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Standing Wave on String Question

  1. May 16, 2015 #1
    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. The problem statement, all variables and given/known data

    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.

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

    3. 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.
     
    Last edited: May 16, 2015
  2. jcsd
  3. May 16, 2015 #2

    CWatters

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    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?
     
  4. May 16, 2015 #3
    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.
     
  5. May 16, 2015 #4

    haruspex

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    It says resonant frequency of 'the wave', i.e. the three loop configuration, not the resonant frequency of the string.
     
  6. May 16, 2015 #5
    Oh, right. Ok so its just 37.5Hz then? Thanks for your help.
     
  7. May 16, 2015 #6

    haruspex

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    Yes. You're welcome.
     
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