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


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

where $$\nabla^2$$ 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


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

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


 * $$\lim_{r \to \infty} r^{\frac{n-1}{2}} \left( \frac{\partial}{\partial r} - ik \right) u(r \hat {x}) = 0$$

uniformly in $$\hat {x}$$ with $$|\hat {x}|=1$$, 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


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

(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


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

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


 * $$G(x) = \frac{ie^{ik|x|}}{2k}$$

for $$n=1,$$


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

for $$n=2$$, where $$H^{(1)}_0$$ is a Hankel function, and


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

for $$n=3.$$