
10.3. Helmholtz Equation ∆
3
w + λw = −Φ(x) 909
where the µ
n
and ν
m
are positive roots of the transcendental equations
tan(µa) =
(k
1
+ k
2
)µ
µ
2
− k
1
k
2
, tan(νb) =
(k
3
+ k
4
)ν
ν
2
− k
3
k
4
.
◮ Domain: 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z < ∞. Mixed boundary value problems.
1
◦
. A semiinfinite cylindrical domain of a rectangular cross-section is considered. Bound-
ary conditions are prescribed:
w = f
1
(y, z) at x = 0, w = f
2
(y, z) at x = a,
w = f
3
(x, z) at y = 0, w = f
4
(x, z) at y = b,
∂
z
w = f
5
(x, y) at z = 0.
Solution:
w(x, y, z) =
Z
b
0
Z
∞
0
f
1
(η, ζ)
∂
∂ξ
G(x, y, z, ξ, η, ζ)
ξ=0
dζ dη
−
Z
b
0
Z
∞
0
f
2
(η, ζ)
∂
∂ξ
G(x, y, z, ξ, η, ζ)
ξ=a
dζ dη
+
Z
a
0
Z
∞
0
f
3
(ξ, ζ)
∂
∂η
G(x, y, z, ξ, η, ζ)
η=0
dζ dξ
−
Z
a
0
Z
∞
0
f
4
(ξ,