Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 78

2.6.2 Spherical Domain

For a spherical domain, the boundary conditions are not separable for x , y and z in
a Cartesian coordinate system, like in the case of a circular domain in Section 2.5.2.
A spherical coordinate transformation of
⎧
⎨

x = r sin θ cos ϕ , 0 ≤ ϕ ≤ 2 π ,

y = r sin θ sin ϕ , 0 ≤ θ ≤ π ,

⎩

z = r cos θ , 0 < r ≤ a 0

is required before applying the method of separation of variables. In the spherical
coordinate system, PDS (2.52) reads
⎧
⎨

a 2

u tt = Δ u ( r , θ , ϕ , t )+ F ( r , θ , ϕ , t )

0 < θ < π , 0 < r < a 0 , 0 < t ,

| u ( 0 , θ , ϕ , t ) | < ∞ , (2.54)

⎩

L ( u , u r ) | r = a 0 = 0 , u ( r , θ , ϕ + 2 π , t )= u ( r , θ , ϕ , t ) ,

u ( r , θ , ϕ , 0 )= Φ ( r , θ , ϕ ) , u t ( r , θ , ϕ , 0 )= Ψ ( r , θ , ϕ ) ,

where the Laplacian is

r 2 sin

2
∂ϕ r 2 ∂

1 ∂

∂

1
r 2 sin

∂
∂θ

∂
∂θ

1
r 2 sin 2

∂

Δ =

+ +

2 .

θ

∂ r r θ

θ

It can be shown by following a similar approach to that in Section 2.5.2 that the
solution structure theorem is also valid in a spherical coordinate system. We thus
focus on seeking u

= W Ψ ( r , θ , ϕ , t ) , the solution for the case Φ = F = 0.

Separation of Variables

Assume a solution of type u . Substituting it into the wave equation

in PDS (2.54) and denoting the separation constant

= v ( r , θ , ϕ ) T ( t )

k 2 , we obtain −

k 2 v Δ v + = 0 , v ( r , θ , ϕ + 2 π )= v ( r , θ , ϕ ) ,

(2.55)

| v ( 0 , θ , ϕ ) | < ∞ , 0 < r < a 0 , 0 < θ < π ,

T (

2 T

t )+( ka ) ( t )= 0 .