experiments involved having people who were in on the experiment pretend to be regular participants alongside those who were actual, unaware subjects of the study. Those that were in on the experiment would behave in certain ways to see if their actions had an influence on the actual experimental participants.

In each experiment, a naive student participant was placed in a room with several other confederates who were in on the experiment. The subjects were told that they were taking part in a "vision test." All told, a total of 50 students were part of Asch’s experimental condition.

The confederates were all told what their responses would be when the line task was presented. The naive participant, however, had no inkling that the other students were not real participants. After the line task was presented, each student verbally announced which line (either 1, 2, or 3) matched the target line.

Critical Trials

There were 18 different trials in the experimental condition , and the confederates gave incorrect responses in 12 of them, which Asch referred to as the "critical trials." The purpose of these critical trials was to see if the participants would change their answer in order to conform to how the others in the group responded.

During the first part of the procedure, the confederates answered the questions correctly. However, they eventually began providing incorrect answers based on how they had been instructed by the experimenters.

Control Condition

The study also included 37 participants in a control condition . In order to ensure that the average person could accurately gauge the length of the lines, the control group was asked to individually write down the correct match. According to these results, participants were very accurate in their line judgments, choosing the correct answer 99% of the time.

Results of the Asch Conformity Experiments

Nearly 75% of the participants in the conformity experiments went along with the rest of the group at least one time.

After combining the trials, the results indicated that participants conformed to the incorrect group answer approximately one-third of the time.

The experiments also looked at the effect that the number of people present in the group had on conformity. When just one confederate was present, there was virtually no impact on participants' answers. The presence of two confederates had only a tiny effect. The level of conformity seen with three or more confederates was far more significant.

Asch also found that having one of the confederates give the correct answer while the rest of the confederates gave the incorrect answer dramatically lowered conformity. In this situation, just 5% to 10% of the participants conformed to the rest of the group (depending on how often the ally answered correctly). Later studies have also supported this finding, suggesting that having social support is an important tool in combating conformity.

At the conclusion of the Asch experiments, participants were asked why they had gone along with the rest of the group. In most cases, the students stated that while they knew the rest of the group was wrong, they did not want to risk facing ridicule. A few of the participants suggested that they actually believed the other members of the group were correct in their answers.

These results suggest that conformity can be influenced both by a need to fit in and a belief that other people are smarter or better informed.

Given the level of conformity seen in Asch's experiments, conformity can be even stronger in real-life situations where stimuli are more ambiguous or more difficult to judge.

Asch went on to conduct further experiments in order to determine which factors influenced how and when people conform. He found that:

• Conformity tends to increase when more people are present . However, there is little change once the group size goes beyond four or five people.
• Conformity also increases when the task becomes more difficult . In the face of uncertainty, people turn to others for information about how to respond.
• Conformity increases when other members of the group are of a higher social status . When people view the others in the group as more powerful, influential, or knowledgeable than themselves, they are more likely to go along with the group.
• Conformity tends to decrease, however, when people are able to respond privately . Research has also shown that conformity decreases if people have support from at least one other individual in a group.

Criticisms of the Asch Conformity Experiments

One of the major criticisms of Asch's conformity experiments centers on the reasons why participants choose to conform. According to some critics, individuals may have actually been motivated to avoid conflict, rather than an actual desire to conform to the rest of the group.

Another criticism is that the results of the experiment in the lab may not generalize to real-world situations.

Many social psychology experts believe that while real-world situations may not be as clear-cut as they are in the lab, the actual social pressure to conform is probably much greater, which can dramatically increase conformist behaviors.

Asch SE. Studies of independence and conformity: I. A minority of one against a unanimous majority . Psychological Monographs: General and Applied . 1956;70(9):1-70. doi:10.1037/h0093718

Morgan TJH, Laland KN, Harris PL. The development of adaptive conformity in young children: effects of uncertainty and consensus . Dev Sci. 2015;18(4):511-524. doi:10.1111/desc.12231

Asch SE. Effects of group pressure upon the modification and distortion of judgments . In: Guetzkow H, ed.  Groups, Leadership and Men; Research in Human Relations. Carnegie Press. 1951:177–190.

Britt MA. Psych Experiments: From Pavlov's Dogs to Rorschach's Inkblots . Adams Media. 

Myers DG. Exploring Psychology (9th ed.). Worth Publishers.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Everything about the way we stand in lines is wrong

Each time I wait in the snaking checkout line at my local Trader Joe's, I have to remind myself of an important point:  The line may be long, but it's saving me time.

The truth is, human beings hate standing in lines, but we love being at the front of them.

According to a growing body of evidence in "queueing theory," that's a recipe for inefficiency.

Whether they're checkout lines or the lines outside the Apple store before a launch, inefficient line designs are wasting precious hours of our lives.

Take the supermarket.

Most major stores offer a dozen or so checkout lanes, all of which call upon the cart-pushing shopper to make a snap judgment about which line will move fastest. This is comforting psychologically because we like to feel in control, but it actually ends up increasing the average wait time, says Bill Hammack, an engineer at the University of Illinois, Urbana.

Hammack, who hosts a YouTube channel  where he explains how everyday stuff works, says  the supermarket problem  comes down to simple probability.

If there are three cash registers, there's only a one in three chance you'll pick the fastest one. Two-thirds of the time, you'll see the other lines gliding by while you're stuck waiting, watching your ice cream thaw.

The solution: Abolish individual checkout lanes and replace them with one long line that feeds to registers as they open up. 

For a store with three registers, Hammack says, it ends up being about three times faster. My local Trader Joe's already does this, as do Best Buy and TJ Maxx. While customers no longer get the rush of victory from picking the fastest line, ultimately they leave the store earlier.

In typical cases, a delayed register wastes the time of everyone in that lane. In a single queue, a delayed register only affects one person; other registers will still open up.

This is a different kind of line than supermarket checkouts, and it can also be improved.

In separate studies, one in 2012 and another in 2014 , Danish researchers found that, if stores want to minimize wait times, they should flip the script: The people who arrive last should be the ones who get served first.

In the 2014 study, an experiment involving 144 participants revealed that giving preference to both the earliest arrivers and people at random increased average wait times. Giving preference to those who showed up last, however, made it pointless for people to set up shop a week ahead of time, so they didn't.

But adopting this set-up en masse would irritate a lot of people.

When we banish typical line rules, research has found, the sense of fairness that accompanies them  goes out the window . People feel robbed.

People who stand in line longer feel they are more deserving of service because they paid for it with their time. (If you don't buy that theory, the next time there's an iPhone release, cut the first person in line and see what happens.)

The first person in line might also irrationally assume that being first somehow signifies to the universe that they  must be most deserving; otherwise, they wouldn't be first.

Like the thrill of buying our fruits and vegetables before those less cunning, the first-come-first-served model gets people thinking in terms of rewards instead of logistics. The line to buy the thing becomes more important than the thing itself.

If the research on lines tells us anything, it's that everyone will be better off if those irrational practices are abandoned. It's something to think about the next time you're setting up your tent on a sidewalk.

Watch: Here's what everyone gets wrong about geniuses

• Main content

What really drives you crazy about waiting in line (it actually isn’t the wait at all)

The Washington Post :

If the people who study the psychology of waiting in line — yes, there is such a thing — have an origin story, it’s this:

It was the 1950s, and a high-rise office building in Manhattan had a problem. The tenants complained of an excessively long wait for the elevator when people arrived in the morning, took their lunch break, and left at night. Engineers examined the building and determined that nothing could be done to speed up the service.

Desperate to keep his tenants, the building manager turned to his staff for suggestions. One employee noted that people were probably just bored and recommended installing floor-to-ceiling mirrors near the elevators, so people could look at themselves and each other while waiting. This was done, and complaints dropped to nearly zero.

Another important factor is the speed and pacing of the line. Research by Daniel Kahneman, the psychologist whose work sparked a broad rethinking in economics, argues that consumers waiting in line experience a dual response: They become gradually demoralized as they wait but have a positive response to each forward movement of the queue. Their overall feeling about the experience depends on how these two responses balance out.

Other research by Kahneman on how people remember unpleasant activities suggests that the way we remember a line is heavily influenced by how the experience ends. A line that starts slow and speeds up is very different, and psychologically preferable, from waiting in a line that starts fast and then slows to a crawl.

Read the whole story: The Washington Post

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Privacy Overview

The Cost of Cutting in Line

No one likes to waste time standing in line. So why don't more people try to bribe their way to the front? Should companies allow some customers to move to the front of the line for a hefty fee? Is there a market for time?

Felix Oberholzer-Gee began to ponder this issue as he was, of course, waiting in line at the airport. Later, he decided to conduct a field experiment to explore the question. He and a team of experimenters equipped with small bills approached 500 people in lines and offered a cash payment of up to $10 to cut in. Would the bribe be accepted? How much would it take to jump the queue? And how would social norms and a sense of fairness play out along the line?

The results were quite surprising.

As might be expected, the higher the amount of payment offered, the more likely individuals were to allow a stranger to cut ahead of them. The surprise? The line-holders allowed the person to cut in but most wouldn't accept the money in return. (Students and women were more likely to pocket the cash.) Oberholzer-Gee took this to mean that people will allow cuts if they perceive the queue jumper has a real need to save time, though most people felt it inappropriate to cash in on that need. For line-holders, a higher bribe meant the jumper was more desperate.

But there were limits to that generosity. When Oberholzer-Gee tried to cut into the same line a second time, the crowd grew hostile and he felt forced to retreat to protect his safety. "The angry reactions suggest that even those who had accepted payment during the first encounter did not view the transaction as an ordinary exchange. Rather, the willingness to let someone cut in seems to be based on seeing the situation as exceptional."

He wrote up the results of his experiment as a paper, "A Market for Time: Fairness and Efficiency in Waiting Lines." In the following e-mail Q&A, Oberholzer-Gee discusses the research and what it might mean for companies who might want to, in fact, create a market for time.

Sean Silverthorne: What gave you the idea to conduct this field experiment?

Felix Oberholzer-Gee: Companies can purchase almost every input they need: labor, office space, a great brand. But sometimes, the price system breaks down and there is no market. In my research, I have been interested in this missing-markets problem for quite a while. Some of my earlier studies show, for example, that companies must not try to buy local approval if they plan to open a new facility that the local community does not welcome. If Wal-Mart plans to open a new store and the town does not like it, the worst thing the company could do is try to bribe its way into this market.

One day, standing in line at the airport, it occurred to me that waiting lines appeared to be another example for a missing-markets problem. Why do I have to wait at airports? Why don't the airlines offer a service that would allow me to pay $20 and move to the head of the line? To find out, I conducted this field experiment.

Q: What were the motives of people who allowed your experimenters to cut in line? Why didn't many of them accept payment? And if money wasn't an issue, why did higher payments correlate with a willingness by line-holders to allow a stranger jump the queue?

A: The data clearly show that you are more likely to be able to jump the queue if you offer more money. So first I thought that this was not an example of a missing-markets problem at all. But I was wrong. You can "purchase" a position in the line, but the people who let you cut in will not accept your money. Their behavior is motivated by a norm that says you should help others when they are in need, but you must not exploit this situation. Monetary incentives "work" in this instance because people read them as a sign for the needs of others. How hurried are you, really? If you offer $20, you must be really hard pressed for time.

Q: Why was the reaction to you so hostile when you tried to jump the same line a second time? What were the reactions like from the people you approached?

A: The same persons who let me cut in the first time got very angry when I approached them again. All fifteen individuals rejected my request, most of them appeared upset, some angry, a few outright hostile, suggesting that it was probably not safe to continue the experiment.

The data clearly show that you are more likely to be able to jump the queue if you offer more money.

The reason is that the helping norm applies to unusual circumstances. Other research shows that individuals are more likely to help if the person in need bears no responsibility for his situation. But what I did is clearly different. Arriving late at a train station and banking on others' willingness to help me cut in line violates the norm. From a social point of view, this is quite ingenious. In groups with helping norms, there will always be some people who try to exploit the friendliness of others. The anger displayed in the experiment protects the group from such exploitation.

Q: What can product and service providers learn from your research in terms of creating waiting lists?

A: Take waiting at the airport as an example. Charging passengers for the right to jump the queue is problematic. First, the airlines have some control over the length of waiting lines. If they hired additional staff, lines would be shorter. In this context, customers would feel exploited if they had to pay to make their flights. In addition, my experiments show that individuals don't feel comfortable cutting in line when jumping the queue results in longer waiting periods for others.

Given the results of the field experiment, it is easy to see why airlines don't offer an opportunity to move to the head of the line. But my research also suggests alternative systems that are more acceptable. For instance, airlines could install an emergency counter for passengers whose flights depart within thirty minutes. Using the emergency counter would result in a hefty fee. My research predicts that this system is likely to be seen as fair. First, there are no negative externalities; other passengers don't have to wait longer because someone else arrives late. Second, the fee prevents passengers from misusing the emergency system. Third, installing and operating these emergency counters is costly.

In all research on the fairness of prices, customers agree that it is fair to reimburse companies for the cost of providing extra services.

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The Everything Guide to Standing on Line

Why we like it, and what’s worth the wait right now..

• By Belle Cushing, Melissa Dahl, Nicholas Gill , Trupti Rami, Katy Schneider, Lauren Schwartzberg, Erica Schwiegershausen & David Wallace-Wells
• Published Nov 1, 2015

Because It Makes Us Feel Like Connoisseurs Among Connoisseurs

Until very recently, people tended to think of the perpetual line as a feature of failed states, where “state” is a catchall term for the social order and its deprivations — think of Depression bread lines, Battle of Britain ration queues, Perestroika tent-city awaiting Moscow’s first McDonald’s. But if you live in a place like New York, you probably now think instead of the ones that snake into Blue Bottle Coffee , or outside sneaker stores to collect shoes that may never be worn, only displayed, or toward a very public audience with Marina Abramovic . These are lines not as deprivation case studies but consumer passion plays. And from the look of things, people seem extremely happy to stand in line (or on line, as New Yorkers so stubbornly say), for almost anything, for almost any amount of time.

These days, the DMV works pretty well, you can hire a TaskRabbit to hold your spot for the Manolo Blahnik sample sale, and there isn’t much you can’t summon from Amazon Prime. But then there are all the lines we’re improbably opting into, since the culture of yuppie abundance that should’ve brought about the end of waiting has actually triggered something like its opposite — a whole new, cultlike fervor for it. Lines are no longer even really a cost to be avoided; they’re an incitement and part of the tribal appeal. What possible other explanation could there be for spending four innings of a Mets playoff game shuffling toward Shake Shack ? (Especially when you know, as any fan should, that every other concession stand at Citi Field also serves burgers forged from Pat LaFrieda beef .)

When, early this fall, Chick-fil-A invited customers to camp out in front of what was billed as the chain’s first New York outpost (actually it was the second — there’s another inside an NYU building), it was just the opening gala for what will truly be an unprecedented line-stuffed fall in the city. When Star Wars: The Force Awakens opens in December, it will surely be preceded by days of camping out, and this month, streetwear legend Supreme had to cancel its launch event for its Supreme Jordan 5 because it was so terrified of the inevitable stampede. Dominique Ansel launched his Cronut way back in 2012 — a seminal 18-month period for lines that included the arrival of the “ Rain Room ,” Mission Chinese , Pok Pok Ny , and Christian Marclay’s The Clock — but every morning the pilgrims are still circling around Spring Street like a noose.

Another way to describe those 18 months is “the life span of the iPhone 5.” Smartphones changed everything about waiting — allowing us to do the only thing we really wanted to do (play on our phones Pavlovishly) while getting credit for the things we think we ought to be doing (accumulating experiences IRL). On the one hand, the new line-lust is just a gullible response to a marketing ploy — introducing scarcity to make people really want something. But, as Pierre Bourdieu could mansplain if you’d pulled him into a movie line, waiting is also an anxious performance of status — signaling that one has an endless amount of time to devote to consumer connoisseurship (invariably of art, food, fashion) while confirming to yourself that you are, in fact, in the right place. At least a right place.

“The act of stepping into line is like putting a stake in the ground: submissive to the people in front of you, dominant to the people behind you,” David Andrews writes in his upcoming book Why Does the Other Line Always Move Faster? — a pretty delicious work of trail-mix pop social science that doesn’t really answer that question but is formatted to fit in a line-waiter’s jacket pocket. Andrews’s view of line culture is playfully paranoid — he sees it as a form of social control, “and if you don’t believe me, recall the sweat that goes into teaching rambunctious kindergartners to pipe down, take turns, and line up already.” But he is sharp on the psychology of it, especially on the way lines turn time into a precious resource (one stockpiled very efficiently by the creative class, which suits Danny Bowien just fine). And then there’s the Zen-surrender (or is it lemminglike?) appeal: “Wouldn’t it be nice to have a switch to turn off the part of our brain not currently needed for the task at hand?” Andrews asks. “As far as social conventions like line-standing go, humans aspire to … the condition of robot.”

As usual, for New York, the robot Warhol got there first: “I like the idea that people in New York now have to wait in line for movies,” he wrote in The Philosophy of Andy Warhol. “The idea for waiting for something makes it more exciting anyway. Never getting in is the most exciting, but after that waiting to get in is the most exciting.” — By David Wallace-Wells

Some Advice From the City’s Seasoned Queuers Watch out for German bloggers.

