Inhomogeneous Helmholtz equation

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

${\mathrm{\nabla }}^{2}u+k^{2}u=-f\text{ in }{\mathbb{ℝ}}^n$

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

Derivation from the wave equation

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

$\frac{{\partial }^{2}U}{\partial t^2}-c^2{\mathrm{\nabla }}^{2}U=F$

where $U=U(x,t)$ and $F=F(x,t)$ are real-valued functions of $n$ spatial variables, $x=(x_1,x_2,\dots ,x_n),$ and one time variable, $t.$ $F$ is given, the source of waves, and $U$ 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

$U(x,t)=e^{i\omega t}u(x)$

with

$F(x,t)=e^{i\omega t}f(x)$

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

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

$\underset{r\to \mathrm{\infty }}{\lim }{r}^{\frac{n-1}{2}}(\frac{\partial }{\partial r}-ik)u(r\hat{x})=0$

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

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

$u(x)=(G*f)(x)=\underset{{\mathbb{ℝ}}^n}{\int }G(x-y)f(y)dy$

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

$\mathrm{\nabla }G+k^{2}G=-\delta \text{ in }{\mathbb{ℝ}}^n.$

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

$G(x)=\frac{i{e}^{ik|x|}}{2k}$

for $n=1,$

$G(x)=\frac{i}{4}{H}_0^{(1)}(k|x|)$

for $n=2$, where $H_0^{(1)}$ is a Hankel function, and

$G(x)=\frac{e^{ik|x|}}{4\pi |x|}$

for $n=3.$

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.