San José State University |
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A Fourier Analysis of theGeneralized Helmholtz Equation |
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The Generalized Helmholtz Equation is

where z repesents location and H is positive. In one dimension this is

One approach to analyzing this equation is to take its Fourier transforms *F*(ω); i.e.,
the Fourier transform of f(x) defined over the interval [−∞, +∞] is

Suppose H(x) is a constant H_{0} and assume that

where A is an arbitrary constant whose value is established by initial conditions. Then

and

d²φ(x)/dx² = β²A·exp(βx)

Therefore for the equation to be satisfied it must be that

and since φ(x)=A·exp(βx)

it follows that

β²=−H

and hence

β =(−H

Thus

where =ω_{0}=(H_{0})^{½}.
Thus for a constant coefficient the spectrum of the solution equals a single value of frequency.

When the Fourier transform is applied to the generalized Helmholtz equation the result is

which reduces to

ω²

The integral on the RHS of the above is known as the *convolution* of *F*_{H} and *F*_{φ}.

The Fourier transform can be interpreted as the decomposition of a function into constant frequency sinusoidal components. That
decomposition is called the *spectrum* of the function. What is desired is an analysis that demonsttrates the relationship of the
spectrum of the solution φ(x) to the coefficient function H(x).

In the above it was found that for the case of a constant H the spectrum of the solution is a single
spike at (H_{0})^{½}.

More generally the convolution is in the nature of a weighted sum. The weighted sum can be repesented as the weight total times a weighted average. Thus

where W= ∫_{−∞}^{+∞}*F*_{H}(ν)dν and

The differential generalized Helmholtz equation is related to an integral equation in the frequency domain for the spectrum
of its solution. The equation has the value of *F*_{φ}(ω) tied to a weighted average of
nearby values this means that ω² is tied to W and hence ω to W^{½}.

The integral equation above is like a system of linear equations and a system of linear equations can be represented
in terms of matrices. Suppose ω can take on only integral values positive, negative and zero. Then *F*_{φ}(ω)
is an infinite order vector denoted as *F*_{φ}. Let Ω be an infinite order diagonal square matrix with ω; on the diagonal.
The LHS of the matrix equation is then Ω²*F*_{φ}.

The construction of the RHS is more involved. Let Γ be the weights of the weighted sum expressed as a row vector and let Λ be the infinite square matrix created by placing Γ on each row centered on the principal diagonal; i.e.,

| .......Γ...............|

| ...........Γ ........,|

| ...............Γ.....|

|.......................|

The matrix equation for the integral equation is

or, equivalently

[Ω² − Λ]F

A solution to this equation would have to be *normalized*.

(To be continued.)

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