“It’s important to remind yourself of all the things you learned in kindergarten while you’re standing in line: Cover your cough. Try not to annoy the people around you. Don’t smoke.” —Geri Strugatz, in line for The Late Show With Stephen Colbert.

“When you learn that your flight’s canceled, people usually start lining up to see the flight attendants to get booked on the next flight. Good luck with that. While you’re standing there, use your time to tweet at the airline ; most of them have really responsive social media, and they can rebook you much quicker than the people standing there trying to help 100 people standing in front of them all at once.” —Brian Kelly, founder and CEO of the Points Guy , airport expert.

“The only situation in which it’s okay to cut is if it’s something you’re really passionate about and this is your one chance. I cut to the front of a line at a Robert Plant concert because I wanted to get front and center. Other people there, who were maybe in their 50s and 60s, they already had their time. I figured this is my time.” —Natalie Torn, on line for a Grumpy Cat book signing at the Strand .

“Offer to buy the security guard coffee; that way, if I have to go to the bathroom, I can just tell security to save my spot.” —Corey Arevalo, on line at Supreme .

“Be aware of who’s listening to your conversation. On the Apple lines, there are a lot of bloggers. One time I saw this German blogger kind of eyeing me and writing down what I was saying to publish without asking me. When I confronted him about it, his answer was ‘This is America.’ ” —Greg Packer, regular Apple-product waiter.

“If a friend shows up late and joins you in line, just continue talking and act like you’re picking up a conversation where it left off, so people think he or she is just coming back from the bathroom. Be like, ‘So, what did she tell you when you guys broke up?’ ” —Aaron Yazdian, on line for a ramen burger at Smorgasburg .

“A bathroom emergency is the only legitimate excuse to ask someone to hold your place while you leave a line and then come back. For hunger, I think you have to wait.” —Jane Park, on line at Shake Shack . — Reporting by Trupti Rami, Lauren Schwartzberg, and Erica Schwiegershausen

“I’ve Been Waiting on Line for 39 Years”

“Since 1976, I’ve been going to Russ & Daughters on Sundays for Gaspé and Scottish smoked salmon. Up until 15 years ago, if you got there at 7:55 for the 8 a.m. opening, you’d be first or second. Now you might be 20th. So to be one of the first five, you have to get there at 7:45 . Back then, we didn’t form a line. When the doors opened, you just proceeded inside the store in the order in which you arrived. A lot of the young people aren’t as comfortable getting into a conversation. The line has become not unfriendly but not as intimate as it was when it was just regulars. But there wasn’t really ever a hell of a lot of conversation. It’s early in the morning, and people are there for a specific purpose.” —Larry Fuchsman, photographer — As told to Belle Cushing

How to Pick the Fastest Line

Pick a line that’s mostly men. Guys are more easily irritated by long waits than women and are more likely to “be less accepting of their inevitability,” according to researchers at the University of Surrey. It’s possible, then, that men may abandon a line more easily than women, which may bump you up a spot or two.

Veer left. We drive on the right, we tend to pass people walking toward us on the right, and an estimated 90 percent of the world’s population is right-handed. In his new book Math Geek: From Klein Bottles to Chaos Theory, a Guide to the Nerdiest Math Facts, Theorems, and Equations, science writer Raphael Rosen argues that this directional bias leads most of us to naturally head to our right when choosing a line, leaving the left lanes more open. (Hey, it’s worth a try.)

But don’t overthink it. Just pick the line with the fewest people. At the grocery store in particular, the number of people in a line matters more than the number of items in their basket. That’s according to Dan Meyer of the San Francisco math-education start-up Desmos . In 2009, Meyer studied checkout data at his local grocery store and found that each person in line adds at least 41 seconds to your wait time, regardless of how much stuff they’re purchasing. That number accounts for the time it takes the person to say hello to the cashier, pay for the items, gather their stuff, and leave the register. Each item that each person is holding adds an additional 2.8 seconds to your wait. This means, Meyer says, that a line with fewer people in it will move more quickly, even if each person is carrying a lot of stuff.

Ignore the express lane. Again, every additional person takes up much more time than each additional item being purchased, so the express option often doesn’t end up saving much time. If you are waiting behind two people in an express lane and each of you has just five items, it will take three minutes and six seconds for you to get through the line. Compare that to a scenario in which you picked a regular line that also would’ve put two people ahead of you; let’s also say each of you has ten items. According to Meyer’s estimates, you’d be through the line in three minutes and 48 seconds. So the express lane would’ve saved you a whopping 42 seconds, plus it meant you had to buy fewer items. The idea of an “express” checkout is mostly in your head.

Pick a cash-only line if it’s an option. Meyer’s study of his local grocery store found that when people paid in cash, their transactions tended to be quicker than those paying with credit. This was surprising to him, but it makes sense: One credit-card reader can be vastly different from another, causing people to fumble with their cards. Cash, on the other hand, is always the same; we know how to handle that. The slowest method of payment of all, of course, was checks. Don’t pay with a check.

You’re not imagining it. The line you aren’t in really is more likely to move faster. Let’s say you’re in the multiple-line, multiple-cashier scenario, with three lines to choose from. You choose the left lane, but the line to your right seems to be moving faster. Why does this always happen? Hammack explains that this is a simple question of probability. In a three-line setup, you have a one-in-three chance that you have picked the quickest line; meanwhile, there is a two-in-three chance that one of the other two lines will be faster. The odds, unfortunately, are not in your favor (but the advice above can certainly help). — By Melissa Dahl

7 Things Worth Standing on Line For And Grimaldi’s is not one of them. (According to our experts.)

The line: “The Hungarian author Laszlo Krasznahorkai, who was already a growing force in hipster literary circles before he won this year’s Man Booker International Prize, alongside his fan Salman Rushdie at the 92Y on December 14 . ” —Boris Kachka, contributing editor. When to stand in it: It’s almost sure to be sold out, so come an hour before the 8 p.m. event and wait in the cancellation line for tickets on a first-come basis.

The line: “ Hamilton rush tickets ($10). Plus, the cast has been known to do impromptu performances for the line-waiters.” —Jesse Green, theater critic. When to stand in it: 2.5 hours prior to the show, when the Ham4Ham lottery is held. The musical performance by a member of the cast of Hamilton or another Broadway guest happens two to three times a week, and for that you just have to get lucky.

The line: “The veggie burger at Superiority Burger . ” —Alan Sytsma, food editor, Grub Street. When to stand in it: 7 p.m. They open at six, and the first half-hour is often people who want to be first in the door; after that, it becomes much more manageable.

The line: “ Buzzard Crest stand at the Union Square Greenmarket . Perfectly sweet, intensely aromatic indigenous grapes that will ruin you for supermarket imports.” —Rob Patronite, food editor and critic. When to stand in it: 8 a.m., right when they open. The line starts forming at 8:15 a.m. and goes until they run out of grapes, which can be as early as 10 a.m.

The line: “ Di Fara Pizza . The only pizza line that’s worth it.” —R.P. When to stand in it: Weekdays at 2 p.m., when most of the initial line — which starts forming at 11:30 a.m. for the noon opening — has been served.

The line: “ The day-after-Christmas sale at Saks . ” —Diana Tsui, senior market editor, the Cut. When to stand in it: 7 a.m. (by 7:30 a.m., the line has wrapped around the corner). “When the doors open at 8 a.m., it’s a stampede. The trick is to head straight for eighth floor and hit up the shoes because it tends to be 50 to 60 percent off the lowest marked price.”

The line: “ Star Wars: The Force Awakens , which opens on December 18 at AMC Loews Lincoln Square 13, home to the biggest screen in New York.” —Lane Brown, culture editor. When to stand in it: Anytime that’s not the evening and not between December 24 and December 31, when the schools are on break. And of course, buy your tickets online , never at the box office.

The line: “ Top of the Rock at Rockefeller Center, which positions you directly atop the middle of Manhattan, with the city fanning out on all sides. Not the Empire State Building, where the vista doesn’t include the, um, Empire State Building.” —Justin Davidson, architecture critic. When to stand in it: Anytime before noon; it’s busiest during the sunset hour. — By Katy Schneider

You Can Also Hire Someone to Stand on Line for You Waiting for Chick-fil-A with Robert Samuel, founder of Same Ole Line Dudes ( sameolelinedudes.com ).

Is now still a good time to talk? Actually, I’m in line at Chick-fil-A. Hey, watch it! [ Muffled sounds. ] I almost just got run over by a taxi.

You okay? Yeah, I had a lawn chair on me, like a shield. I hit the car right back. Okay, we can talk now. I won’t be inside Chick-fil-A for a minute.

So how did Same Ole Line Dudes start? In 2012, I lost my job at AT&T. When the iPhone 5 came out, I posted an ad on Craigslist to wait for someone’s phone. The customer’s online order ended up going through, so he called me and said, “I don’t need you anymore, but I’ll still pay you the $100.” Then I sold my spot for another $100. I was second in line! I sold a few spots that night and some milk crates for people to sit on. By the time the door opened, I had $325. But it wasn’t until the Cronut that it became a business.

How many jobs do you have per week? I’d say six to eight. I charge $25 for the first hour and $10 each half-hour after that. Two-hour minimum. It’s $5 extra when something falls from the sky. We have 12 to 15 part-time employees, who I just call or text when I need someone.

Are you ever short on line-waiters? A few times, I’ve had to find someone on the street. If one of my people was a no-show or overslept and I have to make a delivery by 9 a.m., I’ll find a neighbor. You’d be surprised; people want to do it. I mean, $25 just for waiting around two hours is a good deal.

Do businesses ever get mad? When the Cronut line first took off, I started selling them on the spot, but Dominique did not like that. I only did that one time.

What’s the hardest line to wait in? Saturday Night Live. They discourage spot holders, so we have to be really careful.

Do you have regulars? One of my best customers, she started out asking for Cronuts. Since then, we waited in the Thanksgiving Day parade for her, I’ve gotten her two iPhones, I’ve gotten her kids toys from GameStop. It’s almost like a best friend that does you favors. Except I charge. — By Belle Cushing

Or, You Can Cut … Ethically questionable tactics from 24-year-old producer Lexi T.

“I do not wait in lines. I see a line, and it doesn’t mean anything to me. What’s the worst that’s going to happen? Someone’s like, ‘Oh, excuse me?’ and you apologize? The end.”

Outside a Fashion Week party : “Just walk straight to the front. When there’s a big crowd, no one really knows what’s going on, and if you look like you know what you’re doing, who’s going to question you? It helps that I’m five-foot-two and smiley.”

Outside a club: “I’ve seen people pretend to drop something, bend down and ‘look’ for it, and then, oops, you’re at the front of the line. Tying your shoe also works.”

In a bar: “I’ll start by assessing where I can break in. I usually like to go about five people from the front because they’re less aware of who’s in front and behind them. It’s also good to target people who are talking, because they normally won’t notice. So I’ll stand near the spot I want to break into, and when the line starts moving, I’ll slot right into the open space.”

At Pret à Manger: “There are usually so many people not paying attention, so I just take the initiative to go to the front and the line will start forming behind me.”

For the bathroom: “Sometimes I’ll go to the men’s room.” — By Lauren Schwartzberg

What New Yorkers Have Other People Wait For

According to TaskRabbit, from most to least frequently: concerts at Mercury Lounge , shows at Upright Citizens Brigade Theatre , Cronuts at Dominique Ansel Bakery , a seat at Mission Chinese Food , beer releases at Other Half Brewing Company , a seat at El Sabroso , shows at Radio City Music Hall , events at Brooklyn Bowl , tickets to The Daily Show , an appointment at the New York Passport Agency, products at the Apple store , cupcakes at Magnolia Bakery , burgers at Shake Shack , a spot to watch the Macy’s Thanksgiving Day Parade , sneakers at the Nike store .

Line-Improvement Theories

1. There is a better way to board an airplane. According to astrophysicist Jason Steffen , boarding should ideally start from the back of the plane but be staggered by every-other row and by seat type. In an experiment, the Steffen method was twice as fast as the standard back-to-front model and up to 30 percent faster than random boarding.

2. The last shall be first. Two Danish researchers compared a typical line to two other methods: In one, people were helped in a random order. In the other, those last in line were served first. As it turned out, the last-come-first-served method got the most people where they needed to go quickest.

3. Just give the people wine. Richard Larson , an MIT professor and queuing expert, has a simple solution for making lines more bearable: “Does anything replace a restaurateur who gives guests a free glass of wine as they peruse the menu while waiting? And, double benefit for the restaurateur, the table time of the party is reduced, because they no longer spend table time figuring out what to order. Everyone is happy.”

On Getting a Global Entry Interview How to avoid the line for avoiding the Customs line at the airport.

1. The online interview-appointment system works a bit like reservations at Momofuku Ko : They are canceled and re-reserved all the time. When you log in, the next available date is usually months away. Take whatever they have, but keep refreshing throughout the day and you can often find one for that week.

2. Because most interviews take place at airports, many travelers plan their appointments during layovers or after scheduled landings. But since plane delays are commonplace, those interviews frequently get missed. So if you have Wi-Fi on your plane, you might be able to snag a slot for the time you land. Others have reported success just walking in.

3. Some locations get more backlogged than others. While Boston-Logan and Newark-Liberty might have extra-long wait times, Albuquerque and Anchorage do not. Going home to New Mexico for Christmas? Book your excuse to get out of that awkward immigration discussion with Uncle John. — By Nicholas Gill

• Table of Contents: Nov 2, 2015 issue of New York | Subscribe!

12 Impatient Facts About Waiting in Line

Unless you live alone in the woods, waiting in line is a near-universal experience—though as any international tourist will find, the etiquette of doing so varies from place to place. Whether you queue politely or wait in line (or “ on line ,” as New Yorkers insist on saying), how you wait and how you feel about waiting is more about perception than the actual time that elapses, as writer David Andrews explains in his recent book Why Does the Other Line Always Move Faster? Here are 12 facts about standing in line to make your next wait feel a little more bearable.

1. THERE’S A SEMANTIC DIFFERENCE IN THE WAY AMERICANS AND THE BRITISH LINE UP.

According to cultural critic Robert J.C. Young [ PDF ], there’s a difference in attitude between “to stand in line” and its British equivalent, “queue,” which can be a verb or a noun. In American usage, a line is “something you have to submit yourself to,” he writes, while “if you queue, you remain linguistically and in some broader, important sense an agent, an active subject, part of a particular social consensus about how to behave in a particular situation that requires some measure of equality and fairness to all.”

2. LINES ARE A MORE RECENT PHENOMENON THAN YOU MIGHT THINK.

As late as 1775, the most exhaustive English dictionary yet written contained no word to describe the act of standing in line. In 1837, in a history of the French Revolution, Thomas Carlyle carefully defined the method through which revolutionaries waited their turn, writing that “the Bakers shops have got their Queues, or Tails; their long strings of purchasers, arranged in a tail, so that the first to come be the first served.” He wasn’t the only one who thought the concept strange and foreign. An American traveling through France in 1854 took care to describe his experience standing “ en queue ” (emphasis in the original) with college students to get into a library.

3. INDUSTRIALIZATION MADE LINES NECESSARY.

Factory employment changed people’s daily schedules. Suddenly, everyone started and ended work around the same time, creating crowds waiting for buses to commute to and from the factory and to punch their time cards. Because people could only shop and run errands during specific off-work hours, banks, shops, and post offices filled up with hordes of off-duty workers trying to get things done on their lunch break or right after clocking out.

4. THERE IS A WAITING-IN-LINE ROBOT.

Xavier was created at Carnegie Mellon University in 1995. He was programmed to stand in line at the coffee shop at the university’s Robotics Institute. His creators tested their own comfort levels in lines to figure out how much personal space Xavier should give people in front of him, and how much space probably constitutes someone just standing around not in a line. He queued correctly 70 percent of the time, occasionally messing up when the line curved too sharply or when he misjudged whether a person was actually queuing.

5. CUTTING IN LINE TAKES A LOT OF MENTAL ENERGY.

Infamous psychologist Stanley Milgram, known for studying people’s willingness to obey authority at a high cost to their personal ethics, also studied line-cutters. In the 1980s, he got student volunteers to cut in line at ticket counters without giving any reason. Half the time, no one protested. However, the students themselves hated the experiment. They felt anxious and embarrassed. Milgram hypothesized that our unwillingness to cut is a logical calculation of social cost. If you cut in line and someone puts up a fuss, you might be verbally or even physically attacked, and will probably end up at the back of the line anyway. If you’re in line, it behooves you to keep quiet when someone cuts ahead, because you’d have to step out of line to shout down that person way up in front, and you might lose the place you were fighting to protect, anyway.

