Helmholtz equation

Solving the Helmholtz equation using separation of variables

The solution to the spatial Helmholtz equation: $${\displaystyle {\mathrm{\nabla }}^{2}A=-k^{2}A}$$ can be obtained for simple geometries using separation of variables.

Vibrating membrane

The two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The Helmholtz equation was solved for many basic shapes in the 19th century: the rectangular membrane by Siméon Denis Poisson in 1829, the equilateral triangle by Gabriel Lamé in 1852, and the circular membrane by Alfred Clebsch in 1862. The elliptical drumhead was studied by Émile Mathieu, leading to Mathieu's differential equation.

If the edges of a shape are straight line segments, then a solution is integrable or knowable in closed-form only if it is expressible as a finite linear combination of plane waves that satisfy the boundary conditions (zero at the boundary, i.e., membrane clamped).

If the domain is a circle of radius a, then it is appropriate to introduce polar coordinates r and θ. The Helmholtz equation takes the form $${\displaystyle \frac{{\mathrm{\partial }}^{2}A}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }A}{\mathrm{\partial }r}+\frac{1}{r^2}\frac{{\mathrm{\partial }}^{2}A}{\mathrm{\partial }{\theta }^2}+k^{2}A=0.}$$

We may impose the boundary condition that A vanishes if r = a; thus $${\displaystyle A(a,\theta )=0.}$$

the method of separation of variables leads to trial solutions of the form $${\displaystyle A(r,\theta )=R(r)\mathrm{\Theta }(\theta ),}$$ where Θ must be periodic of period 2π. This leads to

$${\displaystyle {\mathrm{\Theta }}^{″}+n^2\mathrm{\Theta }=0,}$$ $${\displaystyle r^2{R}^{″}+r{R}^{\prime }+r^2{k}^{2}R-n^{2}R=0;}$$ where $\frac{dR(r)}{dr}\equiv R^{\prime }(r),$ and $\frac{d\mathrm{\Theta }(\theta )}{d\theta }\equiv {\mathrm{\Theta }}^{\prime }(\theta ),$ and so on are also equivalent notations.

It follows from the periodicity condition that $${\displaystyle \mathrm{\Theta }=\alpha \cos (n\theta )+\beta \sin (n\theta ),}$$ and that n must be an integer. The radial component R has the form $${\displaystyle R=\gamma J_n(\rho ),}$$ where the Bessel function J_n(ρ) satisfies Bessel's equation $${\displaystyle z^2{J}_n^{″}+z{J}_n^{\prime }+(z^2-n^2)J_n=0,}$$ and z = k r. The radial function J_n has infinitely many roots for each value of n, denoted by ρ_m,n. The boundary condition that A vanishes where r = a will be satisfied if the corresponding wavenumbers are given by $${\displaystyle k_{m,n}=\frac{1}{a}{\rho }_{m,n}.}$$

The general solution A then takes the form of a generalized Fourier series of terms involving products of J_n(k_m,nr) and the sine (or cosine) of n θ. These solutions are the modes of vibration of a circular drumhead.

Three-dimensional solutions

In spherical coordinates, the solution is:

$${\displaystyle A(r,\theta ,\phi )=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{+\ell }\left[a_{\ell m}{j}_{\ell }(kr)+b_{\ell m}{y}_{\ell }(kr)\right]Y_{\ell }^m(\theta ,\phi ).}$$

This solution arises from the spatial solution of the wave equation and diffusion equation. Here j_ℓ(kr) and y_ℓ(kr) are the spherical Bessel functions, and Y^m
_ℓ(θ, φ) are the spherical harmonics. Note that these forms are general solutions, and require boundary conditions to be specified to be used in any specific case. For infinite exterior domains, a radiation condition may also be required.

Writing r₀ = (x, y, z) function A(r₀) has asymptotics $${\displaystyle A(r_0)⟶\frac{e^{ik{r}_0}}{r_0}f\left(\frac{{\mathbf{r}}_0}{r_0},k,u_0\right)+o\left(\frac{1}{r_0}\right)\text{ as }{r}_0\to \mathrm{\infty }}$$

where function f is called scattering amplitude and u₀(r₀) is the value of A at each boundary point r₀.

Three-dimensional solutions given the function on a 2-dimensional plane

Given a 2-dimensional plane where A is known, the solution to the Helmholtz equation is given by: $${\displaystyle A(x,y,z)=-\frac{1}{2\pi }{\iint }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{A}^{\prime }(x^{\prime },y^{\prime })\frac{e^{ikr}}{r}\frac{z}{r}\left(ik-\frac{1}{r}\right)d{x}^{\prime }d{y}^{\prime },}$$

where

• ${\displaystyle A^{\prime }(x^{\prime },y^{\prime })}$ is the solution at the 2-dimensional plane,
• $r=\sqrt{(x-x^{\prime }{)}^2+(y-y^{\prime }{)}^2+z^2},$

As z approaches zero, all contributions from the integral vanish except for r = 0. Thus ${\displaystyle A(x,y,0)=A^{\prime }(x,y)}$ up to a numerical factor, which can be verified to be 1 by transforming the integral to polar coordinates ${\displaystyle \left(\rho ,\theta \right).}$

This solution is important in diffraction theory, e.g. in deriving Fresnel diffraction.

