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
j = 2
j = 3
j = 4
Bessel functions are employed to answer questions about a wave equation. The solution is detailed and well-presented.