6. IN TRAFFIC, BEING A LINE-CUTTER IS ONLY RIGHT.

Sitting in traffic is a form of standing in line. When drivers wait until the last minute to merge into traffic while trying to avoid a lane closure or exit the highway, a lot of people look at it as if the procrastinating mergers are cutting in line. People who politely moved into the congested lane long before they absolutely had to and waited their turn to exit get angry at cars trying to wriggle in as late as possible. But from a road design standpoint, those late mergers are just doing the most sensible thing. More lanes have more capacity, so if your two-lane highway suddenly turns into a one-lane highway, traffic can flow more quickly if cars merge later and utilize both lanes for longer.

7. THE FOUNDER OF WENDY’S WAS OBSESSIVE ABOUT LINE EQUALITY.

People feel better about standing in line at Wendy’s compared to McDonald’s or Burger King, a study of the three major fast food chains found. That’s because Wendy’s guarantees customers will be served in the order they arrived in, using a single-file line bounded by those crowd control belts also seen in airports. Company founder Dave Thomas abhorred uneven wait times. Meanwhile, the other chains let people line up in front of cash registers willy-nilly, meaning that one line might feel faster than the other. And there’s nothing that pisses people off quite like watching the other line move faster.

8. THOSE COMPLICATED WHOLE FOODS LINES ARE GOOD PSYCHOLOGY.

Rather than trying to suss out the shortest line in a long line of checkout queues, Whole Foods customers in habitually busy stores (such as those in urban areas) line up in front of colored screens that direct them to the next available cashier—essentially, a more tech-heavy version of grabbing a number at the deli. No one gets to find that mystical extra-short line, but no one waits any longer than anyone else. “No more of that emotionally fraught exercise of hunting down what you perceive to be the shortest line, and feeling frustrated when the other lines move faster than yours,” Andrews writes of a future of single-line grocery store check outs. As a result, “people would experience the wait time as shorter,” he says.

8. THERE’S A WAITING IN LINE BOARD GAME.

Kolejka (“queue” in Polish) is a board game created in 2011 by the Polish Institute of National Remembrance. The whole point is to line up in front of various empty shops, trying to be the first player to complete a shopping list as deliveries come in to restock the empty shelves. Based on the reality of living in the Soviet Union, where lines were a ubiquitous part of life, players can draw cards that allow them to jump ahead in line, such as using “carrying a small child” as an excuse or getting the goods under the counter.

9. SOMETIMES LINES ARE A GOOD THING.

You may not think so when you’re there, but at theme parks, lines are a feature, not a bug. If there are no lines whatsoever at a park, you would rush through and become bored much more quickly, spending only an hour or two at the park instead of the whole day. Theme parks’ economic model relies on walking the line between having short enough lines that people still want to wait, and long enough lines that people are forced to hang out (and buy snacks) for hours.

10. LONG LINES AT DISNEY CAN BE AVOIDED WITH A LITTLE RESEARCH.

Statistician Bob Sehlinger began publishing The Unofficial Guide to Walt Disney World in 1984 after two years of research and field trials on the study of lines at the Orlando theme park. Decades later, he and his Unofficial Guide to Walt Disney World and Disneyland co-author, computer scientist Len Testa , founded Touring Plans , a website and app devoted to helping people navigate various theme parks using data and mathematical modeling. Touring Plans and the Unofficial Guides predict wait times in real-time at Disneyland and Disney World algorithmically, using operations research and queuing theory. Touring Plans claims that its itineraries based on the patented scheduling system can save users up to four hours in line per day.

11. THERE ARE MANY WAYS TO MAKE A LINE FEEL LONGER …

Not all waiting experiences are equally terrible, as anyone who’s waited in line alone before being joined by a friend can attest. Waiting alone feels longer than waiting in a group, because you don’t have conversation to distract you. Other factors that can affect how long a wait feels include uncertainty about when the line will end, having no explanation for the wait (like when you hit traffic and can’t tell if it’s because of an accident or construction or just rush hour), and perceiving the line as unfair, such as when you see people who were in line behind you receive their food first.

12. … AND WAYS TO MAKE THEM FEEL MORE COMFORTABLE.

Lines feel a lot worse when there’s some anxiety involved. The more uncertainties about the situation, the less secure you’ll feel about eventually getting to the front. Restaurants are masters at putting people at ease while waiting for a table. There’s a check-in point where you can give the hostess your name, so you know the restaurant knows you’re there and will take care of you eventually. You’ll get an estimated wait time. You can grab a menu to peruse, making the time feel productive. And often, there’s a bar where you can hang around before your table is ready, letting you distract yourself from the waiting experience.

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• Standing Waves and Resonance

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• Alternating Current (AC)

Transmission Lines

• A 50-Ohm Cable?
• Circuits and the Speed of Light
• Characteristic Impedance
• Finite-length Transmission Lines
• “Long’’ and “Short’’ Transmission Lines
• Impedance Transformation

Whenever there is a mismatch of impedance between transmission line and load, reflections will occur. If the incident signal is a continuous AC waveform , these reflections will mix with more of the oncoming incident waveform to produce stationary waveforms called standing waves .

The following illustration shows how a triangle-shaped incident waveform turns into a mirror-image reflection upon reaching the line’s unterminated end. The transmission line in this illustrative sequence is shown as a single, thick line rather than a pair of wires, for simplicity’s sake.

The incident wave is shown traveling from left to right, while the reflected wave travels from right to left: (Figure below)

Incident wave reflects off end of unterminated transmission line.

If we add the two waveforms together, we find that a third, stationary waveform is created along the line’s length: (Figure below)

The sum of the incident and reflected waves is a stationary wave.

This third, “standing” wave, in fact, represents the only voltage along the line, being the representative sum of incident and reflected voltage waves. It oscillates in instantaneous magnitude, but does not propagate down the cable’s length like the incident or reflected waveforms causing it.

Note the dots along the line length marking the “zero” points of the standing wave (where the incident and reflected waves cancel each other), and how those points never change position: (Figure below)

The standing wave does not propgate along the transmission line.

Cases Where a Standing Wave is Produced

Standing waves are quite abundant in the physical world. Consider a string or rope, shaken at one end, and tied down at the other (only one half-cycle of hand motion shown, moving downward): (Figure below)

Standing waves on a rope.

Both the nodes (points of little or no vibration) and the antinodes (points of maximum vibration) remain fixed along the length of the string or rope.

The effect is most pronounced when the free end is shaken at just the right frequency. Plucked strings exhibit the same “standing wave” behavior, with “nodes” of maximum and minimum vibration along their length.

The major difference between a plucked string and a shaken string is that the plucked string supplies its own “correct” frequency of vibration to maximize the standing-wave effect: (Figure below)

Standing waves on a plucked string.

Wind blowing across an open-ended tube also produces standing waves; this time, the waves are vibrations of air molecules (sound) within the tube rather than vibrations of a solid object. Whether the standing wave terminates in a node (minimum amplitude) or an antinode (maximum amplitude) depends on whether the other end of the tube is open or closed: (Figure below)

Standing sound waves in open ended tubes.

A closed tube end must be a wave node, while an open tube end must be an antinode. By analogy, the anchored end of a vibrating string must be a node, while the free end (if there is any) must be an antinode.

Progression of Harmonics of Resonant Frequencies

Note how there is more than one wavelength suitable for producing standing waves of vibrating air within a tube that precisely match the tube’s end points.

This is true for all standing-wave systems: standing waves will resonate with the system for any frequency (wavelength) correlating to the node/antinode points of the system. Another way of saying this is that there are multiple resonant frequencies for any system supporting standing waves.

All higher frequencies are integer-multiples of the lowest (fundamental) frequency for the system. The sequential progression of harmonics from one resonant frequency to the next defines the overtone frequencies for the system: (Figure below)

Harmonics (overtones) in open ended pipes

The actual frequencies (measured in Hertz) for any of these harmonics or overtones depends on the physical length of the tube and the waves’ propagation velocity, which is the speed of sound in air.

Simulating a Transmission Line Resonance using SPICE

Because transmission lines support standing waves, and force these waves to possess nodes and antinodes according to the type of termination impedance at the load end, they also exhibit resonance at frequencies determined by physical length and propagation velocity.

Transmission line resonance, though, is a bit more complex than resonance of strings or of air in tubes, because we must consider both voltage waves and current waves.

This complexity is made easier to understand by way of computer simulation. To begin, let’s examine a perfectly matched source, transmission line, and load. All components have an impedance of 75 Ω: (Figure below)

Perfectly matched transmission line.

Using SPICE to simulate the circuit, we’ll specify the transmission line ( t1 ) with a 75 Ω characteristic impedance ( z0=75 ) and a propagation delay of 1 microsecond ( td=1u ). This is a convenient method for expressing the physical length of a transmission line: the amount of time it takes a wave to propagate down its entire length.

If this were a real 75 Ω cable—perhaps a type “RG-59B/U” coaxial cable, the type commonly used for cable television distribution—with a velocity factor of 0.66, it would be about 648 feet long.

Since 1 µs is the period of a 1 MHz signal, I’ll choose to sweep the frequency of the AC source from (nearly) zero to that figure, to see how the system reacts when exposed to signals ranging from DC to 1 wavelength.

Here is the SPICE netlist for the circuit shown above:

Running this simulation and plotting the source impedance drop (as an indication of current), the source voltage, the line’s source-end voltage, and the load voltage, we see that the source voltage—shown as vm(1)(voltage magnitude between node 1 and the implied ground point of node 0) on the graphic plot—registers a steady 1 volt, while every other voltage registers a steady 0.5 volts: (Figure below)

No resonances on a matched transmission line.

In a system where all impedances are perfectly matched, there can be no standing waves, and therefore no resonant “peaks” or “valleys” in the Bode plot.

Now, let’s change the load impedance to 999 MΩ, to simulate an open-ended transmission line. (Figure below) We should definitely see some reflections on the line now as the frequency is swept from 1 mHz to 1 MHz: (Figure below)

Open ended transmission line.

Resonances on open transmission line.

Here, both the supply voltage vm(1) and the line’s load-end voltage vm(3) remain steady at 1 volt. The other voltages dip and peak at different frequencies along the sweep range of 1 mHz to 1 MHz.

There are five points of interest along the horizontal axis of the analysis: 0 Hz, 250 kHz, 500 kHz, 750 kHz, and 1 MHz. We will investigate each one with regard to voltage and current at different points of the circuit.

At 0 Hz (actually 1 mHz), the signal is practically DC, and the circuit behaves much as it would given a 1-volt DC battery source.

There is no circuit current, as indicated by zero voltage drop across the source impedance (Z source : vm(1,2) ), and full source voltage present at the source-end of the transmission line (voltage measured between node 2 and node 0: vm(2) ). (Figure below)

At f=0: input: V=1, I=0; end: V=1, I=0.

At 250 kHz, we see zero voltage and maximum current at the source-end of the transmission line, yet still full voltage at the load-end: (Figure below)

At f=250 KHz: input: V=0, I=13.33 mA; end: V=1 I=0.

You might be wondering, how can this be? How can we get full source voltage at the line’s open end while there is zero voltage at its entrance?

The answer is found in the paradox of the standing wave. With a source frequency of 250 kHz, the line’s length is precisely right for 1/4 wavelength to fit from end to end. With the line’s load end open-circuited, there can be no current, but there will be voltage.

Therefore, the load-end of an open-circuited transmission line is a current node (zero point) and a voltage antinode (maximum amplitude): (Figure below)

Open end of transmission line shows current node, voltage antinode at open end.

At 500 kHz, exactly one-half of a standing wave rests on the transmission line, and here we see another point in the analysis where the source current drops off to nothing and the source-end voltage of the transmission line rises again to full voltage: (Figure below)

Full standing wave on half wave open transmission line.

At 750 kHz, the plot looks a lot like it was at 250 kHz: zero source-end voltage (vm(2)) and maximum current (vm(1,2)). This is due to 3/4 of a wave poised along the transmission line, resulting in the source “seeing” a short-circuit where it connects to the transmission line, even though the other end of the line is open-circuited: (Figure below)

1 1/2 standing waves on 3/4 wave open transmission line.

When the supply frequency sweeps up to 1 MHz, a full standing wave exists on the transmission line. At this point, the source-end of the line experiences the same voltage and current amplitudes as the load-end: full voltage and zero current. In essence, the source “sees” an open circuit at the point where it connects to the transmission line. (Figure below)

Double standing waves on full wave open transmission line.

In a similar fashion, a short-circuited transmission line generates standing waves, although the node and antinode assignments for voltage and current are reversed: at the shorted end of the line, there will be zero voltage (node) and maximum current (antinode). What follows is the SPICE simulation and illustrations of what happens at all the interesting frequencies: 0 Hz , 250 kHz , 500 kHz , 750 kHz , and 1 MHz . The short-circuit jumper is simulated by a 1 µΩ load impedance:

Shorted transmission line.

Resonances on shorted transmission line

At f=0 Hz: input: V=0, I=13.33 mA; end: V=0, I=13.33 mA.

Half wave standing wave pattern on 1/4 wave shorted transmission line.

Full wave standing wave pattern on half wave shorted transmission line.

1 1/2 standing wavepattern on 3/4 wave shorted transmission line.

Double standing waves on full wave shorted transmission line.

In both these circuit examples, an open-circuited line and a short-circuited line, the energy reflection is total: 100% of the incident wave reaching the line’s end gets reflected back toward the source.

If, however, the transmission line is terminated in some impedance other than an open or a short, the reflections will be less intense, as will be the difference between minimum and maximum values of voltage and current along the line.

Suppose we were to terminate our example line with a 100 Ω resistor instead of a 75 Ω resistor. (Figure below) Examine the results of the corresponding SPICE analysis to see the effects of impedance mismatch at different source frequencies: (Figure below)

Transmission line terminated in a mismatch

Weak resonances on a mismatched transmission line

If we run another SPICE analysis, this time printing numerical results rather than plotting them, we can discover exactly what is happening at all the interesting frequencies:

At all frequencies, the source voltage, v(1) , remains steady at 1 volt, as it should. The load voltage, v(3) , also remains steady, but at a lesser voltage: 0.5714 volts. However, both the line input voltage ( v(2) ) and the voltage dropped across the source’s 75 Ω impedance ( v(1,2) , indicating current drawn from the source) vary with frequency.

At f=0 Hz: input: V=0.57.14, I=5.715 mA; end: V=0.5714, I=5.715 mA.

At f=250 KHz: input: V=0.4286, I=7.619 mA; end: V=0.5714, I=7.619 mA.

At f=500 KHz: input: V=0.5714, I=5.715 mA; end: V=5.714, I=5.715 mA.

At f=750 KHz: input: V=0.4286, I=7.619 mA; end: V=0.5714, I=7.619 mA.

At f=1 MHz: input: V=0.5714, I=5.715 mA; end: V=0.5714, I=0.5715 mA.

At odd harmonics of the fundamental frequency (250 kHz, Figure 3rd-above and 750 kHz, Figure above) we see differing levels of voltage at each end of the transmission line, because at those frequencies the standing waves terminate at one end in a node and at the other end in an antinode.

Unlike the open-circuited and short-circuited transmission line examples, the maximum and minimum voltage levels along this transmission line do not reach the same extreme values of 0% and 100% source voltage, but we still have points of “minimum” and “maximum” voltage.

(Figure 6th-above) The same holds true for current: if the line’s terminating impedance is mismatched to the line’s characteristic impedance, we will have points of minimum and maximum current at certain fixed locations on the line, corresponding to the standing current wave’s nodes and antinodes, respectively.

Standing Wave Ratio

One way of expressing the severity of standing waves is as a ratio of maximum amplitude (antinode) to minimum amplitude (node), for voltage or for current.

When a line is terminated by an open or a short, this standing wave ratio , or SWR is valued at infinity, since the minimum amplitude will be zero, and any finite value divided by zero results in an infinite (actually, “undefined”) quotient.

In this example, with a 75 Ω line terminated by a 100 Ω impedance, the SWR will be finite: 1.333, calculated by taking the maximum line voltage at either 250 kHz or 750 kHz (0.5714 volts) and dividing by the minimum line voltage (0.4286 volts).

Standing wave ratio may also be calculated by taking the line’s terminating impedance and the line’s characteristic impedance, and dividing the larger of the two values by the smaller. In this example, the terminating impedance of 100 Ω divided by the characteristic impedance of 75 Ω yields a quotient of exactly 1.333, matching the previous calculation very closely.

A perfectly terminated transmission line will have an SWR of 1, since voltage at any location along the line’s length will be the same, and likewise for current.

Again, this is usually considered ideal, not only because reflected waves constitute energy not delivered to the load, but because the high values of voltage and current created by the antinodes of standing waves may over-stress the transmission line’s insulation (high voltage) and conductors (high current), respectively.

Also, a transmission line with a high SWR tends to act as an antenna, radiating electromagnetic energy away from the line, rather than channeling all of it to the load. This is usually undesirable, as the radiated energy may “couple” with nearby conductors, producing signal interference.

An interesting footnote to this point is that antenna structures—which typically resemble open- or short-circuited transmission lines—are often designed to operate at high standing wave ratios, for the very reason of maximizing signal radiation and reception.

The following photograph (Figure below) shows a set of transmission lines at a junction point in a radio transmitter system. The large, copper tubes with ceramic insulator caps at the ends are rigid coaxial transmission lines of 50 Ω characteristic impedance.

