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

[math]\displaystyle{ \nabla^2 u + k^2 u = -f \mbox { in } \mathbb R^n }[/math]

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

[math]\displaystyle{ \frac{\partial^2 U}{\partial t^2} - c^2 \nabla^2 U = F }[/math]

where [math]\displaystyle{ U=U(x, t) }[/math] and [math]\displaystyle{ F=F(x, t) }[/math] are real-valued functions of [math]\displaystyle{ n }[/math] spatial variables, [math]\displaystyle{ x=(x_1, x_2, \dots, x_n), }[/math] and one time variable, [math]\displaystyle{ t. }[/math] [math]\displaystyle{ F }[/math] is given, the source of waves, and [math]\displaystyle{ 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]\displaystyle{ U(x, t) = e^{i\omega t}u(x)\, }[/math]

with

[math]\displaystyle{ F(x, t) = e^{i\omega t}f(x)\, }[/math]

(where [math]\displaystyle{ i=\sqrt{-1} }[/math] and [math]\displaystyle{ \omega }[/math] is a real number), the wave equation is reduced to the inhomogeneous Helmholtz equation with [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \hat {x} }[/math] with [math]\displaystyle{ |\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

[math]\displaystyle{ 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]\displaystyle{ f }[/math] has compact support). Here, [math]\displaystyle{ G }[/math] is the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with [math]\displaystyle{ f }[/math] equaling the Dirac delta function, so [math]\displaystyle{ G }[/math] satisfies

[math]\displaystyle{ \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]\displaystyle{ G(x) = \frac{ie^{ik|x|}}{2k} }[/math]

for [math]\displaystyle{ n=1, }[/math]

[math]\displaystyle{ G(x) = \frac{i}{4}H^{(1)}_0(k|x|) }[/math]

for [math]\displaystyle{ n=2 }[/math], where [math]\displaystyle{ H^{(1)}_0 }[/math] is a Hankel function, and

[math]\displaystyle{ G(x) = \frac{e^{ik|x|}}{4\pi |x|} }[/math]

for [math]\displaystyle{ 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.