
9.3. Helmholtz Equation ∆
2
w + λw = −Φ(x) 811
◮ Nonhomogeneous Helmholtz equation with homogeneous boun dary conditions.
Three cases are possible.
1
◦
. If the equation parameter λ is not equal to any one of the eigenvalues, then there exists
the series solution
w =
∞
X
n=1
A
n
λ
n
− λ
w
n
, where A
n
=
1
kw
n
k
2
Z
S
Φw
n
dS, kw
n
k
2
=
Z
S
w
2
n
dS.
2
◦
. If λ is equal to some eigenvalue, λ = λ
m
, then the solution of the nonhomogeneous
problem exists only if the function Φ is orthogonal to w
m
, i.e.,
Z
S
Φw
m
dS = 0.
In this case the system is expressed as
w =
m−1
X
n=1
A
n
λ
n
− λ
m
w
n
+
∞
X
n=m+1
A
n
λ
n
− λ
m
w
n
+ Cw
m
, A
n
=
1
kw
n
k
2
Z
S
Φw
n
dS,
where kw
n
k
2
=
Z
S
w
2
n
dS, and C is an arbitrary constant.
3
◦
. If λ = λ
m
and
Z
S
Φw
m