These lines carry RF power from the radio transmitter circuit to a small, wooden shelter at the base of an antenna structure, and from that shelter on to other shelters with other antenna structures:

Flexible coaxial cables connected to rigid lines.

Flexible coaxial cable connected to the rigid lines (also of 50 Ω characteristic impedance) conduct the RF power to capacitive and inductive “phasing” networks inside the shelter. The white, plastic tube joining two of the rigid lines together carries “filling” gas from one sealed line to the other.

The lines are gas-filled to avoid collecting moisture inside them, which would be a definite problem for a coaxial line. Note the flat, copper “straps” used as jumper wires to connect the conductors of the flexible coaxial cables to the conductors of the rigid lines.

Why flat straps of copper and not round wires? Because of the skin effect, which renders most of the cross-sectional area of a round conductor useless at radio frequencies.

Like many transmission lines, these are operated at low SWR conditions. As we will see in the next section, though, the phenomenon of standing waves in transmission lines is not always undesirable, as it may be exploited to perform a useful function: impedance transformation.

• Standing waves are waves of voltage and current which do not propagate (i.e. they are stationary), but are the result of interference between incident and reflected waves along a transmission line.
• A node is a point on a standing wave of minimum amplitude.
• An antinode is a point on a standing wave of maximum amplitude.
• Standing waves can only exist in a transmission line when the terminating impedance does not match the line’s characteristic impedance. In a perfectly terminated line, there are no reflected waves, and therefore no standing waves at all.
• At certain frequencies, the nodes and antinodes of standing waves will correlate with the ends of a transmission line, resulting in resonance .
• The lowest-frequency resonant point on a transmission line is where the line is one quarter-wavelength long. Resonant points exist at every harmonic (integer-multiple) frequency of the fundamental (quarter-wavelength).
• Standing wave ratio , or SWR , is the ratio of maximum standing wave amplitude to minimum standing wave amplitude. It may also be calculated by dividing termination impedance by characteristic impedance, or vice versa, which ever yields the greatest quotient. A line with no standing waves (perfectly matched: Z load to Z 0 ) has an SWR equal to 1.
• Transmission lines may be damaged by the high maximum amplitudes of standing waves. Voltage antinodes may break down insulation between conductors, and current antinodes may overheat conductors.
• Textbook Index

Lessons in Electric Circuits

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• 1 Basic AC Theory
• 2 Complex Numbers
• 3 Reactance and Impedance—Inductive
• 4 Reactance and Impedance—Capacitive
• 5 Reactance and Impedance—R, L, And C
• 6 Resonance
• 7 Mixed-Frequency AC Signals
• 9 Transformers
• 10 Polyphase AC Circuits
• 11 Power Factor
• 12 AC Metering Circuits
• 13 AC Motors

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• 15 Contributors List
• Semiconductors
• Digital Circuits
• EE Reference
• DIY Electronics Projects
• Advanced Textbooks Practical Guide to Radio-Frequency Analysis and Design

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• Waveguide Calculator (Rectangular)
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• LTSpice Simulation Using WAV Files
• Microstrip Wavelength Calculator
• Wireless Basics: How Radio Waves Work

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16.6 Standing Waves and Resonance

Learning objectives.

By the end of this section, you will be able to:

• Describe standing waves and explain how they are produced
• Describe the modes of a standing wave on a string
• Provide examples of standing waves beyond the waves on a string

Throughout this chapter, we have been studying traveling waves, or waves that transport energy from one place to another. Under certain conditions, waves can bounce back and forth through a particular region, effectively becoming stationary. These are called standing wave s .

Another related effect is known as resonance . In Oscillations , we defined resonance as a phenomenon in which a small-amplitude driving force could produce large-amplitude motion. Think of a child on a swing, which can be modeled as a physical pendulum. Relatively small-amplitude pushes by a parent can produce large-amplitude swings. Sometimes this resonance is good—for example, when producing music with a stringed instrument. At other times, the effects can be devastating, such as the collapse of a building during an earthquake. In the case of standing waves, the relatively large amplitude standing waves are produced by the superposition of smaller amplitude component waves.

Standing Waves

Sometimes waves do not seem to move; rather, they just vibrate in place. You can see unmoving waves on the surface of a glass of milk in a refrigerator, for example. Vibrations from the refrigerator motor create waves on the milk that oscillate up and down but do not seem to move across the surface. Figure 16.25 shows an experiment you can try at home. Take a bowl of milk and place it on a common box fan. Vibrations from the fan will produce circular standing waves in the milk. The waves are visible in the photo due to the reflection from a lamp. These waves are formed by the superposition of two or more traveling waves, such as illustrated in Figure 16.26 for two identical waves moving in opposite directions. The waves move through each other with their disturbances adding as they go by. If the two waves have the same amplitude and wavelength, then they alternate between constructive and destructive interference. The resultant looks like a wave standing in place and, thus, is called a standing wave.

Consider two identical waves that move in opposite directions. The first wave has a wave function of y 1 ( x , t ) = A sin ( k x − ω t ) y 1 ( x , t ) = A sin ( k x − ω t ) and the second wave has a wave function y 2 ( x , t ) = A sin ( k x + ω t ) y 2 ( x , t ) = A sin ( k x + ω t ) . The waves interfere and form a resultant wave

This can be simplified using the trigonometric identity

where α = k x α = k x and β = ω t β = ω t , giving us

which simplifies to

Notice that the resultant wave is a sine wave that is a function only of position, multiplied by a cosine function that is a function only of time. Graphs of y ( x , t ) as a function of x for various times are shown in Figure 16.26 . The red wave moves in the negative x -direction, the blue wave moves in the positive x -direction, and the black wave is the sum of the two waves. As the red and blue waves move through each other, they move in and out of constructive interference and destructive interference.

Initially, at time t = 0 , t = 0 , the two waves are in phase, and the result is a wave that is twice the amplitude of the individual waves. The waves are also in phase at the time t = T 2 . t = T 2 . In fact, the waves are in phase at any integer multiple of half of a period:

At other times, the two waves are 180 ° ( π radians ) 180 ° ( π radians ) out of phase, and the resulting wave is equal to zero. This happens at

Notice that some x -positions of the resultant wave are always zero no matter what the phase relationship is. These positions are called node s . Where do the nodes occur? Consider the solution to the sum of the two waves

Finding the positions where the sine function equals zero provides the positions of the nodes.

There are also positions where y oscillates between y = ± A y = ± A . These are the antinode s . We can find them by considering which values of x result in sin ( k x ) = ± 1 sin ( k x ) = ± 1 .

What results is a standing wave as shown in Figure 16.27 , which shows snapshots of the resulting wave of two identical waves moving in opposite directions. The resulting wave appears to be a sine wave with nodes at integer multiples of half wavelengths. The antinodes oscillate between y = ± 2 A y = ± 2 A due to the cosine term, cos ( ω t ) cos ( ω t ) , which oscillates between ± 1 ± 1 .

The resultant wave appears to be standing still, with no apparent movement in the x -direction, although it is composed of one wave function moving in the positive, whereas the second wave is moving in the negative x -direction. Figure 16.27 shows various snapshots of the resulting wave. The nodes are marked with red dots while the antinodes are marked with blue dots.

A common example of standing waves are the waves produced by stringed musical instruments. When the string is plucked, pulses travel along the string in opposite directions. The ends of the strings are fixed in place, so nodes appear at the ends of the strings—the boundary conditions of the system, regulating the resonant frequencies in the strings. The resonance produced on a string instrument can be modeled in a physics lab using the apparatus shown in Figure 16.28 .

The lab setup shows a string attached to a string vibrator, which oscillates the string with an adjustable frequency f . The other end of the string passes over a frictionless pulley and is tied to a hanging mass. The magnitude of the tension in the string is equal to the weight of the hanging mass. The string has a constant linear density (mass per length) μ μ and the speed at which a wave travels down the string equals v = F T μ = m g μ v = F T μ = m g μ Equation 16.7 . The symmetrical boundary conditions (a node at each end) dictate the possible frequencies that can excite standing waves. Starting from a frequency of zero and slowly increasing the frequency, the first mode n = 1 n = 1 appears as shown in Figure 16.29 . The first mode, also called the fundamental mode or the first harmonic, shows half of a wavelength has formed, so the wavelength is equal to twice the length between the nodes λ 1 = 2 L λ 1 = 2 L . The fundamental frequency , or first harmonic frequency, that drives this mode is

where the speed of the wave is v = F T μ . v = F T μ . Keeping the tension constant and increasing the frequency leads to the second harmonic or the n = 2 n = 2 mode. This mode is a full wavelength λ 2 = L λ 2 = L and the frequency is twice the fundamental frequency:

The next two modes, or the third and fourth harmonics, have wavelengths of λ 3 = 2 3 L λ 3 = 2 3 L and λ 4 = 2 4 L , λ 4 = 2 4 L , driven by frequencies of f 3 = 3 v 2 L = 3 f 1 f 3 = 3 v 2 L = 3 f 1 and f 4 = 4 v 2 L = 4 f 1 . f 4 = 4 v 2 L = 4 f 1 . All frequencies above the frequency f 1 f 1 are known as the overtone s . The equations for the wavelength and the frequency can be summarized as:

The standing wave patterns that are possible for a string, the first four of which are shown in Figure 16.29 , are known as the normal mode s , with frequencies known as the normal frequencies. In summary, the first frequency to produce a normal mode is called the fundamental frequency (or first harmonic). Any frequencies above the fundamental frequency are overtones. The second frequency of the n = 2 n = 2 normal mode of the string is the first overtone (or second harmonic). The frequency of the n = 3 n = 3 normal mode is the second overtone (or third harmonic) and so on.

The solutions shown as Equation 16.15 and Equation 16.16 are for a string with the boundary condition of a node on each end. When the boundary condition on either side is the same, the system is said to have symmetric boundary conditions. Equation 16.15 and Equation 16.16 are good for any symmetric boundary conditions, that is, nodes at both ends or antinodes at both ends.

Example 16.7

Standing waves on a string.

• The velocity of the wave can be found using v = F T μ . v = F T μ . The tension is provided by the weight of the hanging mass.
• The standing waves will depend on the boundary conditions. There must be a node at each end. The first mode will be one half of a wave. The second can be found by adding a half wavelength. That is the shortest length that will result in a node at the boundaries. For example, adding one quarter of a wavelength will result in an antinode at the boundary and is not a mode which would satisfy the boundary conditions. This is shown in Figure 16.31 .
• Begin with the velocity of a wave on a string. The tension is equal to the weight of the hanging mass. The linear mass density and mass of the hanging mass are given: v = F T μ = m g μ = 2 kg ( 9.8 m s ) 0.006 kg m = 57.15 m/s . v = F T μ = m g μ = 2 kg ( 9.8 m s ) 0.006 kg m = 57.15 m/s .
• The frequencies of the first three modes are found by using f = v w λ . f = v w λ . f 1 = v w λ 1 = 57.15 m/s 4.00 m = 14.29 Hz f 2 = v w λ 2 = 57.15 m/s 2.00 m = 28.58 Hz f 3 = v w λ 3 = 57.15 m/s 1.333 m = 42.87 Hz f 1 = v w λ 1 = 57.15 m/s 4.00 m = 14.29 Hz f 2 = v w λ 2 = 57.15 m/s 2.00 m = 28.58 Hz f 3 = v w λ 3 = 57.15 m/s 1.333 m = 42.87 Hz

Significance

Interactive.

Engage the Phet simulation below to play with a 1D or 2D system of coupled mass-spring oscillators. Vary the number of masses, set the initial conditions, and watch the system evolve. See the spectrum of normal modes for arbitrary motion. See longitudinal or transverse modes in the 1D system.

Check Your Understanding 16.7

The equations for the wavelengths and the frequencies of the modes of a wave produced on a string:

were derived by considering a wave on a string where there were symmetric boundary conditions of a node at each end. These modes resulted from two sinusoidal waves with identical characteristics except they were moving in opposite directions, confined to a region L with nodes required at both ends. Will the same equations work if there were symmetric boundary conditions with antinodes at each end? What would the normal modes look like for a medium that was free to oscillate on each end? Don’t worry for now if you cannot imagine such a medium, just consider two sinusoidal wave functions in a region of length L , with antinodes on each end.

The free boundary conditions shown in the last Check Your Understanding may seem hard to visualize. How can there be a system that is free to oscillate on each end? In Figure 16.32 are shown two possible configuration of a metallic rods (shown in red) attached to two supports (shown in blue). In part (a), the rod is supported at the ends, and there are fixed boundary conditions at both ends. Given the proper frequency, the rod can be driven into resonance with a wavelength equal to length of the rod, with nodes at each end. In part (b), the rod is supported at positions one quarter of the length from each end of the rod, and there are free boundary conditions at both ends. Given the proper frequency, this rod can also be driven into resonance with a wavelength equal to the length of the rod, but there are antinodes at each end. If you are having trouble visualizing the wavelength in this figure, remember that the wavelength may be measured between any two nearest identical points and consider Figure 16.33 .

Note that the study of standing waves can become quite complex. In Figure 16.32 (a), the n = 2 n = 2 mode of the standing wave is shown, and it results in a wavelength equal to L . In this configuration, the n = 1 n = 1 mode would also have been possible with a standing wave equal to 2 L . Is it possible to get the n = 1 n = 1 mode for the configuration shown in part (b)? The answer is no. In this configuration, there are additional conditions set beyond the boundary conditions. Since the rod is mounted at a point one quarter of the length from each side, a node must exist there, and this limits the possible modes of standing waves that can be created. We leave it as an exercise for the reader to consider if other modes of standing waves are possible. It should be noted that when a system is driven at a frequency that does not cause the system to resonate, vibrations may still occur, but the amplitude of the vibrations will be much smaller than the amplitude at resonance.

A field of mechanical engineering uses the sound produced by the vibrating parts of complex mechanical systems to troubleshoot problems with the systems. Suppose a part in an automobile is resonating at the frequency of the car’s engine, causing unwanted vibrations in the automobile. This may cause the engine to fail prematurely. The engineers use microphones to record the sound produced by the engine, then use a technique called Fourier analysis to find frequencies of sound produced with large amplitudes and then look at the parts list of the automobile to find a part that would resonate at that frequency. The solution may be as simple as changing the composition of the material used or changing the length of the part in question.

There are other numerous examples of resonance in standing waves in the physical world. The air in a tube, such as found in a musical instrument like a flute, can be forced into resonance and produce a pleasant sound, as we discuss in Sound .

At other times, resonance can cause serious problems. A closer look at earthquakes provides evidence for conditions appropriate for resonance, standing waves, and constructive and destructive interference. A building may vibrate for several seconds with a driving frequency matching that of the natural frequency of vibration of the building—producing a resonance resulting in one building collapsing while neighboring buildings do not. Often, buildings of a certain height are devastated while other taller buildings remain intact. The building height matches the condition for setting up a standing wave for that particular height. The span of the roof is also important. Often it is seen that gymnasiums, supermarkets, and churches suffer damage when individual homes suffer far less damage. The roofs with large surface areas supported only at the edges resonate at the frequencies of the earthquakes, causing them to collapse. As the earthquake waves travel along the surface of Earth and reflect off denser rocks, constructive interference occurs at certain points. Often areas closer to the epicenter are not damaged, while areas farther away are damaged.

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Course: physics archive   >   unit 8.

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Standing waves on strings

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Video transcript

16.6 Standing Waves and Resonance

Learning objectives.

By the end of this section, you will be able to:

• Describe standing waves and explain how they are produced
• Describe the modes of a standing wave on a string
• Provide examples of standing waves beyond the waves on a string

Throughout this chapter, we have been studying traveling waves, or waves that transport energy from one place to another. Under certain conditions, waves can bounce back and forth through a particular region, effectively becoming stationary. These are called standing waves .

Another related effect is known as resonance . In Oscillations , we defined resonance as a phenomenon in which a small-amplitude driving force could produce large-amplitude motion. Think of a child on a swing, which can be modeled as a physical pendulum. Relatively small-amplitude pushes by a parent can produce large-amplitude swings. Sometimes this resonance is good—for example, when producing music with a stringed instrument. At other times, the effects can be devastating, such as the collapse of a building during an earthquake. In the case of standing waves, the relatively large amplitude standing waves are produced by the superposition of smaller amplitude component waves.

Standing Waves

Sometimes waves do not seem to move; rather, they just vibrate in place. You can see unmoving waves on the surface of a glass of milk in a refrigerator, for example. Vibrations from the refrigerator motor create waves on the milk that oscillate up and down but do not seem to move across the surface. (Figure) shows an experiment you can try at home. Take a bowl of milk and place it on a common box fan. Vibrations from the fan will produce circular standing waves in the milk. The waves are visible in the photo due to the reflection from a lamp. These waves are formed by the superposition of two or more traveling waves, such as illustrated in (Figure) for two identical waves moving in opposite directions. The waves move through each other with their disturbances adding as they go by. If the two waves have the same amplitude and wavelength, then they alternate between constructive and destructive interference. The resultant looks like a wave standing in place and, thus, is called a standing wave.

