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

102 Solution to the Helmholtz and wave equations
and substituting into the Helmholtz equation, we find that it is satisfied if
k 2 x + k2 y + k2 z = k2 P
(8.47)
The general solution to the scalar Helmholtz equation is then
(x, y, z,ω) = ( a 1 e ikxx + a 2 e −ikxx)( a 3 e iky y + a 4 e −iky y)( a 5 e ikzz + a 6 e −ikzz) (8.48)
in which the a i are arbitrary constants. In combination with the implied factor e iωt , and for
real wavenumbers k x , k y , k z , this solution for represents compressional (P) waves that
propagate in the (locally homogeneous) medium with velocity α.
For two-dimensional problems in which the wave velocity α does not change in horizontal
planes (i.e., for a laterally homogeneous medium), and considering only waves that
propagate and/or decay in the positive x direction, the solution simplifies to
(x, z,ω) = ( a 1 e ikzz + a 2 e −ikzz) √
e −ikxx , k z = kP 2 − k2 x (8.49)
8.3 Vector Helmholtz equation in Cartesian coordinates
Consider the vector Helmholtz equation
∇ 2 Ψ + k 2 S Ψ = 0 (8.50)
with parameter k s = ω/β. More precisely,
∇ · ∇Ψ + k 2 S Ψ = 0 (8.51)
which involves the vector Ψ = ψ x î + ψ y ĵ + ψ z ˆk. 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
Hence
=−∇×∇×Ψ (8.52)
∇×∇×Ψ = k 2 S Ψ (8.53)
This equation admits solutions of the form
Ψ = Ψ 1 + Ψ 2
= ψ ˆk + 1 k 2 S
∇
( ) ∂ψ
+ 1 ( )
∇× χ ˆk
∂z k S
(8.54)
in which ψ, χ are solutions to the scalar Helmholtz equations
∇ 2 ψ + k 2 S ψ = 0 and ∇2 χ + k 2 S χ = 0 (8.55)
To prove that this is indeed the solution, we make use of some well-known vector operation
identities, namely ∇×∇f = 0, ∇ · ∇×f = 0, which are true for any scalar and vector