322 Chapter 5 Higher Dimensions and Other Coordinates
v(r, 0 )=f(r), 0 <r<a, (3)
∂v
∂t(r, 0 )=g(r), 0 <r<a. (4)We start immediately with separation of variables, assuming v(r,t)=
φ(r)T(t). The differential equation (1) becomes
1
r(rφ′)′T=^1
c^2 φT′′,
and the variables may be separated by dividing byφT.Thenwefind
(rφ′(r))′
rφ(r)=T
′′(t)
c^2 T(t).
The two sides must both be equal to a constant (say,−λ^2 ), yielding two linked,
ordinary differential equations
T′′+λ^2 c^2 T= 0 , 0 <t, (5)
(rφ′)′+λ^2 rφ= 0 , 0 <r<a. (6)The boundary condition Eq. (2) is satisfied if
φ(a)= 0. (7)Of course, becauser=0 is a singular point of the differential equation (6), we
add the requirement
∣∣
φ(r)
∣∣
bounded atr= 0 , (8)which is equivalent to requiring that |v(r,t)|be bounded atr=0. We
recognize that Eq. (6) is Bessel’s equation, of which the function φ(r)=
J 0 (λr)is the solution bounded atr=0. In order to satisfy the boundary con-
ditionEq.(7),wemusthave
J 0 (λa)= 0 ,which implies that
λn=αan, n= 1 , 2 ,..., (9)whereαnare the zeros of the functionJ 0. Thus the eigenfunctions and eigen-
values of Eqs. (6)–(8) are
φn(r)=J 0 (λnr), λ^2 n=(α
n
a