Consider the displacement of ,u(r,,t) , a "pie-shaped" membrane of radius a and angle /3 that satisfies:
utt = c22u
Assume that >0. Determine the natural frequencies of oscillation if the boundary conditions are:
a) u(r, 0, t) = 0, u(r, /3, t) = 0, ur(a, , t) = 0
b) u(r, 0, t) = 0, u(r, /3, t) = 0, u(a, , t) = 0
The wave equation on circular domain is given (in polar coordinates):
With the boundary conditions
We want to find the modes (frequencies) of the oscillation.
If we set:
The boundary conditions become
Plugging (1.4) into (1.1) we get:
The left hand side depends on t while the right hand side is a function of the spatial variables , hence both sides must be equal the same constant:
The negative sign is right now meaningless, since is an arbitrary constant, but it will help us later to recognize the radial equation.
The solution shows how to get the modes of the waves on a circular wedge with different radial boundary conditions.