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How to prove a wave is travelling ?

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  1. Jun 4, 2016 #1
    1. The problem statement, all variables and given/known data
    This is not a homework of any form, I am simply interested in proving a wave equation of the form f(x,t)=A•sin(k•x-ω•t) is a travelling wave, preferably an algebraic proof. Thanks heaps!

    2. Relevant equations
    f(x,t)=A•sin(k•x-ω•t)

    3. The attempt at a solution
    I was able to proof this by using graphs, but I need to know the algebra behind it.
     
  2. jcsd
  3. Jun 5, 2016 #2

    andrewkirk

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    Think of the wave as a 'waveform' (which is a fixed shape) that is travelling left or right.

    Write an equation for the location of a particular point in the waveform, as a function of time.
    A point in the waveform is uniquely determined by the argument to the sine function. So to fix the point in the waveform, set the phase equal to any constant C. So you have ##C=kx-\omega t##. Now solve for location ##x## as a function of ##t##. That tells you that the location of that point in the waveform is a linear function of time. It's easiest to visualise this if the point is a crest or a trough, where C is ##(2m+\frac12)\pi## or ##(2m-\frac12)\pi## respectively (##m## being an integer). But it's just as true for any point on the waveform.
     
  4. Jun 5, 2016 #3
    Here's something to get you started. Look at the wave at a single point (for simplicity, look at a zero of the function). What can you deduce about the argument of the sine function? What does this tell you about the motion of the wave?

    EDIT. Just got beat to it :frown:
     
  5. Jun 5, 2016 #4
    To be a more accurate what you have here is a function ##f(x,t)## that satisfies the one dimensional wave equation https://en.wikipedia.org/wiki/Wave_equation

    Any function that satisfies the wave equation is a travelling wave. (or in some cases a standing wave).

    PS. You ll need to know a bit of differential calculus to understand the Wikipedia article.
     
    Last edited: Jun 5, 2016
  6. Jun 5, 2016 #5

    nrqed

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    I guess we are talking about waves traveling without changing shape (otherwise we have to discuss dispersion, group velocity vs phase velocity, etc).

    In general, if you have a function of the combination ##kx- \omega t## or of ##kx+ \omega t##, then you have a wave traveling to the right (or to the left in the second case) at a speed equal to ##v=\omega/k##. In other words, you can tell by using the following trick: if you set ##kx=\omega t## in the function and magically all dependence on x and t disappears, you have a wave traveling to the right without changing shape. If you set ##kx=-\omega t## in the function and magically all dependence on x and t disappears, you have a wave traveling to the left without changing shape.
     
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