# Wave Equations : Comparison Between Circular and Square Vibrating Membranes

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

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