# 2D wave equation on a wedge

7-4

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:

Problem a.

a) u(r, 0, t) = 0, u(r, /3, t) = 0, ur(a, , t) = 0

problem b.

b) u(r, 0, t) = 0, u(r, /3, t) = 0, u(a, , t) = 0

https://brainmass.com/math/fourier-analysis/wave-equation-wedge-625952

#### Solution Preview

The wave equation on circular domain is given (in polar coordinates):

(1.1)

With the boundary conditions

(1.2)

And

(1.3)

We want to find the modes (frequencies) of the oscillation.

If we set:

(1.4)

The boundary conditions become

(1.5)

(1.6)

(1.7)

Plugging (1.4) into (1.1) we get:

(1.8)

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:

(1.9)

The negative sign is right now meaningless, since is an arbitrary constant, but it will help us later to recognize the radial equation.

Furthermore:

(1.10)

Now ...

#### Solution Summary

The solution shows how to get the modes of the waves on a circular wedge with different radial boundary conditions.