Inhomogeneous Helmholtz equation

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The inhomogeneous Helmholtz equation is an important elliptic partial differential equation arising in acoustics and electromagnetism. It models time-harmonic wave propagation in free space due to a localized source.

More specifically, the inhomogeneous Helmholtz equation is the equation

where is the Laplace operator, is a constant, called the wavenumber, is the unknown solution, is a given function with compact support, and (theoretically, can be any positive integer, but since stands for the dimension of the space in which the waves propagate, only the cases with are physical).

Derivation from the wave equation

Wave propagation in free space due to a source is modeled by the wave equation

where and are real-valued functions of spatial variables, and one time variable, is given, the source of waves, and is the unknown wave function.

By taking the Fourier transform of this equation in the time variable, or equivalently, by looking for time-harmonic solutions of the form

with

(where and is a real number), the wave equation is reduced to the inhomogeneous Helmholtz equation with

Solution of the inhomogeneous Helmholtz equation

In order to solve the inhomogeneous Helmholtz equation uniquely, one needs to specify a boundary condition at infinity, which is typically the Sommerfeld radiation condition

uniformly in with , where the vertical bars denote the Euclidean norm. Physically, this states that energy travels from the source away to infinity, and not the other way around.

With this condition, the solution to the inhomogeneous Helmholtz equation is the convolution

(notice this integral is actually over a finite region, since has compact support). Here, is the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with equaling the Dirac delta function, so satisfies

The expression for the Green's function depends on the dimension of the space. One has

for

for , where is a Hankel function, and

for

References

  • Howe, M. S. (1998). Acoustics of fluid-structure interactions. Cambridge; New York: Cambridge University Press. ISBN 0-521-63320-6.
  • A. Sommerfeld, Partial Differential Equations in Physics, Academic Press, New York, New York, 1949.

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