# Standing Wave on String Question

In summary, the conversation is a discussion between two individuals regarding a question from a past paper for an upcoming exam. The question involves two waves on a string and asks for the resonant frequency in Hz. The second part of the question asks for the values of amplitude, wave number, and angular frequency for the second wave, which may be a trick question. The third part asks about the sign in front of the angular frequency. The conversation also includes a discussion about terminology and assumptions made in solving the problem. Ultimately, the resonant frequency is determined to be 37.5 Hz for the three-loop standing wave configuration.
This is not coursework; I am preparing for an exam and this question is from a past paper. We have access to past papers but we are not given the answers to them.

1. Homework Statement

Two waves are generated on a string of length 2m, to produce a three-loop standing wave with an amplitude of 2cm. The wave speed is 50m/s.

A: What is the resonant frequency of the wave in Hz.
B: If the equation for one of the waves is of the form $y(x,t)=y_m \sin(kx+ \omega t)$ , what are the values of $y_m$ , $k$ and $\omega$ for the second wave?
C: What is the sign in front of $\omega$ for the second wave.

## Homework Equations

$L=\frac{n \lambda}{2} \\ k=\frac{2 \pi}{\lambda} \\ v= \lambda f = \frac{\omega}{k} \\$
Other related equations

## The Attempt at a Solution

The question does not specify whether the ends are open or closed (fixed) so I am assuming they are both open ends.

A:
I have no come across the terminology "Three-loop" before but after searching the web I think it means the same thing as being in the third harmonic, if so then this is what I have done.

$L=\frac{n \lambda}{2} \\ 2=\frac{3 \lambda}{2} \\ \lambda = (\frac{2}{3})(2) = \frac{4}{3}m \\ f = \frac{v}{\lambda} = \frac{50}{4/3}=37.5Hz$
And that is the frequency of the third harmonic so the first harmonic would be 37.5/3=12.5 Hz

EDIT: I just noticed I didn't need to use n=3 and could have just done it with n=1 from the beginning. Why is there any need to tell me its in the third-harmonic?

B:
Bit unsure of part B, I think it may be a bit of a trick question as throughout the course we have only dealt with standing waves where the two constituent waves have the same magnitude of amplitude wave-number and angular frequency, only have the phases differed.

So if its a trick question and they're both the same then I think this part is no problem. Oh and for part C, wouldn't it be the opposite, i.e. it would be negative.

Last edited:
The question does not specify whether the ends are open or closed (fixed) so I am assuming they are both open ends.

Is that likely for a string? How would it be tensioned?

I have no come across the terminology "Three-loop" before but after searching the web I think it means the same thing as being in the third harmonic, if so then this is what I have done.

You will have see drawings like this..
http://session.masteringphysics.com/problemAsset/1013932/10/1013932D.jpg
I would take it to show 1, 2 and 3 "loops".

=37.5Hz

I agree up to that point.

And that is the frequency of the third harmonic so the first harmonic would be 37.5/3=12.5 Hz

What does part A ask for?

CWatters said:
Is that likely for a string? How would it be tensioned?
You will have see drawings like this..
http://session.masteringphysics.com/problemAsset/1013932/10/1013932D.jpg
I would take it to show 1, 2 and 3 "loops".
I agree up to that point.
What does part A ask for?

Ah ok, yeah I can see why it would be both be fixed ends. I was used to doing problems with standing sound waves and the realted equations. What I really was getting at that I was assuming the situation to be $L=\frac{n \lambda}{2}$

Part A asks for the "resonant frequency", which I thought meant the "fundamental frequency" i.e. first harmonic, i.e n=1, the lowest frequency that the standing wave can be generated at. Which (again this might be me used to doing standing sound wave problems) I always though that third harmonic is 3 times the frequency of the first, the fourth four times etc.

Ah ok, yeah I can see why it would be both be fixed ends. I was used to doing problems with standing sound waves and the realted equations. What I really was getting at that I was assuming the situation to be $L=\frac{n \lambda}{2}$

Part A asks for the "resonant frequency", which I thought meant the "fundamental frequency" i.e. first harmonic, i.e n=1, the lowest frequency that the standing wave can be generated at. Which (again this might be me used to doing standing sound wave problems) I always though that third harmonic is 3 times the frequency of the first, the fourth four times etc.
It says resonant frequency of 'the wave', i.e. the three loop configuration, not the resonant frequency of the string.

haruspex said:
It says resonant frequency of 'the wave', i.e. the three loop configuration, not the resonant frequency of the string.

Oh, right. Ok so its just 37.5Hz then? Thanks for your help.

Oh, right. Ok so its just 37.5Hz then? Thanks for your help.
Yes. You're welcome.

## What is a standing wave on a string?

A standing wave on a string is a type of wave that forms when two identical waves with equal amplitude and wavelength travel in opposite directions and interfere with each other. This creates a pattern of nodes and antinodes along the string, where the string does not move and where it moves with the greatest amplitude, respectively.

## What factors affect the formation of standing waves on a string?

The main factors that affect the formation of standing waves on a string are the tension of the string, the length of the string, and the frequency of the waves. Higher tension and shorter string length result in higher frequencies and shorter wavelengths, which in turn create more nodes and antinodes on the string.

## How is the wavelength of a standing wave on a string determined?

The wavelength of a standing wave on a string is determined by the distance between two consecutive nodes or antinodes. It can also be calculated using the formula λ = 2L/n, where L is the length of the string and n is the number of nodes or antinodes.

## What is the relationship between frequency and wavelength in a standing wave on a string?

In a standing wave on a string, the frequency and wavelength are inversely proportional. This means that as the frequency increases, the wavelength decreases, and vice versa. This relationship can be seen in the formula f = nv/2L, where f is the frequency, n is the number of nodes or antinodes, v is the speed of the wave, and L is the length of the string.

## How does the amplitude of a standing wave on a string change?

The amplitude of a standing wave on a string remains constant along the string. However, the amplitude at the antinodes is twice as large as the amplitude at the nodes. This is because the two waves that form the standing wave are in phase at the antinodes and out of phase at the nodes, resulting in constructive and destructive interference, respectively.

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