
9.3. Helmholtz Equation ∆
2
w + λw = −Φ(x) 821
Two forms of representation of the Green’s function:
G(x, y, ξ, η) =
2
a
∞
X
n=1
sin(p
n
x) sin(p
n
ξ)
β
n
sinh(β
n
b)
H
n
(y, η) =
2
b
∞
X
m=1
sin(q
m
y) sin(q
m
η)
µ
m
sinh(µ
m
a)
Q
m
(x, ξ),
where
p
n
=
πn
a
, β
n
=
p
p
2
n
− λ, q
m
=
πm
b
, µ
m
=
p
q
2
m
− λ,
H
n
(y, η) =
(
sinh(β
n
η) sinh[β
n
(b −y)] for b ≥ y > η ≥ 0,
sinh(β
n
y) sinh[β
n
(b − η)] for b ≥ η > y ≥ 0,
Q
m
(x, ξ) =
(
sinh(µ
m
ξ) sinh[µ
m
(a − x)] for a ≥ x > ξ ≥ 0,
sinh(µ
m
x) sinh[µ
m
(a − ξ)] for a ≥ ξ > x ≥ 0.
Alternatively, the Green’s function can be written as the double series
G(x, y, ξ, η)=
4
ab
∞
X
n=1
∞
X
m=1
sin(p
n
x) sin(q
m
y) sin(p
n
ξ) sin (q
m
η)
p
2
n
+q
2
m
−λ
, p
n
=
πn
a
, q
m
=
πm
b
.
◮ Domain: 0 ≤ x ≤ a, 0 ≤ y ≤ b. Second bound ary ...