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2D wave equation on a wedge

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7-4

Consider the displacement of ,u(r,,t) , a "pie-shaped" membrane of radius a and angle /3 that satisfies:

utt = c22u

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

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

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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.

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