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Reduction of a wave aquation

  1. Oct 30, 2009 #1
    I'm having trouble with the following related questions. Any help is appreciated.

    (a) Show that a change of variables of the form [tex]\xi = px + qt[/tex], [tex]\eta = rx + st[/tex] can be used to reduce the one dimensional wave equation [tex]\frac1{c^2} u_{tt} = u_{xx}[/tex] to an equation of the form [tex]\frac{\partial^2 U}{\partial\xi \partial\eta} = 0[/tex]. Hence show that the general solution of the wave equation is of the form [tex]u(x,t) = F(x + ct) + G(x - ct)[/tex], where F,G are arbitrary twice differentiable functions.

    (b) Show that the solution of the wave equation for the infinite domain [tex]-\infty < x < \infty[/tex] subject to [tex]u(x,0) = f(x)[/tex] and [tex]u_t(x,0) = g(x)[/tex] can be written as [tex]u(x,t) = \frac12 [f(x + ct) + f(x - ct)] + \frac1{2c} \int_{x - ct}^{x + ct} g(y) dy.[/tex]

    This is called the D'Alembert solution.
     
  2. jcsd
  3. Oct 30, 2009 #2

    HallsofIvy

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    What have you done on this yourself? It is a pretty direct, though tedious, exercise in using the chain rule to change variables in a differential equation.

    [tex]\frac{\partial U}{\partial x}= \frac{\partial \xi}{\partial x}\frac{\partial U}{\partial \xi}+ \frac{\partial \eta}{\partial x}\frac{\partial U}{\partial \eta}[/tex]
     
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