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2nd order diff eq, 3 dim. and wierd boundry condition

  1. Mar 20, 2010 #1
    I need help. For the following problem, can someone suggest how I should start on this question. I only have one quarter of diff eq classes plus a few classes in fourier analysis. I'm out of my league.

    Consider a box with length, width and height given by L. The box encompasses the region
    described by 0<=x<=L, 0<=y<=L, and 0<=z<=L. A scalar field inside the box satisfies the
    differential equation:

    ∇^2 ψ = −aψ

    Here a is a positive constant that is equal to 30/L2.
    The field is 0 on the surfaces y=0, y=L, x=0, x=L and finally the surface z=L.
    On the surface z=0, the field has the functional form:

    ψ(x, y) = ( 1-|x-2/L|*2/L)(1- |x-2/L|*2/L)

    Solve for ψ as a function of x, y, and z inside the box. Your final solution must be
    an analytic expression, though it can involve an infinite sum.


    What I don't understand is that ψ is time-independant wave, correct? and if so, how do I solve this when the boundry on one face is clearly not a wave. Please point me in the right direction to figure out how to do this. thanks.
     
  2. jcsd
  3. Mar 20, 2010 #2

    LCKurtz

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    Re: help? pls. point me in the right direction

    Well, this isn't a wave equation since it doesn't involve t.

    Have you tried separation of variables: [itex]\Psi(x,y,z) = X(x)Y(y)Z(z)[/itex]?
     
  4. Mar 21, 2010 #3
    Is that not a Helmholtz equation, and is a Helmholtz not a time-independant wave?
     
  5. Mar 21, 2010 #4

    LCKurtz

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    Yes it is. And apparently it is referred to as a wave equation.
     
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