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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

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