Helmholtz equation

Template:Short description In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: <math display="block">\nabla^2 f = -k^2 f,</math> where Template:Math is the Laplace operator, Template:Math is the eigenvalue, and Template:Mvar is the (eigen)function. When the equation is applied to waves, Template:Mvar is known as the wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle.

In optics, the Helmholtz equation is the wave equation for the electric field.Template:Sfn

The equation is named after Hermann von Helmholtz, who studied it in 1860.<ref>Helmholtz Equation, from the Encyclopedia of Mathematics.</ref>

Motivation and usesEdit

The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis.

For example, consider the wave equation <math display="block">\left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right) u(\mathbf{r},t)=0.</math>

Separation of variables begins by assuming that the wave function Template:Math is in fact separable: <math display="block">u(\mathbf{r},t) =A (\mathbf{r}) T(t).</math>

Substituting this form into the wave equation and then simplifying, we obtain the following equation: <math display="block">\frac{\nabla^2 A}{A} = \frac{1}{c^2 T} \frac{\mathrm{d}^2 T}{\mathrm{d} t^2}.</math>

Notice that the expression on the left side depends only on Template:Math, whereas the right expression depends only on Template:Mvar. As a result, this equation is valid in the general case if and only if both sides of the equation are equal to the same constant value. This argument is key in the technique of solving linear partial differential equations by separation of variables. From this observation, we obtain two equations, one for Template:Math, the other for Template:Math <math display="block">\frac{\nabla^2 A}{A} = -k^2</math> <math display="block">\frac{1}{c^2 T} \frac{\mathrm{d}^2 T}{\mathrm{d}t^2} = -k^2,</math>

where we have chosen, without loss of generality, the expression Template:Math for the value of the constant. (It is equally valid to use any constant Template:Mvar as the separation constant; Template:Math is chosen only for convenience in the resulting solutions.)

Rearranging the first equation, we obtain the (homogeneous) Helmholtz equation: <math display="block">\nabla^2 A + k^2 A = (\nabla^2 + k^2) A = 0.</math>

Likewise, after making the substitution Template:Math, where Template:Mvar is the wave number, and Template:Mvar is the angular frequency (assuming a monochromatic field), the second equation becomes

<math display="block">\frac{\mathrm{d}^2 T}{\mathrm{d}t^2} + \omega^2T = \left( \frac{\mathrm{d}^2}{\mathrm{d}t^2} + \omega^2 \right) T = 0.</math>

We now have Helmholtz's equation for the spatial variable Template:Math and a second-order ordinary differential equation in time. The solution in time will be a linear combination of sine and cosine functions, whose exact form is determined by initial conditions, while the form of the solution in space will depend on the boundary conditions. Alternatively, integral transforms, such as the Laplace or Fourier transform, are often used to transform a hyperbolic PDE into a form of the Helmholtz equation.Template:Sfn

Because of its relationship to the wave equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology, and acoustics.

Solving the Helmholtz equation using separation of variablesEdit

The solution to the spatial Helmholtz equation: <math display="block"> \nabla^2 A = -k^2 A </math> can be obtained for simple geometries using separation of variables.

Vibrating membraneEdit

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 Template:Mvar, then it is appropriate to introduce polar coordinates Template:Mvar and Template:Mvar. The Helmholtz equation takes the form <math display="block">\ A_{rr} + \frac{1}{r} A_r + \frac{1}{r^2}A_{\theta\theta} + k^2 A = 0 ~.</math>

We may impose the boundary condition that Template:Mvar vanishes if Template:Math; thus <math display="block">\ A(a,\theta) = 0 ~.</math>

the method of separation of variables leads to trial solutions of the form <math display="block">\ A(r,\theta) = R(r)\Theta(\theta)\ ,</math> where Template:Math must be periodic of Template:Nobr This leads to

<math display="block">\ \Theta + n^2 \Theta = 0\ ,</math> <math display="block">\ r^2 R + r R' + r^2 k^2 R - n^2 R = 0 ~.</math>