Figure 16.25 Standing waves are formed on the surface of a bowl of milk sitting on a box fan. The vibrations from the fan causes the surface of the milk of oscillate. The waves are visible due to the reflection of light from a lamp.

figure 16.26 Time snapshots of two sine waves. The red wave is moving in the −x-direction and the blue wave is moving in the +x-direction. The resulting wave is shown in black. Consider the resultant wave at the points [latex] x=0\,\text{m},3\,\text{m},6\,\text{m},9\,\text{m},12\,\text{m},15\,\text{m} [/latex] and notice that the resultant wave always equals zero at these points, no matter what the time is. These points are known as fixed points (nodes). In between each two nodes is an antinode, a place where the medium oscillates with an amplitude equal to the sum of the amplitudes of the individual waves.

Consider two identical waves that move in opposite directions. The first wave has a wave function of [latex] {y}_{1}(x,t)=A\,\text{sin}(kx-\omega t) [/latex] and the second wave has a wave function [latex] {y}_{2}(x,t)=A\,\text{sin}(kx+\omega t) [/latex]. The waves interfere and form a resultant wave

This can be simplified using the trigonometric identity

where [latex] \alpha =kx [/latex] and [latex] \beta =\omega t [/latex], giving us

which simplifies to

Notice that the resultant wave is a sine wave that is a function only of position, multiplied by a cosine function that is a function only of time. Graphs of y ( x , t ) as a function of x for various times are shown in (Figure) . The red wave moves in the negative x -direction, the blue wave moves in the positive x -direction, and the black wave is the sum of the two waves. As the red and blue waves move through each other, they move in and out of constructive interference and destructive interference.

Initially, at time [latex] t=0, [/latex] the two waves are in phase, and the result is a wave that is twice the amplitude of the individual waves. The waves are also in phase at the time [latex] t=\frac{T}{2}. [/latex] In fact, the waves are in phase at any integer multiple of half of a period:

At other times, the two waves are [latex] 180\text{°}(\pi \,\text{radians}) [/latex] out of phase, and the resulting wave is equal to zero. This happens at

Notice that some x -positions of the resultant wave are always zero no matter what the phase relationship is. These positions are called nodes . Where do the nodes occur? Consider the solution to the sum of the two waves

Finding the positions where the sine function equals zero provides the positions of the nodes.

There are also positions where y oscillates between [latex] y=\text{±}A [/latex]. These are the antinodes . We can find them by considering which values of x result in [latex] \text{sin}(kx)=\text{±}1 [/latex].

What results is a standing wave as shown in (Figure) , which shows snapshots of the resulting wave of two identical waves moving in opposite directions. The resulting wave appears to be a sine wave with nodes at integer multiples of half wavelengths. The antinodes oscillate between [latex] y=\text{±}2A [/latex] due to the cosine term, [latex] \text{cos}(\omega t) [/latex], which oscillates between [latex] ±1 [/latex].

The resultant wave appears to be standing still, with no apparent movement in the x -direction, although it is composed of one wave function moving in the positive, whereas the second wave is moving in the negative x -direction. (Figure) shows various snapshots of the resulting wave. The nodes are marked with red dots while the antinodes are marked with blue dots.

Figure 16.27 When two identical waves are moving in opposite directions, the resultant wave is a standing wave. Nodes appear at integer multiples of half wavelengths. Antinodes appear at odd multiples of quarter wavelengths, where they oscillate between [latex] y=\text{±}A. [/latex] The nodes are marked with red dots and the antinodes are marked with blue dots.

A common example of standing waves are the waves produced by stringed musical instruments. When the string is plucked, pulses travel along the string in opposite directions. The ends of the strings are fixed in place, so nodes appear at the ends of the strings—the boundary conditions of the system, regulating the resonant frequencies in the strings. The resonance produced on a string instrument can be modeled in a physics lab using the apparatus shown in (Figure) .

Figure 16.28 A lab setup for creating standing waves on a string. The string has a node on each end and a constant linear density. The length between the fixed boundary conditions is L. The hanging mass provides the tension in the string, and the speed of the waves on the string is proportional to the square root of the tension divided by the linear mass density.

The lab setup shows a string attached to a string vibrator, which oscillates the string with an adjustable frequency f . The other end of the string passes over a frictionless pulley and is tied to a hanging mass. The magnitude of the tension in the string is equal to the weight of the hanging mass. The string has a constant linear density (mass per length) [latex] \mu [/latex] and the speed at which a wave travels down the string equals [latex] v=\sqrt{\frac{{F}_{T}}{\mu }}=\sqrt{\frac{mg}{\mu }} [/latex] (Figure) . The symmetrical boundary conditions (a node at each end) dictate the possible frequencies that can excite standing waves. Starting from a frequency of zero and slowly increasing the frequency, the first mode [latex] n=1 [/latex] appears as shown in (Figure) . The first mode, also called the fundamental mode or the first harmonic, shows half of a wavelength has formed, so the wavelength is equal to twice the length between the nodes [latex] {\lambda }_{1}=2L [/latex]. The fundamental frequency , or first harmonic frequency, that drives this mode is

where the speed of the wave is [latex] v=\sqrt{\frac{{F}_{T}}{\mu }}. [/latex] Keeping the tension constant and increasing the frequency leads to the second harmonic or the [latex] n=2 [/latex] mode. This mode is a full wavelength [latex] {\lambda }_{2}=L [/latex] and the frequency is twice the fundamental frequency:

Figure 16.29 Standing waves created on a string of length L. A node occurs at each end of the string. The nodes are boundary conditions that limit the possible frequencies that excite standing waves. (Note that the amplitudes of the oscillations have been kept constant for visualization. The standing wave patterns possible on the string are known as the normal modes. Conducting this experiment in the lab would result in a decrease in amplitude as the frequency increases.)

The next two modes, or the third and fourth harmonics, have wavelengths of [latex] {\lambda }_{3}=\frac{2}{3}L [/latex] and [latex] {\lambda }_{4}=\frac{2}{4}L, [/latex] driven by frequencies of [latex] {f}_{3}=\frac{3v}{2L}=3{f}_{1} [/latex] and [latex] {f}_{4}=\frac{4v}{2L}=4{f}_{1}. [/latex] All frequencies above the frequency [latex] {f}_{1} [/latex] are known as the overtones . The equations for the wavelength and the frequency can be summarized as:

The standing wave patterns that are possible for a string, the first four of which are shown in (Figure) , are known as the normal modes , with frequencies known as the normal frequencies. In summary, the first frequency to produce a normal mode is called the fundamental frequency (or first harmonic). Any frequencies above the fundamental frequency are overtones. The second frequency of the [latex] n=2 [/latex] normal mode of the string is the first overtone (or second harmonic). The frequency of the [latex] n=3 [/latex] normal mode is the second overtone (or third harmonic) and so on.

The solutions shown as (Equation) and (Equation) are for a string with the boundary condition of a node on each end. When the boundary condition on either side is the same, the system is said to have symmetric boundary conditions. (Equation) and (Equation) are good for any symmetric boundary conditions, that is, nodes at both ends or antinodes at both ends.

Standing Waves on a String

Consider a string of [latex] L=2.00\,\text{m}. [/latex] attached to an adjustable-frequency string vibrator as shown in (Figure) . The waves produced by the vibrator travel down the string and are reflected by the fixed boundary condition at the pulley. The string, which has a linear mass density of [latex] \mu =0.006\,\text{kg/m,} [/latex] is passed over a frictionless pulley of a negligible mass, and the tension is provided by a 2.00-kg hanging mass. (a) What is the velocity of the waves on the string? (b) Draw a sketch of the first three normal modes of the standing waves that can be produced on the string and label each with the wavelength. (c) List the frequencies that the string vibrator must be tuned to in order to produce the first three normal modes of the standing waves.

Figure 16.30 A string attached to an adjustable-frequency string vibrator.

• The velocity of the wave can be found using [latex] v=\sqrt{\frac{{F}_{T}}{\mu }}. [/latex] The tension is provided by the weight of the hanging mass.
• The standing waves will depend on the boundary conditions. There must be a node at each end. The first mode will be one half of a wave. The second can be found by adding a half wavelength. That is the shortest length that will result in a node at the boundaries. For example, adding one quarter of a wavelength will result in an antinode at the boundary and is not a mode which would satisfy the boundary conditions. This is shown in (Figure) .

Figure 16.31 (a) The figure represents the second mode of the string that satisfies the boundary conditions of a node at each end of the string. (b)This figure could not possibly be a normal mode on the string because it does not satisfy the boundary conditions. There is a node on one end, but an antinode on the other.

• Begin with the velocity of a wave on a string. The tension is equal to the weight of the hanging mass. The linear mass density and mass of the hanging mass are given: [latex] v=\sqrt{\frac{{F}_{T}}{\mu }}=\sqrt{\frac{mg}{\mu }}=\sqrt{\frac{2\,\text{kg}(9.8\frac{\text{m}}{\text{s}})}{0.006\frac{\text{kg}}{\text{m}}}}=57.15\,\text{m/s}. [/latex]

• The frequencies of the first three modes are found by using [latex] f=\frac{{v}_{w}}{\lambda }. [/latex] [latex] \begin{array}{}\\ {f}_{1}=\frac{{v}_{w}}{{\lambda }_{1}}=\frac{57.15\,\text{m/s}}{4.00\,\text{m}}=14.29\,\text{Hz}\hfill \\ {f}_{2}=\frac{{v}_{w}}{{\lambda }_{2}}=\frac{57.15\,\text{m/s}}{2.00\,\text{m}}=28.58\,\text{Hz}\hfill \\ {f}_{3}=\frac{{v}_{w}}{{\lambda }_{3}}=\frac{57.15\,\text{m/s}}{1.333\,\text{m}}=42.87\,\text{Hz}\hfill \end{array} [/latex]

Significance

The three standing modes in this example were produced by maintaining the tension in the string and adjusting the driving frequency. Keeping the tension in the string constant results in a constant velocity. The same modes could have been produced by keeping the frequency constant and adjusting the speed of the wave in the string (by changing the hanging mass.)

Visit this simulation to play with a 1D or 2D system of coupled mass-spring oscillators. Vary the number of masses, set the initial conditions, and watch the system evolve. See the spectrum of normal modes for arbitrary motion. See longitudinal or transverse modes in the 1D system.

Check Your Understanding

The equations for the wavelengths and the frequencies of the modes of a wave produced on a string:

were derived by considering a wave on a string where there were symmetric boundary conditions of a node at each end. These modes resulted from two sinusoidal waves with identical characteristics except they were moving in opposite directions, confined to a region L with nodes required at both ends. Will the same equations work if there were symmetric boundary conditions with antinodes at each end? What would the normal modes look like for a medium that was free to oscillate on each end? Don’t worry for now if you cannot imagine such a medium, just consider two sinusoidal wave functions in a region of length L , with antinodes on each end.

Yes, the equations would work equally well for symmetric boundary conditions of a medium free to oscillate on each end where there was an antinode on each end. The normal modes of the first three modes are shown below. The dotted line shows the equilibrium position of the medium.

Note that the first mode is two quarters, or one half, of a wavelength. The second mode is one quarter of a wavelength, followed by one half of a wavelength, followed by one quarter of a wavelength, or one full wavelength. The third mode is one and a half wavelengths. These are the same result as the string with a node on each end. The equations for symmetrical boundary conditions work equally well for fixed boundary conditions and free boundary conditions. These results will be revisited in the next chapter when discussing sound wave in an open tube.

The free boundary conditions shown in the last Check Your Understanding may seem hard to visualize. How can there be a system that is free to oscillate on each end? In (Figure) are shown two possible configuration of a metallic rods (shown in red) attached to two supports (shown in blue). In part (a), the rod is supported at the ends, and there are fixed boundary conditions at both ends. Given the proper frequency, the rod can be driven into resonance with a wavelength equal to length of the rod, with nodes at each end. In part (b), the rod is supported at positions one quarter of the length from each end of the rod, and there are free boundary conditions at both ends. Given the proper frequency, this rod can also be driven into resonance with a wavelength equal to the length of the rod, but there are antinodes at each end. If you are having trouble visualizing the wavelength in this figure, remember that the wavelength may be measured between any two nearest identical points and consider (Figure) .

Figure 16.32 (a) A metallic rod of length L (red) supported by two supports (blue) on each end. When driven at the proper frequency, the rod can resonate with a wavelength equal to the length of the rod with a node on each end. (b) The same metallic rod of length L (red) supported by two supports (blue) at a position a quarter of the length of the rod from each end. When driven at the proper frequency, the rod can resonate with a wavelength equal to the length of the rod with an antinode on each end.

Figure 16.33 A wavelength may be measure between the nearest two repeating points. On the wave on a string, this means the same height and slope. (a) The wavelength is measured between the two nearest points where the height is zero and the slope is maximum and positive. (b) The wavelength is measured between two identical points where the height is maximum and the slope is zero.

Note that the study of standing waves can become quite complex. In (Figure) (a), the [latex] n=2 [/latex] mode of the standing wave is shown, and it results in a wavelength equal to L . In this configuration, the [latex] n=1 [/latex] mode would also have been possible with a standing wave equal to 2 L . Is it possible to get the [latex] n=1 [/latex] mode for the configuration shown in part (b)? The answer is no. In this configuration, there are additional conditions set beyond the boundary conditions. Since the rod is mounted at a point one quarter of the length from each side, a node must exist there, and this limits the possible modes of standing waves that can be created. We leave it as an exercise for the reader to consider if other modes of standing waves are possible. It should be noted that when a system is driven at a frequency that does not cause the system to resonate, vibrations may still occur, but the amplitude of the vibrations will be much smaller than the amplitude at resonance.

A field of mechanical engineering uses the sound produced by the vibrating parts of complex mechanical systems to troubleshoot problems with the systems. Suppose a part in an automobile is resonating at the frequency of the car’s engine, causing unwanted vibrations in the automobile. This may cause the engine to fail prematurely. The engineers use microphones to record the sound produced by the engine, then use a technique called Fourier analysis to find frequencies of sound produced with large amplitudes and then look at the parts list of the automobile to find a part that would resonate at that frequency. The solution may be as simple as changing the composition of the material used or changing the length of the part in question.

There are other numerous examples of resonance in standing waves in the physical world. The air in a tube, such as found in a musical instrument like a flute, can be forced into resonance and produce a pleasant sound, as we discuss in Sound .

At other times, resonance can cause serious problems. A closer look at earthquakes provides evidence for conditions appropriate for resonance, standing waves, and constructive and destructive interference. A building may vibrate for several seconds with a driving frequency matching that of the natural frequency of vibration of the building—producing a resonance resulting in one building collapsing while neighboring buildings do not. Often, buildings of a certain height are devastated while other taller buildings remain intact. The building height matches the condition for setting up a standing wave for that particular height. The span of the roof is also important. Often it is seen that gymnasiums, supermarkets, and churches suffer damage when individual homes suffer far less damage. The roofs with large surface areas supported only at the edges resonate at the frequencies of the earthquakes, causing them to collapse. As the earthquake waves travel along the surface of Earth and reflect off denser rocks, constructive interference occurs at certain points. Often areas closer to the epicenter are not damaged, while areas farther away are damaged.

• A standing wave is the superposition of two waves which produces a wave that varies in amplitude but does not propagate.
• Nodes are points of no motion in standing waves.
• An antinode is the location of maximum amplitude of a standing wave.
• Normal modes of a wave on a string are the possible standing wave patterns. The lowest frequency that will produce a standing wave is known as the fundamental frequency. The higher frequencies which produce standing waves are called overtones.

Key Equations

Conceptual Questions

A truck manufacturer finds that a strut in the engine is failing prematurely. A sound engineer determines that the strut resonates at the frequency of the engine and suspects that this could be the problem. What are two possible characteristics of the strut can be modified to correct the problem?

It may be as easy as changing the length and/or the density a small amount so that the parts do not resonate at the frequency of the motor.

Why do roofs of gymnasiums and churches seem to fail more than family homes when an earthquake occurs?

Wine glasses can be set into resonance by moistening your finger and rubbing it around the rim of the glass. Why?

Energy is supplied to the glass by the work done by the force of your finger on the glass. When supplied at the right frequency, standing waves form. The glass resonates and the vibrations produce sound.

Air conditioning units are sometimes placed on the roof of homes in the city. Occasionally, the air conditioners cause an undesirable hum throughout the upper floors of the homes. Why does this happen? What can be done to reduce the hum?

Consider a standing wave modeled as [latex] y(x,t)=4.00\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x)\text{cos}(4\,{\text{s}}^{-1}t). [/latex] Is there a node or an antinode at [latex] x=0.00\,\text{m}? [/latex] What about a standing wave modeled as [latex] y(x,t)=4.00\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x+\frac{\pi }{2})\text{cos}(4\,{\text{s}}^{-1}t)? [/latex] Is there a node or an antinode at the [latex] x=0.00\,\text{m} [/latex] position?

