5.7 Vibrations of a Circular Membrane 321
EXERCISES
- Use Eq. (18) to find an expression for the functionv( 0 ,t)/T 0 .Evaluatethe
function for
kt
a^2
= 0. 1 , 0. 2 , 0. 3.
(The first two terms of the series are sufficient.)- Write out the first three terms of the series in Eq. (18).
- Solve the heat problem consisting of Eqs. (1)–(3) iff(r)is
f(r)=
T 0 , 0 <r<a
2,
0 ,
a
2 <r<a.- Letφ(r)=J 0 (λr)so thatφ(r)satisfies Bessel’s equation of order 0. Multiply
through the differential equation byrφ′and conclude that
d
dr
[
(rφ′)^2]
+λ^2 r^2 d
dr[
φ^2]
= 0.
- Assuming thatλis chosen so thatφ(a)=0, integrate the equation in Exer-
cise 4 over the interval 0<r<ato find
∫a
0φ^2 (r)rdr=1
2 λ^2(
aφ′(a)) 2
- Use Exercise 5 to validate Eq. (16).
5.7 Vibrations of a Circular Membrane
We shall now attempt to solve the problem of describing the displacement of a
circular membrane that is fixed at its edge.
Symmetric Vibrations
To begin with, we treat the simple case in which the initial conditions are in-
dependent ofθ. Thus the displacementv(r,t)satisfies the problem
1
r∂
∂r(
r∂v∂r)
=c^12 ∂(^2) v
∂t^2 ,^0 <r<a,^0 <t, (1)
v(a,t)= 0 , 0 <t, (2)