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Displacement of Transverse Waves HELP

  1. Oct 1, 2008 #1
    Displacement of Transverse Waves HELP!!

    1. The problem statement, all variables and given/known data
    At time t=0, the displacement of a transverse wave pulse is described by y=2/(x^(4) +1), with both x and y in cm. Write an expression for the wavefunction as a function of position x and time t if it is propagating in the positive x direction at 3.0 cm/s


    2. Relevant equations
    I'm not sure if this has to do with partial derivatives...and I don't quite understand partial derivatives.


    3. The attempt at a solution
    I know that v= 3.0 cm/s...
     
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  3. Oct 1, 2008 #2

    gabbagabbahey

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    Re: Displacement of Transverse Waves HELP!!

    Well your given a stationary wave pulse [itex]f(x,0)=y(x)=\frac{2}{x^4+1}[/itex] and you want to find f(x,t) that satisfies the wave equation [itex]f_{xx}(x,t)-\frac{1}{v^2}f_{tt}(x,t)=0[/itex]. You probably know that any function of the form [itex]f(x \pm vt)[/itex] will satisfy the wave equation, and that if you want just the solution that travels forward at speed v, you choose the negative sign (i.e. [itex]f(x - vt)[/itex]). You are given y(x), so what is y(x-vt)?
     
  4. Oct 1, 2008 #3
    Re: Displacement of Transverse Waves HELP!!

    what do you mean by fxx? is that the second derivative of x?
     
  5. Oct 2, 2008 #4

    gabbagabbahey

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    Re: Displacement of Transverse Waves HELP!!

    Yes,

    [tex]f_{xx}(x,t)=\frac{\partial ^2 f(x,t)}{\partial x^2}[/tex]
     
  6. Oct 2, 2008 #5
    Re: Displacement of Transverse Waves HELP!!

    would y(x-vt) be [2/(x^(4) +1)] - 3.0 cm?
     
  7. Oct 2, 2008 #6

    gabbagabbahey

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    Re: Displacement of Transverse Waves HELP!!

    No, just substitute x-vt everywhere you see an x.
     
  8. Oct 2, 2008 #7
    Re: Displacement of Transverse Waves HELP!!

    thank you so much for all of your help!!! i really, really appreciate it!!!

    I'm sorry, but I have another question:

    why do you set f{xx}(x,t)-1/v^(2) * f{tt}(x,t) = 0?
     
  9. Oct 2, 2008 #8

    gabbagabbahey

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    Re: Displacement of Transverse Waves HELP!!

    You mean [itex]f_{xx}(x,t)-\frac{1}{v^2}f_{tt}(x,t)=0[/itex]?

    That's the one-dimensional wave equation; have you not seen it before?

    Would it help if I wrote it like this:
    [tex]\frac{\partial ^2 f(x,t)}{\partial x^2}-\frac{1}{v^2} \frac{\partial ^2 f(x,t)}{\partial t^2}=0[/tex]
     
  10. Oct 2, 2008 #9
    Re: Displacement of Transverse Waves HELP!!

    i've seen the two second derivatives equal to each other, but i never thought of manipulating the equation to move the variables to one side.

    thank you again for all of your help. i really appreciate it!!!

    so just to double-check...the answer would be y(x,t) =2/[(x-3t)^(4)+1] ?
     
  11. Oct 2, 2008 #10

    gabbagabbahey

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    Re: Displacement of Transverse Waves HELP!!

    Yes, you can check your answer yourself too by seeing what happens at t=0, you should get y(x) back. Also, you can take the partial second derivatives and verify that [itex]f_{xx}(x,t)-\frac{1}{v^2}f_{tt}(x,t)=0[/itex]. You also know that the pulse should be traveling at 3cm/s to the right; which means that since the pulse is centered at x=0 for t=0, you should have a pulse that is centered at x=3cm for t=1s. These are checks that you should do to convince yourself that you have the correct answer.
     
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