View source for Inhomogeneous Helmholtz equation

The '''inhomogeneous Helmholtz equation''' is an important [[Elliptic partial differential equation|elliptic partial differential equation]] arising in acoustics and [[Physics:Electromagnetism|electromagnetism]]. It models time-harmonic [[Wave|wave]] propagation in free space due to a localized source. 

More specifically, the inhomogeneous Helmholtz equation is the equation

: <math>\nabla^2 u + k^2 u = -f \mbox { in } \mathbb R^n</math>

where <math>\nabla^2</math> is the [[Laplace operator]], <math>k>0</math> is a constant, called the ''wavenumber'', <math>u:\mathbb R^n\to \mathbb C</math> is the unknown solution, <math>f:\mathbb R^n\to \mathbb C</math> is a given function with compact support, and <math>n=1, 2, 3</math> (theoretically, <math>n</math> can be any positive integer, but since <math>n</math> stands for the dimension of the space in which the waves propagate, only the cases with <math>1\le n\le 3</math> are physical). 

==Derivation from the wave equation==

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

: <math>\frac{\partial^2 U}{\partial t^2} - c^2 \nabla^2 U = F</math>

where <math>U=U(x, t)</math> and <math>F=F(x, t)</math> are [[Real number|real]]-valued functions of <math>n</math> spatial variables, <math>x=(x_1, x_2, \dots, x_n),</math> and one time variable, <math>t.</math> <math>F</math> is given, the ''source'' of waves, and <math>U</math> 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 

: <math>U(x, t) = e^{i\omega t}u(x)\,</math>

with 

:<math>F(x, t) = e^{i\omega t}f(x)\, </math>

(where <math>i=\sqrt{-1}</math> and <math>\omega</math> is a real number), the wave equation is reduced to the inhomogeneous Helmholtz equation with <math>k^2=\omega^2/c^2.</math>

==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]]

: <math>\lim_{r \to \infty} r^{\frac{n-1}{2}} \left( \frac{\partial}{\partial r} - ik \right) u(r \hat {x}) = 0</math>

uniformly in <math>\hat {x}</math> with <math>|\hat {x}|=1</math>, where the vertical bars denote the Euclidean norm. 

With this condition, the solution to the inhomogeneous Helmholtz equation is the [[Convolution|convolution]]

: <math>u(x)=(G*f)(x)=\int\limits_{\mathbb R^n}\! G(x-y)f(y)\,dy</math>

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

: <math>\nabla G + k^2 G = -\delta \mbox { in } \mathbb R^n.</math>

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

: <math>G(x) = \frac{ie^{ik|x|}}{2k}</math>

for <math>n=1,</math>

: <math>G(x) = \frac{i}{4}H^{(1)}_0(k|x|)</math> 

for <math>n=2</math>, where <math>H^{(1)}_0</math> is a Hankel function, and

: <math>G(x) = \frac{e^{ik|x|}}{4\pi |x|}</math> 

for <math>n=3.</math>

==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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