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I Deriving the progressive mechanical wave equation

  1. Jan 12, 2017 #1
    Is it correct to state that a progressive wave, originates when a simple harmonic motion is imparted continuously to adjacent particles from one direction to another moving with a velocity v. Using this idea, substituting (t - x/v) instead of t is the simple harmonic motion function y=Asin(ωt), we obtain the final answer as y= Asin(ωt - kx).

    In another method, drawing a sine wave and finding out the function for a sine wave propagating towards right with a velocity v , substituting (x - vt) instead of x, the propagating wave function is obtained as y= Asin(kx- ωt).

    Why is there a phase difference here?
     
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  3. Jan 12, 2017 #2

    Simon Bridge

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    Notice: the first one gets you ##y(x,t)=-A\sin (kx-\omega t)##
    If you want the same phase, then the first sub should be ##x/v - t \to t##, or use a cosine wave.

    The difference is because of how the wave propagates.
    Think of the physical situation being described in each case: In the first you grab a point, say at x=0, and wave it first up and then down (and suppress the -x propagating solution); in the second you have a wave already and you shove it to one side - so that x=0 goes down first and then up.
    The maths is just describing that correctly.
     
  4. Jan 12, 2017 #3
    Thank you Simon Bridge, the answer was really helpful.
     
  5. Jan 19, 2017 #4
    For the wave travelling towards left, the equations is Asin(kx + ωt). How does the same mathematical equation explain the possibility of two initial conditions. In the case of the wave travelling towards right, Asin(kx - ωt) and Asin(ωt - kx) gives two initial conditions Asin(kx) and -Asin(kx) on substituting t=0. Explaining the possibility of two different initial movements. In the case of waves travelling towards left, this difference doesn't come up. How can we explain this situation physically.
     
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