Physics 318
Electromagnetism
R.G. Palmer
3/26/10
Helmholtz & Laplace Equations — Solution Summary
The Helmholtz equation is
(∇2 + k 2 )ψ = 0
• If k 6= 0, then is called the Helmholtz equation. k 2 is usually positive in most applications.
• If k = 0, then is called the Laplace equation.
• The wave equation and heat/diffusion equations give rise to the Helmholtz equation when time is
2 2
separated out, with time dependence e±ikvt for the wave equation (with speed v) and e−α k t for the
heat/diffusion equation (with thermal diffusivity or diffusion constant α2 ).
• In the following summary, any linear combination of the functions listed together in braces {} is allowed.
• Whenever I write cos x and sin x, you could equally well have e±ix or any linear combination thereof.
Similarly cosh x and sinh x should be taken to include any linear combination of e±x . If there is a zero
separation constant (i.e. α = 0) for cosh αx/sinh αx/cos αx/sin αx, then the functions change to c1 +c2 x
(or y/z/φ) .
• m must be an integer if φ is unrestricted (i.e. φ → φ + 2π).
• l must be an integer if regular behavior is required at θ = 0 and θ = π.
• If m and l are both integers, then −l ≤ m ≤ l for non-vanishing Plm .
• If regular behavior is required at the origin, then only the upper alternative is allowed in the braces {}
below for the r or ρ dependence.
Cartesian Coordinates
cos βy
cos γz
ψ(x, y, z) =
sin βy
sin γz
cosh αx
cos βy
cos γz
ψ(x, y, z) =
sinh αx
sin βy
sin γz
cosh αx
cosh βy
cos γz
ψ(x, y, z) =
sinh αx
sinh βy
sin γz
cosh αx
cosh βy
cosh γz
ψ(x, y, z) =
sinh αx
sinh βy
sinh γz
cos αx
sin αx
(k 2 > 0)
α2 + β 2 + γ 2 = k 2
−α2 + β 2 + γ 2 = k 2
−α2 − β 2 + γ 2 = k 2
(k 2 < 0) −α2 − β 2 − γ 2 = k 2
ψ(x, y, z) = other permutations of (x, y, z)
Cylindrical Coordinates
Jm (Kρ)
cos mφ
cos γz
Nm (Kρ)
sin mφ
sin γz
Jm (Kρ)
cos mφ
cosh γz
ψ(ρ, φ, z) =
Nm (Kρ)
sin mφ
sinh γz
cos γz
Im (Kρ)
cos mφ
ψ(ρ, φ, z) =
sin γz
Km (Kρ)
sin mφ
Im (Kρ)
cos mφ
cosh γz
ψ(ρ, φ, z) =
Km (Kρ)
sin mφ
sinh γz
m ρ
cos mφ
cos kz
ψ(ρ, φ, z) =
ρ−m
sin mφ
sin kz
(k 2 > 0)
ψ(ρ, φ, z) =
1
K 2 + γ 2 = k2
K 2 − γ 2 = k2
−K 2 + γ 2 = k 2
(k 2 < 0) −K 2 − γ 2 = k 2
(k 2 ≥ 0)
→ c1 + c2 ln ρ for m = 0)
±m
(ρ
Physics 318
Electromagnetism
R.G. Palmer
3/26/10
Spherical Coordinates
jl (kr)
nl (kr)
Plm (cos θ)
Qm
l (cos θ)
cos mφ
sin mφ
il (Kr)
kl (Kr)
Plm (cos θ)
Qm
l (cos θ)
cos mφ
sin mφ
ψ(r, θ, φ) =
ψ(r, θ, φ) =
ψ(r, θ, φ) =
rl
r−l−1
Plm (cos θ)
Qm
l (cos θ)
cos mφ
sin mφ
(k 2 > 0)
(k 2 < 0) −K 2 = k 2
(k 2 = 0)
Note that for integer l and m, Plm (cos θ)eimφ may also be written Ylm (θ, φ) besides a
numerical prefactor. For the case m = 0, Plm (cos θ) reduces to Pl (cos θ), a Legendre
polynomial (or Legendre function for non-integer l).
2
(Laplace)