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Homework Help: Wave equation, D'Alembert's Solution

  1. Jun 19, 2006 #1
    I am having trouble understanding the solution to the wave equation:

    735128d3de55366340a2e5fb647c470e.png




    this is thought of as the final solution to the PDE:



    img250.gif




    but I see that:


    img246.gif

    is a solution to the function. But what I dont get is why D'Alembert's Solution isn't in terms of sines and cosines like that solution right above.

    Is it because D'Albemerts is a gereneral solution, and the other is a specific solution? If so, still how come the general solution to the problem isn't expressed as a wave, and instead of some arbitrary function g(x-ct)?
     
  2. jcsd
  3. Jun 20, 2006 #2

    HallsofIvy

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    Science Advisor

    The general solution is a wave. But a wave is not necessarily periodic like sine or cosine. Imagine a taut rope that is distorted initially in a shape that consists of a bulge, say a triangle between x= -1 and x= 1, given by y= x+1 for x<0, y= 1-x for x>=0 and then released from rest at t=0. "g" splits into two parts which move right and left: that's what (1/2)g(x-ct) and (1/2)g(x+ct) are. The h integral allows non-zero speed at t= 0 also.
     
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