San José State University
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|The Helmholtz Equation and Its Solution|
An equation of the form
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
With separation of variables the Helmholtz Equation becomes
The next step in the solution is to multiply through by r² to get
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
The solution of this equation is
where A and θ0 are constants. In order for this to be a valid solution it must be that
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.
The function R(r) must satisfy
This is closely related to the Bessel equation
To transform the radial equation into the Bessel equation let ρ=√λr. Then
Thus the radial equation is transformed into
This has the solution
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
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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