How to Solve Vibrations of a Rectangular Membrane with a Hole?

[VktGS]

Homework Statement

I'm trying to solve the vibrations of a rectangular membrane with a rectangular hole inside. Both the inner and outer edges are fixed. I know i have to use the wave equation, but how do i write the boundary conditions in orden to incluye the hole? Any ideas?

F.Y.I. I already know how to solve a rectangular membrane, the thing is that i don't know how to include the hole

Homework Equations

$$\frac{d^2u}{dt^2} = c^2 \nabla^2 u$$

The Attempt at a Solution

I know that some of the boundary conditions must be

u(0,y,t)=u(x,0,t)=u(a,y,t)=u(x,b,t)=0

With a the length of the membrane and b it's width. It should also contain a hole inside of length c and width d. I don't know what to do from here. My guess is that i have to put some kind of boundary conditions. Any help?

 

[tiny-tim]

Welcome to PF!

Hi VktGSWelcome to PF!

Why not put the origin at the centre?

 

[clamtrox]

If the inner edges of the hole are also fixed, then you should just use the Dirichlet boundary conditions (that is, u=0 at the boundary)

 

[VktGS]

Thanks for the centre idea, tiny-tim. Very useful.

And, F.Y.I. clamtox, i know what Dirichlet boundary conditions are, just didn't know how to express them in this problem.