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Wave Equations : Comparison Between Circular and Square Vibrating Membranes

Please see the attached file for the fully formatted problems.

Consider a circular membrane of radius a and a square membrane
Assume the two membranes
(i) have the same area.
....
(ii) obey the same wave equation...
(iii) Have the same boundary conditions phi = 0 at their boundaries.
A) TABULATE (i) the 3 lowest frequencies for each of the two membranes
(ii) all the concomitant normal modes.
B) For each of the normal modes of the circular membrane DRAW a picture of the nodes, i.e. the locus of points where = 0. LABEL each of the normal mode pictures by
(a) its frequency and
(b) its formula for the amplitude.
C) DO the same for the other membrane. (Caution: Watch out for degeneracies!)
D) CAN ONE HEAR THE SHAPE of a DRUM?
Roots: is the jth root of the Bessel Functions
n=0 n=1 n=2 n=3 n=4
j = 1
2.405
3.832
5.136
6.380
7.586
j = 2
5.520
7.016
8.417
9.760
11.064
j = 3
8.654
10.173
11.620
13.017
14.373
j = 4
11.792
13.324
14.796
16.223
16.223

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Solution Summary

Bessel functions are employed to answer questions about a wave equation. The solution is detailed and well-presented.

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