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The Helmholtz Equation and Its Solution

An equation of the form

∇²ψ + λψ = 0

is known as a Helmholtz equation. It may have solutions for only discrete values of λ. Those values of λ would be called eigenvalues. Generally the relevant values of λ are positive.

Suppose ψ is a function of the polar coordinates (r, θ). Furthermore let us assume there is a separation of variables; i.e., ψ(r, θ)=R(r)Θ(θ).

The Laplacian ∇² for polar coordinates

(∂²/∂r²) + (1/r)(∂/∂r) + (1/r²)(∂²/∂θ²)

With separation of variables the Helmholtz Equation becomes

R"(r)Θ(θ) + (1/r)R'(r)Θ(θ) + (1/r²)R(r)Θ"(θ) + λR(r)Θ(θ)
which upon division by RΘ becomes
R"(r)/R + (1/r)R'(r)/R + (1/r²)(Θ"(θ)/Θ + λ = 0

The next step in the solution is to multiply through by r² to get

r²R"(r)/R + rR'(r)/R + λr² = −(Θ"(θ)/Θ

The LHS of the above is a function only of r and the RHS is a function only of θ. Therefore their common value must be a constant, say N.

This means that Θ(θ) is a solution to the equation

Θ"(θ) + NΘ(θ) = 0

The solution of this equation is

Θ(θ) = A·cos(√N(θ−θ0))

where A and θ0 are constants. In order for this to be a valid solution it must be that

Θ(θ+2π) = Θ(θ)

This requires that √N(2π)=n(2π), where n is an integer. Thus N=n²; i.e., the constant N is the square of an integer.

With a proper orientation of the polar coordinate system θ0 can be made equal to zero.

Solution of the Radial Equation

The function R(r) must satisfy

r²R"(r)/R + rR'(r)/R + λr² = n²
or, upon multiplying through by R
r²R"(r) + rR'(r) + (λr² − n²)R = 0

This is closely related to the Bessel equation

ρ²Jn"(ρ) + ρJn'(ρ) + (ρ²−n²)Jn(ρ) = 0\

To transform the radial equation into the Bessel equation let ρ=√λr. Then

R'(r) = dR/dr = (dR/dρ)(dρ/dr) = (dR/dρ)√λ
R"(r) = d(dR/dr)/dr = (d[(dR/dρ)√λ]/dρ)(dρ/dr) = λ(d²R/dρ²) = λR"(ρ)

Thus the radial equation is transformed into

ρ²R"(ρ) + ρR'(ρ) + (ρ²−n²)Jn(ρ) = 0\

This has the solution

R(ρ) = Jn(ρ)
and hence
R(r) = Jn(√λr)

where Jn(ρ) is a Bessel function of index n. There are two types of Bessel functions.

The general solution to the Helmholtz equation is then

ψ(r, θ) = (A· Jn(√λr)+B· Kn(√λr))cos(nθ)

where A and B are constants and Jn(.) and Kn(.) are Bessel functions of the first and second type, both of index n, an integer.

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