For the equation [latex] y(x,t)=4.00\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x)\text{cos}(4\,{\text{s}}^{-1}t), [/latex] there is a node because when [latex] x=0.00\,\text{m} [/latex], [latex] \text{sin}(3\,{\text{m}}^{-1}(0.00\,\text{m}))=0.00, [/latex] so [latex] y(0.00\,\text{m},t)=0.00\,\text{m} [/latex] for all time. For the equation [latex] y(x,t)=4.00\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x+\frac{\pi }{2})\text{cos}(4\,{\text{s}}^{-1}t), [/latex] there is an antinode because when [latex] x=0.00\,\text{m} [/latex], [latex] \text{sin}(3\,{\text{m}}^{-1}(0.00\,\text{m})+\frac{\pi }{2})=+1.00 [/latex], so [latex] y(0.00\,\text{m},t) [/latex] oscillates between + A and − A as the cosine term oscillates between +1 and -1.

A wave traveling on a Slinky® that is stretched to 4 m takes 2.4 s to travel the length of the Slinky and back again. (a) What is the speed of the wave? (b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?

A 2-m long string is stretched between two supports with a tension that produces a wave speed equal to [latex] {v}_{w}=50.00\,\text{m/s}. [/latex] What are the wavelength and frequency of the first three modes that resonate on the string?

[latex] \begin{array}{cc} {\lambda }_{n}=\frac{2.00}{n}L,\quad {f}_{n}=\frac{v}{{\lambda }_{n}}\hfill \\ {\lambda }_{1}=4.00\,\text{m},\quad {f}_{1}=12.5\,\text{Hz}\hfill \\ {\lambda }_{2}=2.00\,\text{m},\quad {f}_{2}=25.00\,\text{Hz}\hfill \\ {\lambda }_{3}=1.33\,\text{m},\quad {f}_{3}=37.59\,\text{Hz}\hfill \end{array} [/latex]

Consider the experimental setup shown below. The length of the string between the string vibrator and the pulley is [latex] L=1.00\,\text{m}. [/latex] The linear density of the string is [latex] \mu =0.006\,\text{kg/m}. [/latex] The string vibrator can oscillate at any frequency. The hanging mass is 2.00 kg. (a)What are the wavelength and frequency of [latex] n=6 [/latex] mode? (b) The string oscillates the air around the string. What is the wavelength of the sound if the speed of the sound is [latex] {v}_{s}=343.00\,\text{m/s?} [/latex]

A cable with a linear density of [latex] \mu =0.2\,\text{kg/m} [/latex] is hung from telephone poles. The tension in the cable is 500.00 N. The distance between poles is 20 meters. The wind blows across the line, causing the cable resonate. A standing waves pattern is produced that has 4.5 wavelengths between the two poles. The air temperature is [latex] T=20\text{°}\text{C}. [/latex] What are the frequency and wavelength of the hum?

Show Answer

Consider a rod of length L , mounted in the center to a support. A node must exist where the rod is mounted on a support, as shown below. Draw the first two normal modes of the rod as it is driven into resonance. Label the wavelength and the frequency required to drive the rod into resonance.

Consider two wave functions [latex] y(x,t)=0.30\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x-4\,{\text{s}}^{-1}t) [/latex] and [latex] y(x,t)=0.30\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x+4\,{\text{s}}^{-1}t) [/latex]. Write a wave function for the resulting standing wave.

[latex] y(x,t)=[0.60\,\text{cm}\,\text{sin}(3\,{\text{m}}^{-1}x)]\text{cos}(4\,{\text{s}}^{-1}t) [/latex]

A 2.40-m wire has a mass of 7.50 g and is under a tension of 160 N. The wire is held rigidly at both ends and set into oscillation. (a) What is the speed of waves on the wire? The string is driven into resonance by a frequency that produces a standing wave with a wavelength equal to 1.20 m. (b) What is the frequency used to drive the string into resonance?

A string with a linear mass density of 0.0062 kg/m and a length of 3.00 m is set into the [latex] n=100 [/latex] mode of resonance. The tension in the string is 20.00 N. What is the wavelength and frequency of the wave?

[latex] \begin{array}{cc} {\lambda }_{100}=0.06\,\text{m}\hfill \\ \\ v=56.8\,\text{m/s,}\quad {f}_{n}=n{f}_{1},\quad n=1,2,3,4,5\text{…}\hfill \\ {f}_{100}=947\,\text{Hz}\hfill \end{array} [/latex]

A string with a linear mass density of 0.0075 kg/m and a length of 6.00 m is set into the [latex] n=4 [/latex] mode of resonance by driving with a frequency of 100.00 Hz. What is the tension in the string?

Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions along a string producing a standing wave. The linear mass density of the string is [latex] \mu =0.075\,\text{kg/m} [/latex] and the tension in the string is [latex] {F}_{T}=5.00\,\text{N}. [/latex] The time interval between instances of total destructive interference is [latex] \text{Δ}t=0.13\,\text{s}. [/latex] What is the wavelength of the waves?

[latex] T=2\text{Δ}t,\quad v=\frac{\lambda }{T},\quad \lambda =2.12\,\text{m} [/latex]

A string, fixed on both ends, is 5.00 m long and has a mass of 0.15 kg. The tension if the string is 90 N. The string is vibrating to produce a standing wave at the fundamental frequency of the string. (a) What is the speed of the waves on the string? (b) What is the wavelength of the standing wave produced? (c) What is the period of the standing wave?

A string is fixed at both end. The mass of the string is 0.0090 kg and the length is 3.00 m. The string is under a tension of 200.00 N. The string is driven by a variable frequency source to produce standing waves on the string. Find the wavelengths and frequency of the first four modes of standing waves.

[latex] \begin{array}{cc} {\lambda }_{1}=6.00\,\text{m},\quad {\lambda }_{2}=3.00\,\text{m},\quad {\lambda }_{3}=2.00\,\text{m},\quad {\lambda }_{4}=1.50\,\text{m}\hfill \\ v=258.20\,\text{m/s}=\lambda f\hfill \\ {f}_{1}=43.03\,\text{Hz},\quad {f}_{2}=86.07\,\text{Hz},\quad {f}_{3}=129.10\,\text{Hz},\quad {f}_{4}=172.13\,\text{Hz}\hfill \end{array} [/latex]

The frequencies of two successive modes of standing waves on a string are 258.36 Hz and 301.42 Hz. What is the next frequency above 100.00 Hz that would produce a standing wave?

A string is fixed at both ends to supports 3.50 m apart and has a linear mass density of [latex] \mu =0.005\,\text{kg/m}. [/latex] The string is under a tension of 90.00 N. A standing wave is produced on the string with six nodes and five antinodes. What are the wave speed, wavelength, frequency, and period of the standing wave?

[latex] v=134.16\,\text{ms},\lambda =1.4\,\text{m},f=95.83\,\text{Hz},T=0.0104\,\text{s} [/latex]

Sine waves are sent down a 1.5-m-long string fixed at both ends. The waves reflect back in the opposite direction. The amplitude of the wave is 4.00 cm. The propagation velocity of the waves is 175 m/s. The [latex] n=6 [/latex] resonance mode of the string is produced. Write an equation for the resulting standing wave.

Additional Problems

Ultrasound equipment used in the medical profession uses sound waves of a frequency above the range of human hearing. If the frequency of the sound produced by the ultrasound machine is [latex] f=30\,\text{kHz,} [/latex] what is the wavelength of the ultrasound in bone, if the speed of sound in bone is [latex] v=3000\,\text{m/s?} [/latex]

[latex] \lambda =0.10\,\text{m} [/latex]

Shown below is the plot of a wave function that models a wave at time [latex] t=0.00\,\text{s} [/latex] and [latex] t=2.00\,\text{s} [/latex]. The dotted line is the wave function at time [latex] t=0.00\,\text{s} [/latex] and the solid line is the function at time [latex] t=2.00\,\text{s} [/latex]. Estimate the amplitude, wavelength, velocity, and period of the wave.

The speed of light in air is approximately [latex] v=3.00\,×\,{10}^{8}\,\text{m/s} [/latex] and the speed of light in glass is [latex] v=2.00\,×\,{10}^{8}\,\text{m/s} [/latex]. A red laser with a wavelength of [latex] \lambda =633.00\,\text{nm} [/latex] shines light incident of the glass, and some of the red light is transmitted to the glass. The frequency of the light is the same for the air and the glass. (a) What is the frequency of the light? (b) What is the wavelength of the light in the glass?

a. [latex] f=4.74\,×\,{10}^{14}\,\text{Hz;} [/latex] b. [latex] \lambda =422\,\text{nm} [/latex]

A radio station broadcasts radio waves at a frequency of 101.7 MHz. The radio waves move through the air at approximately the speed of light in a vacuum. What is the wavelength of the radio waves?

A sunbather stands waist deep in the ocean and observes that six crests of periodic surface waves pass each minute. The crests are 16.00 meters apart. What is the wavelength, frequency, period, and speed of the waves?

[latex] \lambda =16.00\,\text{m},\quad f=0.10\,\text{Hz},\quad T=10.00\,\text{s},\quad v=1.6\,\text{m/s} [/latex]

A tuning fork vibrates producing sound at a frequency of 512 Hz. The speed of sound of sound in air is [latex] v=343.00\,\text{m/s} [/latex] if the air is at a temperature of [latex] 20.00\text{°}\text{C} [/latex]. What is the wavelength of the sound?

A motorboat is traveling across a lake at a speed of [latex] {v}_{b}=15.00\,\text{m/s}. [/latex] The boat bounces up and down every 0.50 s as it travels in the same direction as a wave. It bounces up and down every 0.30 s as it travels in a direction opposite the direction of the waves. What is the speed and wavelength of the wave?

[latex] \lambda =({v}_{b}+v){t}_{b},\quad v=3.75\,\text{m/s,}\quad \lambda =3.00\,\text{m} [/latex]

Use the linear wave equation to show that the wave speed of a wave modeled with the wave function [latex] y(x,t)=0.20\,\text{m}\,\text{sin}(3.00\,{\text{m}}^{-1}x+6.00\,{\text{s}}^{-1}t) [/latex] is [latex] v=2.00\,\text{m/s}. [/latex] What are the wavelength and the speed of the wave?

Given the wave functions [latex] {y}_{1}(x,t)=A\,\text{sin}(kx-\omega t) [/latex] and [latex] {y}_{2}(x,t)=A\,\text{sin}(kx-\omega t+\varphi ) [/latex] with [latex] \varphi \ne \frac{\pi }{2} [/latex], show that [latex] {y}_{1}(x,t)+{y}_{2}(x,t) [/latex] is a solution to the linear wave equation with a wave velocity of [latex] v=\sqrt{\frac{\omega }{k}}. [/latex]

[latex] \begin{array}{cc} \frac{{\partial }^{2}({y}_{1}+{y}_{2})}{\partial {t}^{2}}=\text{−}A{\omega }^{2}\,\text{sin}(kx-\omega t)-A{\omega }^{2}\,\text{sin}(kx-\omega t+\varphi )\hfill \\ \frac{{\partial }^{2}({y}_{1}+{y}_{2})}{\partial {x}^{2}}=\text{−}A{k}^{2}\,\text{sin}(kx-\omega t)-A{k}^{2}\,\text{sin}(kx-\omega t+\varphi )\hfill \\ \frac{{\partial }^{2}y(x,t)}{\partial {x}^{2}}=\frac{1}{{v}^{2}}\,\frac{{\partial }^{2}y(x,t)}{\partial {t}^{2}}\hfill \\ \\ -A{\omega }^{2}\,\text{sin}(kx-\omega t)-A{\omega }^{2}\,\text{sin}(kx-\omega t+\varphi )=(\frac{1}{{v}^{2}})(\text{−}A{k}^{2}\,\text{sin}(kx-\omega t)-A{k}^{2}\,\text{sin}(kx-\omega t+\varphi ))\hfill \\ v=\frac{\omega }{k}\hfill \end{array} [/latex]

A transverse wave on a string is modeled with the wave function [latex] y(x,t)=0.10\,\text{m}\,\text{sin}(0.15\,{\text{m}}^{-1}x+1.50\,{\text{s}}^{-1}t+0.20) [/latex]. (a) Find the wave velocity. (b) Find the position in the y -direction, the velocity perpendicular to the motion of the wave, and the acceleration perpendicular to the motion of the wave, of a small segment of the string centered at [latex] x=0.40\,\text{m} [/latex] at time [latex] t=5.00\,\text{s}. [/latex]

A sinusoidal wave travels down a taut, horizontal string with a linear mass density of [latex] \mu =0.060\,\text{kg/m}. [/latex] The magnitude of maximum vertical acceleration of the wave is [latex] {a}_{y\,\text{max}}=0.90\,{\text{cm/s}}^{2} [/latex] and the amplitude of the wave is 0.40 m. The string is under a tension of [latex] {F}_{T}=600.00\,\text{N} [/latex]. The wave moves in the negative x -direction. Write an equation to model the wave.

[latex] y(x,t)=0.40\,\text{m}\,\text{sin}(0.015\,{\text{m}}^{-1}x+1.5\,{\text{s}}^{-1}t) [/latex]

A transverse wave on a string [latex] (\mu =0.0030\,\text{kg/m}) [/latex] is described with the equation [latex] y(x,t)=0.30\,\text{m}\,\text{sin}(\frac{2\pi }{4.00\,\text{m}}(x-16.00\frac{\text{m}}{\text{s}}t)). [/latex] What is the tension under which the string is held taut?

A transverse wave on a horizontal string [latex] (\mu =0.0060\,\text{kg/m}) [/latex] is described with the equation [latex] y(x,t)=0.30\,\text{m}\,\text{sin}(\frac{2\pi }{4.00\,\text{m}}(x-{v}_{w}t)). [/latex] The string is under a tension of 300.00 N. What are the wave speed, wave number, and angular frequency of the wave?

[latex] v=223.61\,\text{m/s},\,k=1.57\,{\text{m}}^{-1},\,\omega =142.43\,{\text{s}}^{-1} [/latex]

A student holds an inexpensive sonic range finder and uses the range finder to find the distance to the wall. The sonic range finder emits a sound wave. The sound wave reflects off the wall and returns to the range finder. The round trip takes 0.012 s. The range finder was calibrated for use at room temperature [latex] T=20\text{°}\text{C} [/latex], but the temperature in the room is actually [latex] T=23\text{°}\text{C}. [/latex] Assuming that the timing mechanism is perfect, what percentage of error can the student expect due to the calibration?

A wave on a string is driven by a string vibrator, which oscillates at a frequency of 100.00 Hz and an amplitude of 1.00 cm. The string vibrator operates at a voltage of 12.00 V and a current of 0.20 A. The power consumed by the string vibrator is [latex] P=IV [/latex]. Assume that the string vibrator is [latex] 90\text{%} [/latex] efficient at converting electrical energy into the energy associated with the vibrations of the string. The string is 3.00 m long, and is under a tension of 60.00 N. What is the linear mass density of the string?

[latex] \begin{array}{cc} P=\frac{1}{2}{A}^{2}{(2\pi f)}^{2}\sqrt{\mu {F}_{T}}\hfill \\ \mu =2.00\,×\,{10}^{-4}\,\text{kg/m}\hfill \end{array} [/latex]

A traveling wave on a string is modeled by the wave equation [latex] y(x,t)=3.00\,\text{cm}\,\text{sin}(8.00\,{\text{m}}^{-1}x+100.00\,{\text{s}}^{-1}t). [/latex] The string is under a tension of 50.00 N and has a linear mass density of [latex] \mu =0.008\,\text{kg/m}. [/latex] What is the average power transferred by the wave on the string?

A transverse wave on a string has a wavelength of 5.0 m, a period of 0.02 s, and an amplitude of 1.5 cm. The average power transferred by the wave is 5.00 W. What is the tension in the string?

[latex] P=\frac{1}{2}\mu {A}^{2}{\omega }^{2}\frac{\lambda }{T},\,\mu =0.0018\,\text{kg/m} [/latex]

(a) What is the intensity of a laser beam used to burn away cancerous tissue that, when [latex] 90.0\text{%} [/latex] absorbed, puts 500 J of energy into a circular spot 2.00 mm in diameter in 4.00 s? (b) Discuss how this intensity compares to the average intensity of sunlight (about) and the implications that would have if the laser beam entered your eye. Note how your answer depends on the time duration of the exposure.

Consider two periodic wave functions, [latex] {y}_{1}(x,t)=A\,\text{sin}(kx-\omega t) [/latex] and [latex] {y}_{2}(x,t)=A\,\text{sin}(kx-\omega t+\varphi ). [/latex] (a) For what values of [latex] \varphi [/latex] will the wave that results from a superposition of the wave functions have an amplitude of 2 A ? (b) For what values of [latex] \varphi [/latex] will the wave that results from a superposition of the wave functions have an amplitude of zero?

a. [latex] {A}_{R}=2A\,\text{cos}(\frac{\varphi }{2}),\,\text{cos}(\frac{\varphi }{2})=1,\,\varphi =0,2\pi ,4\pi \text{,…} [/latex]; b. [latex] {A}_{R}=2A\,\text{cos}(\frac{\varphi }{2}),\,\text{cos}(\frac{\varphi }{2})=0,\,\varphi =0,\pi ,3\pi ,5\pi \text{…} [/latex]

Consider two periodic wave functions, [latex] {y}_{1}(x,t)=A\,\text{sin}(kx-\omega t) [/latex] and [latex] {y}_{2}(x,t)=A\,\text{cos}(kx-\omega t+\varphi ) [/latex]. (a) For what values of [latex] \varphi [/latex] will the wave that results from a superposition of the wave functions have an amplitude of 2 A ? (b) For what values of [latex] \varphi [/latex] will the wave that results from a superposition of the wave functions have an amplitude of zero?

