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Exact Solutions > Linear Partial Differential Equations >Second-Order Elliptic Partial Differential Equations > Helmholtz Equation

3.3. Helmholtz Equation ∆w + λw = –Φ(x)

Many problems related to steady-state oscillations (mechanical, acoustical, thermal, electromag-netic) lead to the two-dimensional Helmholtz equation. Forλ < 0, this equation describes masstransfer processes with volume chemical reactions of the first order.

The two-dimensional Helmholtz equation has the following form:

∂2w

∂x2 +∂2w

∂y2 + λw = −Φ(x,y) in the Cartesian coordinate system,

1r

∂

∂r

(r∂w

∂r

)+

1r2

∂2w

∂ϕ2 + λw = −Φ(r,ϕ) in the polar coordinate system.

3.3-1. Particular solutions of the homogeneous Helmholtz equation withΦ ≡ 0.

1. Particular solutions of the homogeneous Helmholtz equation in the Cartesian coordinate system:

w = (A cosµ1x + B sinµ1x)(C cosµ2y + D sinµ2y), λ = µ21 + µ2

2,

w = (A cosµ1x + B sinµ1x)(C coshµ2y + D sinhµ2y), λ = µ21 − µ2

2,

w = (A coshµ1x + B sinhµ1x)(C cosµ2y + D sinµ2y), λ = −µ21 + µ2

2,

w = (A coshµ1x + B sinhµ1x)(C coshµ2y + D sinhµ2y), λ = −µ21 − µ2

2,

whereA, B, C, andD are arbitrary constants.

2. Particular solutions of the homogeneous Helmholtz equation in the polar coordinate system:

w = [AJm(µr) + BYm(µr)](C cosmϕ + D sinmϕ), λ = µ2,

w = [AIm(µr) + BKm(µr)](C cosmϕ + D sinmϕ), λ = −µ2,

wherem = 1, 2, . . . ; A, B, C, D are arbitrary constants; theJm(µ) andYm(µ) are the Besselfunctions; and theIm(µ) andKm(µ) are the modified Bessel functions.

3.3-2. Domain: –∞ < x < ∞, –∞ < y < ∞.

1. Solution forλ = −s2 < 0:

w(x,y) =1

2π

∫ ∞

−∞

∫ ∞

−∞Φ(ξ,η)K0(s%) dξ dη, % =

√(x − ξ)2 + (y − η)2.

2. Solution forλ = k2 > 0:

w(x,y) = −i

4

∫ ∞

−∞

∫ ∞

−∞Φ(ξ,η)H (2)

0 (k%) dξ dη, % =√

(x − ξ)2 + (y − η)2,

whereH (2)0 (z) is the Hankel function of the second kind of order0. The radiation conditions

(Sommerfeld conditions) at infinity were used to obtain this solution.

3.3-3. Domain: –∞ < x < ∞, 0 ≤ y < ∞. First boundary value problem.

A half-plane is considered. A boundary condition is prescribed:

w = f (x) at y = 0.

1

2 HELMHOLTZ EQUATION

Solution:

w(x,y) =∫ ∞

−∞f (ξ)

[∂

∂ηG(x,y, ξ,η)

]

η=0dξ +

∫ ∞

0

∫ ∞

−∞Φ(ξ,η)G(x,y, ξ,η) dξ dη.

1. The Green’s function forλ = −s2 < 0:

G(x,y, ξ,η) =1

2π

[K0(s%1) − K0(s%2)

],

%1 =√

(x − ξ)2 + (y − η)2, %2 =√

(x − ξ)2 + (y + η)2.

2. The Green’s function forλ = k2 > 0:

G(x,y, ξ,η) = −i

4[H (2)

0 (k%1) − H (2)0 (k%2)

].

3.3-4. Domain: 0≤ x ≤ a, 0 ≤ y ≤ b. First boundary value problem.

A rectangle is considered. Boundary conditions are prescribed:

w = f1(y) at x = 0, w = f2(y) at x = a,

w = f3(x) at y = 0, w = f4(x) at y = b.

1. Eigenvalues of the homogeneous problem withΦ ≡ 0 (it is convenient to label them with adouble subscript):

λnm = π2(

n2

a2 +m2

b2

); n = 1, 2, . . . ; m = 1, 2, . . .

Eigenfunctions and the norm squared:

wnm = sin

(nπx

a

)sin

(mπy

b

), ‖wnm‖2 =

ab

4.

2. Solution forλ ≠ λnm:

w(x,y) =∫ a

0

∫ b

0Φ(ξ,η)G(x,y, ξ,η) dη dξ

+∫ b

0f1(η)

[∂

∂ξG(x,y, ξ,η)

]

ξ=0dη −

∫ b

0f2(η)

[∂

∂ξG(x,y, ξ,η)

]

ξ=a

dη

+∫ a

0f3(ξ)

[∂

∂ηG(x,y, ξ,η)

]

η=0dξ −

∫ a

0f4(ξ)

[∂

∂ηG(x,y, ξ,η)

]

η=b

dξ.

Two forms of representation of the Green’s function:

G(x,y, ξ,η) =2a

∞∑

n=1

sin(pnx) sin(pnξ)βn sinh(βnb)

Hn(y,η) =2b

∞∑

m=1

sin(qmy) sin(qmη)µm sinh(µma)

Qm(x, ξ),

where

pn =πn

a, βn =

√p2

n − λ, Hn(y,η) =

sinh(βnη) sinh[βn(b − y)] for b ≥ y > η ≥ 0,

sinh(βny) sinh[βn(b − η)] for b ≥ η > y ≥ 0,

qm =πm

b, µm =

√q2m − λ, Qm(x, ξ) =

sinh(µmξ) sinh[µm(a − x)] for a ≥ x > ξ ≥ 0,

sinh(µmx) sinh[µm(a − ξ)] for a ≥ ξ > x ≥ 0.

ReferencesBudak, B. M., Samarskii, A. A., and Tikhonov, A. N., Collection of Problems on Mathematical Physics[in Russian],

Nauka, Moscow, 1980.Tikhonov, A. N. and Samarskii, A. A., Equations of Mathematical Physics, Dover Publ., New York, 1990.Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC,

2002.

Helmholtz Equation

Copyright c© 2004 Andrei D. Polyanin http://eqworld.ipmnet.ru/en/solutions/lpde/lpde303.pdf