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

• johnnymopo

#### johnnymopo

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.

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

Have you tried separation of variables: $\Psi(x,y,z) = X(x)Y(y)Z(z)$?

Is that not a Helmholtz equation, and is a Helmholtz not a time-independant wave?

Is that not a Helmholtz equation, and is a Helmholtz not a time-independant wave?

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