Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

118 Solution to the Helmholtz and wave equations
∂ 2 [
G ∂G
+ cot φ
∂φ2 ∂φ + m(m + 1) −
n2
sin 2 φ
The solution to these two equations is
or
]
G = 0 (assoc. Legendre equation) (8.164)
F(R) = c 3 h (1)
m (kR) + c 4 h (2)
m (kR) (spherical Hankel functions) (8.165)
and
F(R) = c 3 j m (kR) + c 4 y m (kR) (spherical Bessel functions) (8.166)
G(φ) = c 5 P n m + c 6 Q n m (assoc. Legendre functions) (8.167)
in which again c 3 , c 4 , c 5 , c 6 are constants that may depend on the parameters m, n It follows
that in spherical coordinates, particular solutions to the Helmholtz equation exist that are
the form
(
)
(R,φ,θ) = (c 1 cos nθ + c 2 sin nθ) c 3 h (1)
m (kR) + c 4 h (2)
m (kR) (c 5 Pm n + c 6 Qm n) (8.168)
In most cases, c 6 will be zero, because Q n m is singular at φ = 0,π (i.e., at the north and
south poles), so for problems that include the full sphere, it must be excluded on physical
grounds. For infinite media, c 3 may also be zero, to satisfy the radiation condition at
R =∞, whereas media that include the origin may need to be formulated in terms of the
conventional spherical Bessel functions j n , to avoid the singularity of the spherical Hankel
functions (or y n )atR = 0.
8.9 Vector Helmholtz equation in spherical coordinates
Consider the vector Helmholtz equation
∇ 2 Ψ + k 2 Ψ = 0 (8.169)
or more precisely
∇ · ∇Ψ + k 2 Ψ = 0 (8.170)
which involves the vector Ψ = ψ r ˆr + ψ φ ŝ + ψ θ ˆt. We shall assume Ψ to be solenoidal, which
means that it satisfies the gaugecondition ∇ · Ψ = 0. Hence, only two of the three components
of Ψ are independent, so the solution can only contain two independent functions.
However,
∇ · ∇Ψ =∇(∇ · Ψ) −∇×∇×Ψ
=−∇×∇×Ψ (8.171)
Hence
∇×∇×Ψ = k 2 Ψ (8.172)