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.

 

[paranormal]

Hello, it seems like you are on the right track with your boundary conditions for the rectangular membrane with a hole. To include the hole in your solution, you will need to add additional boundary conditions that describe the behavior of the membrane at the edges of the hole. These conditions will depend on the shape and size of the hole, as well as the material properties of the membrane.

One approach you can take is to divide the membrane into smaller rectangular sections, including the hole, and treat each section separately. This will allow you to write specific boundary conditions for each section, taking into account the effects of the hole on the surrounding membrane.

Another approach is to use a mathematical technique called separation of variables, which allows you to solve the wave equation for different regions of the membrane separately. This can be a more complex method, but it may provide a more accurate solution for the membrane with a hole.

I would also recommend consulting with a textbook or a colleague who has experience with solving similar problems to gain a better understanding of the techniques and methods used for this type of problem. Good luck with your solution!