5.8 Some Applications of Bessel Functions 331
Thea’s andb’s are determined from Eqs. (4) and (5) by using the orthogonality
relation ∫
a
0
J 0 (λnr)J 0 (λmr)rdr= 0 , n=m.B. Spherical Waves
In spherical(ρ,θ,φ)coordinates (see Section 5.9), the Laplacian operator∇^2
becomes
∇^2 u=ρ^12 ∂ρ∂(
ρ^2 ∂∂ρu)
+ρ (^2) sin^1 (φ)∂φ∂
(
sin(φ)∂φ∂u)
+^1
ρ^2 sin^2 (φ)∂^2 u
∂θ^2.Consider a wave problem in a sphere when the initial conditions depend only
on the radial coordinateρ:
1
ρ^2∂
∂ρ(
ρ^2 ∂u
∂ρ)
=^1
c^2∂^2 u
∂t^2, 0 <ρ<a, 0 <t,( 13 )u(a,t)= 0 , 0 <t,( 14 )
u(ρ, 0 )=f(ρ), 0 <ρ<a,( 15 )
∂u
∂t(ρ, 0 )=g(ρ), 0 <ρ<a.( 16 )Assumingu(ρ,t)=R(ρ)T(t), we separate variables and find
T′′+λ^2 c^2 T= 0 , (17)
(
ρ^2 R′)′
+λ^2 ρ^2 R= 0 , 0 <ρ<a, (18)
R(a)= 0 , (19)
∣∣
R( 0 )∣∣
bounded. (20)Again, the condition (20) has been added becauseρ=0 is a singular point.
Equation (18) may be put into the form
R′′+ρ^2 R′+λ^2 R= 0 ,and comparison with Eq. (1) shows thatα=− 1 /2,γ=1, andρ= 1 /2; thus
the general solution of Eq. (18) is
R(ρ)=ρ−^1 /^2[
AJ 1 / 2 (λρ)+BY 1 / 2 (λρ)]
.
We know that nearρ=0,
J 1 / 2 (λρ)∼const×ρ^1 /^2 ,
Y 1 / 2 (λρ)∼const×ρ−^1 /^2.