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Showing wave equation?

  1. Oct 10, 2004 #1
    Can someone show me that [tex]f(x, t) = A\cos(K(x-vt) + \phi)[/tex] is in fact a solution of the wave equation?
    I kind of know how to show it by using calculus, but is there other way to show it?

    Thank you very much!
     
  2. jcsd
  3. Oct 10, 2004 #2

    Fredrik

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    Any function f that can be expressed as

    [tex]f(x,t)=g(x-vt)[/tex]

    satisfies the wave equation. You don't even have to know what g is to show it.
     
  4. Oct 10, 2004 #3
    Could you please show me a little more? I don't really get it why that will satisfie the wave equation...
    Thank you very much!
     
  5. Oct 10, 2004 #4

    Fredrik

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    We're supposed to give hints here, not complete answers, but if you compute the second-order partial derivative

    [tex]\frac{\partial^2 f(x,t)}{\partial t^2}[/tex]

    using the formula

    [tex]f(x,t)=g(x-vt)[/tex]

    you're almost there. Does the result look anything like any other second-order partial derivative that appears in the wave equation?
     
  6. Oct 11, 2004 #5
    Thanks, I'll give it a try.
    But I'm just wondering, is there any other way to show it beside using calculus?
    Thanks again!
     
  7. Oct 11, 2004 #6

    Fredrik

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    I don't think so. The wave equation is a partial differential equation, so any explanation would have to involve derivatives in some way.

    I'm pretty sure that there's no easier way to understand the wave equation than the way I suggested. You should note that the graphs of the functions [tex]h_t[/tex], defined by

    [tex]h_t(x)=g(x-vt)[/tex]

    can be thought of as the individual frames of a "movie" that shows the graph of g moving with velocity v.
     
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