Paraxial approximation

In the paraxial approximation of the Helmholtz equation, the complex amplitude A is expressed as $${\displaystyle A(\mathbf{r})=u(\mathbf{r})e^{ikz}}$$ where u represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Then under a suitable assumption, u approximately solves $${\displaystyle {\mathrm{\nabla }}_{\perp }^{2}u+2ik\frac{\mathrm{\partial }u}{\mathrm{\partial }z}\approx 0,}$$ where ${\mathrm{\nabla }}_{\perp }^2\equiv \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}$ is the two dimensional ( x, y ) transverse part of the Laplacian.

This equation has important applications in the science of optics, where it provides solutions that describe the propagation of electromagnetic waves (light) in the form of either paraboloidal waves or Gaussian beams. Most lasers emit beams that take this form.

The assumption under which the paraxial approximation is valid is that the z derivative of the amplitude function u is a slowly varying function of z:

$${\displaystyle \left|\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}\right|\ll \left|k\frac{\mathrm{\partial }u}{\mathrm{\partial }z}\right|.}$$

This condition is equivalent to saying that the angle θ between the wave vector k and the optical axis z is small: θ ≪ 1.

The paraxial form of the Helmholtz equation is found by substituting the above-stated expression for the complex amplitude into the general form of the Helmholtz equation as follows:

$${\displaystyle {\mathrm{\nabla }}^2(u\left(x,y,z\right)e^{ikz})+k^{2}u\left(x,y,z\right)e^{ikz}=0.}$$

Expansion and cancellation yields the following:

$${\displaystyle \left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}\right)u(x,y,z)e^{ikz}+\left(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}u(x,y,z)\right)e^{ikz}+2\left(\frac{\mathrm{\partial }}{\mathrm{\partial }z}u(x,y,z)\right)ik{e}^{ikz}=0.}$$

Because of the paraxial inequality stated above, we can choose to neglect the $$⁠∂²u/∂z²⁠ term as compared to the much larger k ⁠∂u/∂z⁠ term. Dropping the smaller term produces the paraxial Helmholtz equation. Substituting u(r) = A(r) e^−ikz then gives the paraxial equation for the original complex amplitude A:

$${\displaystyle {\mathrm{\nabla }}_{\perp }^{2}A+2ik\frac{\mathrm{\partial }A}{\mathrm{\partial }z}+2k^{2}A=0.}$$

The Fresnel diffraction integral is an exact solution to the paraxial Helmholtz equation.

Inhomogeneous Helmholtz equation

The inhomogeneous Helmholtz equation is the equation $${\displaystyle {\mathrm{\nabla }}^{2}A(\mathbf{x})+k^{2}A(\mathbf{x})=-f(\mathbf{x}),\mathrm{\forall }\mathbf{x}\in {\mathbb{R}}^n,}$$ where ƒ : Rⁿ → C is a function with compact support, and n = 1, 2, 3. This equation is very similar to the screened Poisson equation, and would be identical if the plus sign (in front of the k term) were switched to a minus sign.

Solution

In order to solve this equation uniquely, one needs to specify a boundary condition at infinity, which is typically the Sommerfeld radiation condition $${\displaystyle \underset{r\to \mathrm{\infty }}{lim}{r}^{\frac{n-1}{2}}\left(\frac{\mathrm{\partial }}{\mathrm{\partial }r}-ik\right)A(\mathbf{x})=0,}$$ in ${\displaystyle n}$ spatial dimensions, for all angles (i.e. any value of ${\displaystyle \theta ,\varphi }$). Here $${\displaystyle r=\sqrt{\sum _{i=1}^n{x}_i^2}}$$ where ${\displaystyle x_i}$ are the coordinates of the vector ${\displaystyle \mathbf{x}.}$

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

$${\displaystyle A(\mathbf{x})={\int }_{{\mathbb{R}}^n}G(\mathbf{x},{\mathbf{x}}^{\prime })f({\mathbf{x}}^{\prime })\mathrm{d}{\mathbf{x}}^{\prime }}$$

(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

$${\displaystyle {\mathrm{\nabla }}^{2}G(\mathbf{x},{\mathbf{x}}^{\prime })+k^{2}G(\mathbf{x},{\mathbf{x}}^{\prime })=-\delta (\mathbf{x},{\mathbf{x}}^{\prime })\in {\mathbb{R}}^n.}$$

The expression for the Green's function depends on the dimension n of the space. One has $${\displaystyle G(x,x^{\prime })=\frac{i{e}^{ik|x-x^{\prime }|}}{2k}}$$ for n = 1,

$${\displaystyle G(\mathbf{x},{\mathbf{x}}^{\prime })=\frac{i}{4}{H}_0^{(1)}\left(k\left|\mathbf{x}-{\mathbf{x}}^{\prime }\right|\right)}$$ for n = 2, where H⁽¹⁾
₀ is a Hankel function, and $${\displaystyle G(\mathbf{x},{\mathbf{x}}^{\prime })=\frac{e^{ik\left|\mathbf{x}-{\mathbf{x}}^{\prime }\right|}}{4\pi \left|\mathbf{x}-{\mathbf{x}}^{\prime }\right|}}$$ for n = 3. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for |x| → ∞.

Finally, for general n,

$${\displaystyle G(\mathbf{x},{\mathbf{x}}^{\prime })=c_d{k}^p\frac{H_p^{(1)}\left(k\left|\mathbf{x}-{\mathbf{x}}^{\prime }\right|\right)}{{\left|\mathbf{x}-{\mathbf{x}}^{\prime }\right|}^p}}$$

where $p=\frac{n}{2}-1$ and ${\displaystyle c_d=\frac{i}{4(2\pi {)}^p}.}$