Eigenvalues and eigenfunctions

[ted12]

Hello!

 

I don't know how to solve this problem: find eigenvalues and eigenfunctions of quadratic membrane which is fixed in three edges. Fourth edge is flexible bended in the middle (at this edge membrane is in the shape of triangular).

 

So I have u(x,y,t). Boundary conditions are: u(x,0,t)=u(x,a,t)=u(0,y,t)=0 and u(a,y,t)=k(t)*y for y < a/2 and u(a,y,t) = k(t)(a-y) for a/2 < y < a, where k(t)<<1 (because of that I don't have to account change of length of edge). Surface tension of membrane is γ, mass of edge is m.

 

If anybody can help me?

Edited August 29, 2012 by ted12

[ted12]

Till now I put first three boundary conditons in solved wave equation and got u(x,y,t) = sin(k1*x)*sin(k2*y)*exp(-i*omega*t), where k2 = m*pi/a.

I don't know how to write 4. boundary condition (triangular) to get equation for k1.