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The Solution of a Generalized Helmholtz
Equation of Two Dimensions

The Helmholtz equation is usually expressed as

∇²φ = −k²φ

with k being a constant. If k is a function of location it is a generalized Helmholtz equation. For now let k be a constant.

The Laplacian operator is equal to the divergence of the gradient operator. Thus the Helmholtz equation is more properly expressed as

∇·:(∇φ) = −k²φ

The Laplacian operator in polar coordinants (r, θ) is

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

Separation of Variables

For a solution to the Helmholtz equation assume the dependent variable has the property known as separation of the variables; i.e., φ(r, θ) = R(r)Φ(θ).

Therefore the Helmholtz equation reduces to the form

R"(r)Φ(θ) + (1/r)R'(r)Φ(θ) + (1/r²)(R(r)Φ"(θ) = − k²R(r)Φ(θ)

If this equation is divided by RΦ and multiplied by r² the result is

r²(R"(r)/R) + r(R'(r)/R) + Φ"(θ)/Φ = −k²r²

This can be rearranged so that all of the functions of r are on one side of the equation and all the functions of θ are on the other. The common value of the two sides must be a constant; i.e.,

r²(R"(r)/R) + r(R'(r)/R) + k²r² = −Φ"(θ)/Φ = λ²

The Angular Component

The terms involving θ can be considered first.

Φ"(θ) = − λ²Φ(θ)

The general solution is Φ(θ) = A·sin(λθ) + B·cos(λθ), where A and B are constants. It must be that Φ(2π)=Φ(0). This means that λ must equal an integer, say n. With the proper choice of the orientation of the coordinate system the solution can be represented as

Φ(θ) = B·cos(nθ)

The Radial Component

The radial equation is the

r²R"(r) + rR'(r) + (k²r²− n²)R = 0
R"(r) + (1/r)R'(r) + (k²− (n/r)²)R = 0

This equation is closely related to the Bessel differential equation

y"(x) + (1/x)y'(x) + (1 − (m/x)²) = 0
where m is a nonnegative integer

which has the solution

y(x) = C·Jm(x) + D·Ym(x)

where Jm(x) and Ym(x) are Bessel functions of the first and second kind of index m, respectively, and C and D are constants.

A change of independent variable will transform the equation derived above into a Bessel differential equation. Let z=kr, Then

R'(r) = R'(z)(dz/dr) = R'(z)k
R"(r) = R"(z)k²


k²R"(z) + (k/z)kR'(z) + (k²− (nk/z)²)R(z) = 0
which reduces to
k²R"(z) + k²(1/z)R'(z) + (k²− k²(n/z)²)R(z) = 0
and hence
R"(z) + (1/z)R'(z) + (1− (n/z)²)R = 0

The solution is then

R(z) = C·Jn(z) + D·Yn(z)
or, equivalently
R(r) = C·Jn(kr) + D·C

Yn(z) has a singularity that may prelude it being part of the solution.

The overall solution may be of the form

φ(r, θ) = E·cos(nθ)Yn(kr)

where n is a positive integer and E is a real constant.

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