Standing Wave on String Question

[FaraDazed]

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 [itex]y(x,t)=y_m \sin(kx+ \omega t) [/itex] , what are the values of [itex]y_m[/itex] , [itex]k[/itex] and [itex]\omega[/itex] for the second wave?
C: What is the sign in front of [itex]\omega[/itex] for the second wave.

Homework Equations

[itex]
L=\frac{n \lambda}{2} \\
k=\frac{2 \pi}{\lambda} \\
v= \lambda f = \frac{\omega}{k} \\
[/itex]
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.

[itex]
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
[/itex]
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.

 

[CWatters]

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?

 

[FaraDazed]

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 [itex]L=\frac{n \lambda}{2}[/itex]

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.

 

[haruspex]

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 [itex]L=\frac{n \lambda}{2}[/itex]

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.

 

[FaraDazed]

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.

 

[haruspex]

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

Yes. You're welcome.