A trough with dimensions 10.00 meters by 0.10 meters by 0.10 meters is partially filled with water. Small-amplitude surface water waves are produced from both ends of the trough by paddles oscillating in simple harmonic motion. The height of the water waves are modeled with two sinusoidal wave equations, [latex] {y}_{1}(x,t)=0.3\,\text{m}\,\text{sin}(4\,{\text{m}}^{-1}x-3\,{\text{s}}^{-1}t) [/latex] and [latex] {y}_{2}(x,t)=0.3\,\text{m}\,\text{cos}(4\,{\text{m}}^{-1}x+3\,{\text{s}}^{-1}t-\frac{\pi }{2}). [/latex] What is the wave function of the resulting wave after the waves reach one another and before they reach the end of the trough (i.e., assume that there are only two waves in the trough and ignore reflections)? Use a spreadsheet to check your results. ( Hint: Use the trig identities [latex] \text{sin}(u±v)=\text{sin}\,u\,\text{cos}\,v±\text{cos}\,u\,\text{sin}\,v [/latex] and [latex] \text{cos}(u±v)=\text{cos}\,u\,\text{cos}\,v\mp \text{sin}\,u\,\text{sin}\,v) [/latex]

[latex] {y}_{R}(x,t)=0.6\,\text{m}\,\text{sin}(4\,{\text{m}}^{-1}x)\text{cos}(3\,{\text{s}}^{-1}t) [/latex]

A seismograph records the S- and P-waves from an earthquake 20.00 s apart. If they traveled the same path at constant wave speeds of [latex] {v}_{S}=4.00\,\text{km/s} [/latex] and [latex] {v}_{P}=7.50\,\text{km/s}, [/latex] how far away is the epicenter of the earthquake?

Consider what is shown below. A 20.00-kg mass rests on a frictionless ramp inclined at [latex] 45\text{°} [/latex]. A string with a linear mass density of [latex] \mu =0.025\,\text{kg/m} [/latex] is attached to the 20.00-kg mass. The string passes over a frictionless pulley of negligible mass and is attached to a hanging mass ( m ). The system is in static equilibrium. A wave is induced on the string and travels up the ramp. (a) What is the mass of the hanging mass ( m )? (b) At what wave speed does the wave travel up the string?

Consider the superposition of three wave functions [latex] y(x,t)=3.00\,\text{cm}\,\text{sin}(2\,{\text{m}}^{-1}x-3\,{\text{s}}^{-1}t), [/latex] [latex] y(x,t)=3.00\,\text{cm}\,\text{sin}(6\,{\text{m}}^{-1}x+3\,{\text{s}}^{-1}t), [/latex] and [latex] y(x,t)=3.00\,\text{cm}\,\text{sin}(2\,{\text{m}}^{-1}x-4\,{\text{s}}^{-1}t). [/latex] What is the height of the resulting wave at position [latex] x=3.00\,\text{m} [/latex] at time [latex] t=10.0\,\text{s?} [/latex]

A string has a mass of 150 g and a length of 3.4 m. One end of the string is fixed to a lab stand and the other is attached to a spring with a spring constant of [latex] {k}_{s}=100\,\text{N/m}. [/latex] The free end of the spring is attached to another lab pole. The tension in the string is maintained by the spring. The lab poles are separated by a distance that stretches the spring 2.00 cm. The string is plucked and a pulse travels along the string. What is the propagation speed of the pulse?

[latex] {F}_{T}=12\,\text{N,}\,v=16.49\,\text{m/s} [/latex]

A standing wave is produced on a string under a tension of 70.0 N by two sinusoidal transverse waves that are identical, but moving in opposite directions. The string is fixed at [latex] x=0.00\,\text{m} [/latex] and [latex] x=10.00\,\text{m}. [/latex] Nodes appear at [latex] x=0.00\,\text{m,} [/latex] 2.00 m, 4.00 m, 6.00 m, 8.00 m, and 10.00 m. The amplitude of the standing wave is 3.00 cm. It takes 0.10 s for the antinodes to make one complete oscillation. (a) What are the wave functions of the two sine waves that produce the standing wave? (b) What are the maximum velocity and acceleration of the string, perpendicular to the direction of motion of the transverse waves, at the antinodes?

A string with a length of 4 m is held under a constant tension. The string has a linear mass density of [latex] \mu =0.006\,\text{kg/m}. [/latex] Two resonant frequencies of the string are 400 Hz and 480 Hz. There are no resonant frequencies between the two frequencies. (a) What are the wavelengths of the two resonant modes? (b) What is the tension in the string?

a. [latex] \begin{array}{cc} {f}_{n}=\frac{nv}{2L},\,v=\frac{2L{f}_{n+1}}{n+1},\,\frac{n+1}{n}=\frac{2L{f}_{n+1}}{2L{f}_{n}},\,1+\frac{1}{n}=1.2,\,n=5\hfill \\ {\lambda }_{n}=\frac{2}{n}L,\,{\lambda }_{5}=1.6\,\text{m},\,{\lambda }_{6}=1.33\,\text{m}\hfill \end{array} [/latex]; b. [latex] {F}_{T}=245.76\,\text{N} [/latex]

Challenge Problems

A copper wire has a radius of [latex] 200\,\text{μm} [/latex] and a length of 5.0 m. The wire is placed under a tension of 3000 N and the wire stretches by a small amount. The wire is plucked and a pulse travels down the wire. What is the propagation speed of the pulse? (Assume the temperature does not change: [latex] (\rho =8.96\frac{\text{g}}{{\text{cm}}^{3}},Y=1.1\,×\,{10}^{11}\frac{\text{N}}{\text{m}}).) [/latex]

A pulse moving along the x axis can be modeled as the wave function [latex] y(x,t)=4.00\,\text{m}{e}^{\text{−}{(\frac{x+(2.00\,\text{m/s})t}{1.00\,\text{m}})}^{2}}. [/latex] (a)What are the direction and propagation speed of the pulse? (b) How far has the wave moved in 3.00 s? (c) Plot the pulse using a spreadsheet at time [latex] t=0.00\,\text{s} [/latex] and [latex] t=3.00\,\text{s} [/latex] to verify your answer in part (b).

a. Moves in the negative x direction at a propagation speed of [latex] v=2.00\,\text{m/s} [/latex]. b. [latex] \text{Δ}x=-6.00\,\text{m;} [/latex] c.

A string with a linear mass density of [latex] \mu =0.0085\,\text{kg/m} [/latex] is fixed at both ends. A 5.0-kg mass is hung from the string, as shown below. If a pulse is sent along section A , what is the wave speed in section A and the wave speed in section B ?

Consider two wave functions [latex] {y}_{1}(x,t)=A\,\text{sin}(kx-\omega t) [/latex] and [latex] {y}_{2}(x,t)=A\,\text{sin}(kx+\omega t+\varphi ) [/latex]. What is the wave function resulting from the interference of the two wave? ( Hint: [latex] \text{sin}(\alpha ±\beta )=\text{sin}\,\alpha \,\text{cos}\,\beta ±\text{cos}\,\alpha \,\text{sin}\,\beta [/latex] and [latex] \varphi =\frac{\varphi }{2}+\frac{\varphi }{2} [/latex].)

[latex] \begin{array}{cc} \text{sin}(kx-\omega t)=\text{sin}(kx+\frac{\varphi }{2})\text{cos}(\omega t+\frac{\varphi }{2})-\text{cos}(kx+\frac{\varphi }{2})\text{sin}(\omega t+\frac{\varphi }{2})\hfill \\ \text{sin}(kx-\omega t+\varphi )=\text{sin}(kx+\frac{\varphi }{2})\text{cos}(\omega t+\frac{\varphi }{2})+\text{cos}(kx+\frac{\varphi }{2})\text{sin}(\omega t+\frac{\varphi }{2})\hfill \\ \text{sin}(kx-\omega t)+\text{sin}(kx+\omega t+\varphi )=2\,\text{sin}(kx+\frac{\varphi }{2})\text{cos}(\omega t+\frac{\varphi }{2})\hfill \\ {y}_{R}=2\,A\,\text{sin}(kx+\frac{\varphi }{2})\text{cos}(\omega t+\frac{\varphi }{2})\hfill \end{array} [/latex]

The wave function that models a standing wave is given as [latex] {y}_{R}(x,t)=6.00\,\text{cm}\,\text{sin}(3.00\,{\text{m}}^{-1}x+1.20\,\text{rad}) [/latex] [latex] \text{cos}(6.00\,{\text{s}}^{-1}t+1.20\,\text{rad}) [/latex]. What are two wave functions that interfere to form this wave function? Plot the two wave functions and the sum of the sum of the two wave functions at [latex] t=1.00\,\text{s} [/latex] to verify your answer.

Consider two wave functions [latex] {y}_{1}(x,t)=A\,\text{sin}(kx-\omega t) [/latex] and [latex] {y}_{2}(x,t)=A\,\text{sin}(kx+\omega t+\varphi ) [/latex]. The resultant wave form when you add the two functions is [latex] {y}_{R}=2A\,\text{sin}(kx+\frac{\varphi }{2})\text{cos}(\omega t+\frac{\varphi }{2}). [/latex] Consider the case where [latex] A=0.03\,{\text{m}}^{-1}, [/latex] [latex] k=1.26\,{\text{m}}^{-1}, [/latex] [latex] \omega =\pi \,{\text{s}}^{-1} [/latex], and [latex] \varphi =\frac{\pi }{10} [/latex]. (a) Where are the first three nodes of the standing wave function starting at zero and moving in the positive x direction? (b) Using a spreadsheet, plot the two wave functions and the resulting function at time [latex] t=1.00\,\text{s} [/latex] to verify your answer.

[latex] \begin{array}{cc} \text{sin}(kx+\frac{\varphi }{2})=0,\,kx+\frac{\varphi }{2}=0,\pi ,2\pi ,\,1.26\,{\text{m}}^{-1}x+\frac{\pi }{20}=\pi ,2\pi ,3\pi \hfill \\ x=2.37\,\text{m},4.86\,\text{m},7.35\,\text{m}\hfill \end{array} [/latex];

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Shop Experiment Standing Waves on a String Experiments​

Standing waves on a string.

Experiment #3 from Advanced Physics with Vernier — Beyond Mechanics

Introduction

When you shake a string, a pulse travels down its length. When it reaches the end, the pulse can be reflected. A series of regularly occurring pulses will generate traveling waves that, after reflection from the other end, will interfere with the oncoming waves. When the conditions are right, the superposition of these waves traveling in opposite directions can give rise to something known as a “standing wave.” That is, there appear to be stationary waves on the string with some parts of the string hardly moving at all and other regions where the string experiences a large displacement. In this lab you will investigate the various factors that give rise to this phenomenon.

In this experiment, you will

• Adjust the frequency of the driver so that the string vibrates in the fundamental mode.
• Set up other standing wave patterns on the string.
• Relate the frequency of the various harmonics to that of the fundamental mode of vibration.
• Describe the terms amplitude, frequency, wavelength, node, and antinode as they relate to vibrating strings.
• Determine the velocity of waves in the string.
• Relate wave velocity to the tension of the string and its linear density.

Sensors and Equipment

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Correlations

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This experiment is #3 of Advanced Physics with Vernier — Beyond Mechanics . The experiment in the book includes student instructions as well as instructor information for set up, helpful hints, and sample graphs and data.

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Ohio recreational marijuana sales start: What Columbus customers saw

Recreational marijuana sales are now in full swing in Ohio.

After months of regulatory maneuvering and permit-seeking, 98 marijuana dispensaries received licenses to sell recreational marijuana in Ohio starting Tuesday . While Ohio has had medical marijuana for years, Tuesday marks the first day Ohio residents over 21 can legally purchase the plant recreationally.

While it's up to individual stores if they're open for business, many dispensaries across the state welcomed customers on recreational marijuana's first day.

Here's a sampling of what that looked like:

Ohio's first customer buys in Cincinnati

A Madisonville resident made the first recreational marijuana purchase at a Greater Cincinnati dispensary Tuesday morning, the first day of legal sales in Ohio.

Jeff Riede was first in line at Sunnyside dispensary, according to the Cincinnati Enquirer . The 55-year-old said he’d been waiting in his car in the parking lot since about 6:30 p.m. Monday for the dispensary to open at 7 a.m. He planned to buy edibles and some flower.

“Yeah, I slept in my car,” Riede told reporters. “This is pretty epic to me. I wanted to be the first one here.”

Related article: 'I slept in my car': Cincinnati man camps out overnight for first recreational weed sale

Bloom Medicinals starts selling in Columbus

Bloom Medicinals, 1361 Georgesville Road, opened its door Tuesday around 8 a.m. to a line of about 20 people eager to be some of the first people in Columbus to buy marijuana over the counter without a prescription.

Employees at what began as a medicinal dispensary provided a separate line for their regular customers who buy marijuana for medicinal.

As customers left the store, a handful of them raised their fists in celebration, eliciting cheers from the other customers waiting in line.

Inside the dispensary, customers stood in cordoned-off lines, waiting to approach Bloom’s security station to verify they’re legally allowed to buy marijuana.

In the brightly lit storeroom, customers could use touch screen tablets or chat with employees to browse marijuana products like Black Sheep’s “8 Inch Bagel” marijuana flowers and Camino’s “Freshly Squeezed Oranges” edibles.

While dispensaries can’t play music outside their shops , pop hits like Doja Cat’s “Say So” filtered through Bloom’s speakers in their showroom. 

Most marijuana flower products cost around $40 for 2.83 grams or around a tenth of an ounce. Edibles appear less expensive, with prices ranging from $17 to $100, depending on the size of the containers. 

Gavin McKenney, Bloom Medicinals' general manager, said customers used to buying marijuana in Michigan may have to get used to Ohio's higher prices for now.

"Hopefully in the next few months stuff will go down. With sales tax as of now and product availability, you see what happens," he said.

Jarris Barfield, 66, waited in Bloom Medicinals’ recreational marijuana line early Tuesday morning. He was recently diagnosed with stage 4 prostate cancer, and he said he may only have six months to live.

His marijuana purchase, he said, was meant to help bring him some happiness during the time he has left.

“Life has its ups and downs," Barfield said. "But I’m going to stay on the better side of life and stay happy. And if this is what it takes to get me there, I’ll do it.”

At one Columbus dispensary, line shorter than expected

At Nar Reserve, recreational sales began in force at 10 a.m. after a two-hour priority period for medical customers that began at 8 a.m. While the morning started slow, dozens of customers came and went from Long Street and Grant Avenue dispensary.

Around 10 a.m., there was a short line inside the dispensary. Linda McAlexander, 70, said she has asthma and was most excited to purchase edibles because of her condition. She said she felt like Ohio was finally getting with the times.

"Ohio is so conservative, it's ridiculous," McAlexander said.

David McAlexander, 68, was surprised to see the line was so short.

"No line, can you believe that?" David McAlexander said. "I thought there'd be a big line."

Ali Bazzi, managing partner of Nar Reserve, said while things were starting slow, the store was seeing some traction as word got out of recreational sales starting Tuesday. He said he expects traffic to the dispensary to build over several weeks.

"Once word gets out, as the lines start going, we think it's going to keep getting busier," Bazzi said.

Nar Reserve brought extra staff on hand Tuesday and ensured it had enough product to meet demand.

"It's kind of better it started off slow, just so we can ramp up," Bazzi said. "You don't want the employees walking in with 200 people in line."

Michael Hawkins, 46, said he was excited to see if the product "was worth the wait" of getting dispensaries online. He bought some to sample, saying he paid $40 for 2.8 grams of weed, much higher than he would've seen paying on the black market.

"You don't have to buy from not shady characters no more," Hawkins said. "I know I'm paying for what I'm getting."

Truelieve dispensary in Westerville sees lines around the building

The line at Truelieve in Westerville wrapped around the building, and its parking lot was full late Tuesday morning as customers came to purchase marijuana on their lunch breaks.

"It's just exciting seeing people from literally every walk of life, every different background. People are extremely happy and excited. Everyone's in a really good mood," Kyle Schrader, age not known, said while waiting to enter the store.

Employees sifted through the 30-person line, checking customers in, answering questions and handing out water bottles while the summer sun beat down on the dispensary.

Katie Masko, 22, said while waiting in line that marijuana dulls her nausea, allowing her to eat. It also helps with anxiety, she said.

"It's not fun to be nauseous," Masko said.

Nick Rassler, Trulieve's director of state operations, said the store increased its inventory and staffing levels in anticipation of Tuesday's avalanche of recreational demand. Rassler said the store was consistently busy throughout the morning, with 20 to 30 people circling the building at any given time.

To keep things moving quickly, he recommended customers order online before coming in. Medicinal marijuana patients don't have to worry about the lines; they get to jump to the front, he said.

