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Equation of Two Dimensions
The Helmholtz equation is usually expressed as
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
The Laplacian operator in polar coordinants (r, θ) is
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
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.,
The terms involving θ can be considered first.
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
The radial equation is the
This equation is closely related to the Bessel differential equation
which has the solution
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
The solution is then
Yn(z) has a singularity that may prelude it being part of the solution.
The overall solution may be of the form
where n is a positive integer and E is a real constant.
If such a solution is depicted by displaying the peaks of φ² it would look something like this for n=10
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