"(Medicinal patients have) taken care of us, and we intend to continue to take care of them. We've done that as a company in every state that we have operated as both medical and recreational," he said.

Columbus medicinal marijuana patrons welcome the change

Some medical marijuana patients have expressed apprehension about the launch of recreational, but Robert Coalter, a Tuesday morning customer at Amplify on Hamilton Road, said he’s not too bothered – and he doesn’t expect the crowds to last.

“Everybody's really nice because they're coping with something somehow, and there's an answer inside,” he said.

Recreational launch a 'long time coming,' customers say

The Botanist, part of a chain of dispensaries with a location at 115 Vine St. in Columbus, opened its doors to non-medical patients at 9 a.m. Gyen Musgrave, director of retail operations, said he expected a 100-to-150% increase in traffic on Tuesday. His staff has been preparing for this day since March. 

"That's forever been a difficult hurdle for the medical cannabis program, and hopefully, those restrictions lessen as we approach the adult-use market," he said. "We're awfully busy to where it's hard to appreciate the moment, but it's extremely exciting."

Justin Davis, 28, one of the first recreational marijuana buyers at The Botanist, said: "This is more convenient, and it just lets us know that Ohioans are not so conservative like we think."

Other customers said they appreciate the safety and convenience.

"It feels good to know you can get it, and it's actually safe, you know," said Cameron Gregory, 26. "I ran here right after work."

Related recreational marijuana article: As recreational weed sales begin, here's where you can buy in Columbus

Others viewed Tuesday as something that should have happened a long time ago.

"It feels like I'm being treated as an adult, as opposed to a criminal," Troy Stanley, 50, said while waiting in line at The Botanist. "I've been persecuted and prosecuted for the last 40 years over something that should have been made legal a long time ago," Sean Smith, 59, said. "So I'm glad to not have to look over my shoulder."

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2024 FedEx Cup standings, schedule, PGA Tour leaderboard, purse, prize money for FedEx Cup Playoffs

Your one-stop shop for everything you need to know throughout the 2024 fedex cup playoffs.

Following an exciting regular season, the PGA Tour hosts a three-tournament postseason, the 2024 FedEx Cup Playoffs, which concludes with a massive Tour Championship where the majority of the grand total $100 million purse is dished out to golfers. In fact, the $25 million top prize is tied for the largest payout on the PGA Tour this season with the Players Championship. 

Only 70 golfers advanced to the first round of the playoffs, the FedEx St. Jude Championship, same as last season. With 20 of those players now eliminated following the first event of the playoffs at TPC Southwind in Memphis, some of the biggest names in golf will no longer have a chance to compete for the top prize.

There's plenty of star power in the field, though, with Scottie Scheffler and Xander Schauffele atop the standings and other significant players like Rory McIlroy, Hideki Matsuyama, Collin Morikawa and Wyndham Clark in contention. For Scheffler, a FedEx Cup win would be his first, but it would also serve as a feather in his cap on the back of a stellar season that already includes six PGA Tour victories and an Olympic gold medal. The same goes for Schauffele, a two-time major championship winner this season.

For everyone involved, there will be a ton of money and plenty of accolades at stake over the next three weeks. Let's take a closer look at what to expect from this year's festivities.

2024 FedEx Cup Playoffs schedule

The top 70 in the FedEx Cup standings, via points accumulated throughout the year, played in the St. Jude Championship this week.

All three events are 72-hole, stroke-play tournaments, though the fields gradually get smaller as the playoffs roll on. The points change, too, as everything is quadrupled. During regular-season events, most winners receive 500 FedEx Cup points for finishing first at tournaments (in a handful of events, 600 points went to first place). The winners of the first two FedEx Cup Playoffs events will instead receive 2,000 points each. The point boost goes for every slot on the leaderboard: 300 for second becomes 1,200 and so on. 

Only seven golfers surpassed the 2,000-point total during the entire regular season: Scheffler, Schauffele, McIlroy, Collin Morikawa, Wyndham Clark, Ludvig Åberg and Sahith Theegala. Scheffler opened with nearly a 3,500-point lead on third-place McIlroy, while Schauffele himself was 1,500 points up on the rest of the field.

Still, the FedEx Cup standings can shift quite a bit -- especially for the winners of the first two events -- over the next couple weeks. Winners are disproportionately rewarded and deservedly so given this is the postseason. This provides the opportunity for golfers to go on a hot streak and rocket up the FedEx Cup standings. Regardless of what else happens, the first two playoff winners will be in great spots entering the finale, the Tour Championship at East Lake. Similar to other sports, now that the postseason has begun, almost anything can happen.

The top 50 in the FedEx Cup standings after the St. Jude Championship move on to the BMW Championship. Then the top 30 after that move on to the Tour Championship.

2024 FedEx Cup standings

Scheffler and Schauffele are having extraordinary seasons. They rank No. 1 and No. 3 on the all-time single season money list at $29.1 million and $17.6 million, respectively, following the first playoff event. Those numbers will only go up from there as the second playoff event, the BMW Championship, has a similar $20 million purse to the FedEx St. Jude Championship.

Here's a look at the top 30 in the standings following the FedEx St. Jude Championship.

Even though anything can happen over the next two weeks, players are still rewarded for what they accomplished in the regular season. The lead Scheffler (nearly 3,000 points over third) and Schauffele (1,100 points over third) have built will be difficult to chip away if those two continue performing well. For example, Schauffele is the only player who can mathematically catch Scheffler next week at the BMW Championship, and it will take a win to do it. Matsuyama is the only player who can catch Schauffele for second.

Scheffler and Schauffele are in a great spot to jump into the top two spots at the Tour Championship where they would start at 10 under and 8 under, respectively. That gives them a huge head start on winning the first FedEx Cup of their careers. For either, it would be emblematic of the quality of golf they have played for the last seven months as the 2024 PGA Tour season winds down.

2024 Tour Championship format

Heading into the Tour Championship inside the top five or top 10 in the FedEx Cup standings is important because of how scoring is dispersed. Whoever is first in the FedEx Cup standings after the BMW Championship starts the Tour Championship at 10 under, and the event is played under normal scoring conditions from there. Second starts at 8 under and so on (see full numbers below). 

With so much money at stake (again, $25 million for first place), those margins become more meaningful than even a normal week. The eventual winners of the four FedEx Cups played under this format have all started in the top seven at the Tour Championship.

• 6th to 10th: -4
• 11th to 15th: -3
• 16th to 20th: -2
• 21st to 25th: -1
• 26th to 30th: E

2024 FedEx Cup Playoffs purse, prize money

2024 st. jude championship purse, prize money.

• 1st: $3.6 million
• 2nd: $2.2 million
• 3rd: $1.4 million
• 4th: $960,000
• 5th: $800,000
• 6th: $720,000
• 7th: $670,000
• 8th: $620,000
• 9th: $580,000
• 10th: $540,000

2024 BMW Championship purse, prize money

• 4th: $990,000
• 5th: $830,000
• 6th: $750,000
• 7th: $695,000
• 8th: $640,000
• 9th: $600,000
• 10th: $560,000

2024 Tour Championship purse, prize money

The figures are startling for the finale. The winner of the Tour Championship receive $18 million. If you just make into the final FedEx Cup Playoff event, you're guaranteed $500,000. Here's a look at what the lucrative top 10 will look like at the Tour Championship.

• 1st: $25 million
• 2nd: $12.5 million
• 3rd: $7.5 million
• 4th: $6 million
• 5th: $5 million
• 6th: $3.5 million
• 7th: $2.75 million
• 8th: $2.25 million
• 9th: $2 million
• 10th $1.75 million

Last year, Viktor Hovland won the BMW Championship and then took the Tour Championship and FedEx Cup over Schauffele. Both players shot the same 19-under score at East Lake to end the year, but Hovland started the tournament at 8 under while Schauffele only started it at 3 under so Hovland easily won by five and took home the first prize of what was then $18 million.

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Future of Perth's last Aboriginal housing community at Cullacabardee unclear

By Jon Daly

Topic: Communities

Cullacabardee is the oldest and last remaining Aboriginal community in metropolitan Perth. (August 2024) ( ABC News: Jon Daly )

Planned as a model urban community, Cullacabardee was once filled with Indigenous families and playgrounds, but today it's virtually abandoned, with its few remaining residents facing an uncertain future.

Cul-de-sacs occupied by single homes and paved driveways leading to empty lots are all that remain of Western Australia's forgotten Aboriginal "housing experiment".

Nestled in the bushlands of Perth's northern suburbs on Whadjuk Noongar lands, Cullacabardee was established in the 1980s and once housed more than 100 people.

Now it's the last housing community of its kind in the metropolitan area.

Community in limbo

Sitting at her dimly lit kitchen table, Cullacabardee's Julie Lewis is among about a dozen residents living in one of the last five homes at a settlement which once had 31 dwellings.

"We are still here," she said.

The community's future is uncertain, partly because of asbestos contamination, and the state government is yet to provide any clarity on what its plans are for Cullacabardee.

"We're in big limbo," Ms Lewis said.

Guitar-playing pastor You-Sin Houe is another to bear witness to the community's decline.

For the last 20 years, Mr Houe has delivered his Sunday morning sermon to an increasing number of empty chairs in Cullacabardee's community centre.

"A lot of children and families enjoyed this community," Mr Houe said.

"Now only a few people are here, and all the people are gone.

"I'm very sad because this community is their identity, they really love this land."

Troubled past, unknown future

Cullacabardee was built by the WA and Commonwealth governments as a "community welfare proposal" and "housing experiment", according to media reports of the day.

Aboriginal families were resettled from makeshift camps on Perth's outskirts to conventional, double-bricked homes in Cullacabardee.

Community unrest and violence marred the settlement in the decades that followed.

Public housing was gradually knocked down and never rebuilt.

Ms Lewis said Cullacabardee became a liability and had been left to slowly die.

"That's due to the government wanting no people out here," she said.

The Department of Planning, Lands and Heritage (DPLH) did not respond to direct questions about whether funding and provision of housing at Cullacabardee would be restored.

"DPLH has been working closely with the local community to ensure the current residents of Cullacabardee have better social and economic outcomes," a spokesperson said.

WA Aboriginal Affairs Minister Tony Buti declined the ABC's request for an interview.

The community's fate is similar to the now-demolished Swan Valley Nyungah community in the nearby suburb of Lockridge.

The Swan Valley camp was shut down by an act of parliament in 2003 following an inquiry into sexual abuse in Aboriginal communities.

University of Sydney emeritus professor Peter Phibbs has studied affordable and Aboriginal housing over the last 30 years.

He said the "self-determination" approach to Aboriginal housing prevailed through the 1980s and 1990s, in which Aboriginal housing organisations were the main vehicle for housing delivery.

That was dismantled during the Howard government's interventionist era from 2007.

Dr Phibbs' research found the number of Aboriginal housing organisations almost halved from 616 to 330 between 2001 and 2012.

"I think for many Aboriginal communities, and housing organisations, they found that change quite bewildering, and really a setback in terms of where they were moving," he said.

A slow demise

Bricks and twisted metal strewn in the sand of nearby vacant blocks remind Ms Lewis of the old days, when each street had a communal barbecue and playground.

The Cullacabardee community has since fallen into obscurity.

"It's like we are forgotten," Ms Lewis said.

The few homes left are owned and managed by the Cullacabardee Aboriginal Corporation.

State government funding has all but ceased.

The Cullacabardee land reserve is held by WA's Aboriginal Lands Trust, though the state government said it supported divesting this land to Aboriginal ownership.

Ms Lewis wonders where that leaves the community.

"For most of us, we think one day they'll come out and say 'youse have got to go'. Where do we go?" she asked.

Asbestos and 'the pit'

A dusty dirt road leads to what locals call "the pit" — an illegal landfill littered with bush-basher cars and detritus.

Ms Lewis said authorities fenced off the area about eight years ago, erecting asbestos warning signs.

She claims authorities have left residents in the dark about the extent of the contamination and plans to deal with it.

"Are we exposed to it?" she asked. "Are our kids exposed to it?" "We don't know."

Cliff Weeks, the former director general of the now-amalgamated Department of Aboriginal Affairs addressed the issue at an estimates committee in WA's parliament in May 2016.

"The asbestos is located in a tip that was never approved," he said.

"Without going back too far, when I was working in the Department of Housing, Cullacabardee residents were accepting cash payments for trucks to drop off rubbish."

Mr Weeks said asbestos experts gave advice of "limited" risk around airborne exposure, and the department would go through the process of working out how to "get rid of the asbestos".

But eight years later, the problem remains.

A spokesperson from WA's Department of Planning, Lands and Heritage said the department was "developing a broader strategy" to address contamination and support "divestment".

"[The department] has engaged a qualified environmental consultant to complete further investigation works to inform future remediation requirements," they said. "Environmental investigations have been undertaken to assess the contamination in the area."

Ms Lewis said the findings of those investigations are yet to be provided to the Cullacabardee Aboriginal Corporation.

No native title

Cullacabardee is yet to be included in a landmark $1.3 billion native title settlement covering WA's South West.

Inclusion in the settlement, struck between traditional owners and the WA government in 2021, could provide new avenues of funding for development and housing at the community.

A spokesperson from the Whadjuk Aboriginal Corporation, whose territory encompasses Cullacabardee, said the state government had "recently notified" the corporation the Cullacabardee land parcel was under an "initial review" for inclusion in the settlement.

"The Whadjuk Aboriginal Corporation has not made any decision regarding the acceptance or rejection of this property," the spokesperson said.

'We're still out here'

Whether government or Aboriginal-owned, Ms Lewis said residents just wanted a say in the future of their community.

"What would be good about is that when they make a decision, be mindful, we're still out here," she said.

Looking out her kitchen window at surrounding banksia trees and bushlands, Ms Lewis said life in limbo had its positives.

"Then again, it's nice to be forgotten too, because we can still live out here."

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Cleveland Browns Rookie Named Top Player to Watch in Preseason vs. Vikings

Evan massey | aug 17, 2024.

• Cleveland Browns

The Cleveland Browns are set to take on the Minnesota Vikings this afternoon in their second preseason game. With the regular season right around the corner, players are fighting for roster spots as well as potential playing time to begin the year.

One player who is working hard to get on the field early on during his rookie season is wide receiver Jamari Thrash.

Thrash, a fifth-round rookie out of Louisville, has made a strong impression early on in his tenure with the Browns. There is a chance that he could work his way onto the field early in the season.

He has even been named one of the top players to watch today by Cleveland team writer Patrick Warren .

In his "burning questions" for today's preseason matchup, Warren asked a question about Thrash. Can he continue building on his strong performance from the preseason opener?

During that first preseason game against the Green Bay Packers , Thrash ended up catching three passes for 43 yards. He looked very confident and strong in his NFL debut.

After his first preseason game, quarterback Jameis Winston spoke out about the rookie wideout.

"It's always great seeing young guys when it finally clicks. I was very impressed with him, and I just want to continue to build on that."

Warren believes that his friendship with quarterback Dorian Thompson-Robinson has a chance to be on full display today, with both players having an opportunity to shine.

"While he won't be playing with Watson or Winston on Saturday, Thrash and Thompson-Robinson are good friends off the field. Thompson-Robinson said he thinks their friendship helps their connection on the field despite not having many reps together in practice. This weekend's game should provide several opportunities for the friends to make something happen on the field."

While there are quite a few players to watch today, Thrash will be one of the top names. He has a real chance to make an impact on the team as a rookie and another strong showing will increase his chances of getting on the field and doing just that.

EVAN MASSEY

Stand for Studying Soil Friction

• Published: 17 August 2024
• Volume 45 , pages 101–106, ( 2024 )

Cite this article

• G. V. Makarevich 1 ,
• I. A. Salnikova 1 ,
• V. V. Saskovets 1 &
• E. I. Pavalanski 1  

To study the wear and friction force of a solid surface during friction with the ground, a carousel-type laboratory stand was created. The main task was to measure friction forces under various parameters of model soil and speeds of relative movement close to real ones during field agricultural work. Traditional electronic dynamometers are designed for static or slowly varying loads. Laboratory stands with such sensors have a linear design, a limited friction path (up to 2 m), and very low relative movement speeds (up to 0.15 m/s). The short friction path complicates the running-in process at the beginning of the experiment. Integral friction forces depend on speed. The adhesion component depends entirely on the presence of soil water at the interface and, thus, on the time required for water to move to the friction surface. With a carousel design of the stand, the friction path is infinite, and the speed can be increased by an order of magnitude (up to 1.5 m/s). Since studies usually compare the influence of different materials or soil compositions on the results of experiments, the systematic error due to the difference in the friction path from a straight line is insignificant. To measure rapidly changing loads, a force measuring station was developed based on a flat spring and a small displacement mechatronic sensor. The advantage of mechatronic linear displacement sensors is high sensitivity, ease of switching on, and high reliability. Disadvantages include dependence on temperature conditions and consequently the need for calibration in each experiment.

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Makarevich, G.V., Salnikova, I.A., Saskovets, V.V. et al. Stand for Studying Soil Friction. J. Frict. Wear 45 , 101–106 (2024). https://doi.org/10.3103/S1068366624700156

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