It follows from the periodicity condition that <math display="block">\ \Theta = \alpha \cos n\theta + \beta \sin n\theta\ ,</math> and that Template:Mvar must be an integer. The radial component Template:Mvar has the form <math display="block">\ R = \gamma\ J_n(\rho)\ ,</math> where the Bessel function Template:Math satisfies Bessel's equation <math display="block">\ z^2 J_n + z J_n' + (z^2 - n^2) J_n =0\ ,</math> and Template:Nobr The radial function Template:Math has infinitely many roots for each value of Template:Nobr denoted by Template:Nobr The boundary condition that Template:Mvar vanishes where Template:Math will be satisfied if the corresponding wavenumbers are given by <math display="block">\ k_{m,n} = \frac{1}{a} \rho_{m,n} ~.</math>

The general solution Template:Mvar then takes the form of a generalized Fourier series of terms involving products of Template:Math and the sine (or cosine) of Template:Nobr These solutions are the modes of vibration of a circular drumhead.

Three-dimensional solutionsEdit

In spherical coordinates, the solution is:

<math display="block">\ A( r, \theta, \varphi ) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^{+\ell} \bigl(\ a_{\ell m}\ j_\ell( k r ) + b_{\ell m}\ y_\ell(kr)\ \bigr)\ Y^m_\ell(\theta,\varphi) ~.</math>

This solution arises from the spatial solution of the wave equation and diffusion equation. Here Template:Math and Template:Math are the spherical Bessel functions, and Template:Math are the spherical harmonics (Abramowitz and Stegun, 1964). 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 (Sommerfeld, 1949).

Writing Template:Math function Template:Math has asymptotics <math display="block">A(r_0)=\frac{e^{i k 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\infty</math>

where function Template:Mvar is called scattering amplitude and Template:Math is the value of Template:Mvar at each boundary point Template:Math

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

Given a 2-dimensional plane where A is known, the solution to the Helmholtz equation is given by:Template:Sfn <math display=block> A(x, y, z) = -\frac{ 1 }{ 2\pi } \iint_{-\infty}^{+\infty} A'(x', y')\ \frac{~~ e^{i k r}\ }{ r }\ \frac{\ z\ }{ r } \left(\ i\ k - \frac{1}{r}\ \right)\ \operatorname{d} x'\ \operatorname{d}y'\ ,</math>

where

<math>\ A'(x', y')\ </math> is the solution at the 2-dimensional plane,
<math>\ r = \sqrt{ (x - x')^2 + (y - y')^2 + z^2\ }\ ,</math>

As Template:Mvar approaches zero, all contributions from the integral vanish except for Template:Math Thus <math>\ A(x, y, 0) = A'(x,y)\ </math> up to a numerical factor, which can be verified to be Template:Math by transforming the integral to polar coordinates <math>\ \left( \rho, \theta \right) ~.</math>

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

Paraxial approximationEdit

Template:Further In the paraxial approximation of the Helmholtz equation,Template:Sfn the complex amplitude Template:Mvar is expressed as <math display="block">A(\mathbf{r}) = u(\mathbf{r}) e^{ikz} </math> where Template:Mvar represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Then under a suitable assumption, Template:Mvar approximately solves <math display="block">\nabla_{\perp}^2 u + 2ik\frac{\partial u}{\partial z} = 0,</math> where <math display="inline">\nabla_\perp^2 \overset{\text{ def }}{=} \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}</math> is the 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 Template:Mvar derivative of the amplitude function Template:Mvar is a slowly varying function of Template:Mvar:

<math display="block"> \left| \frac{ \partial^2 u }{ \partial z^2 } \right| \ll \left| k \frac{\partial u}{\partial z} \right| .</math>

This condition is equivalent to saying that the angle Template:Mvar between the wave vector Template:Math and the optical axis Template:Mvar is small: Template:Math.

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:

<math display="block">\nabla^{2}(u\left( x,y,z \right) e^{ikz}) + k^2 u\left( x,y,z \right) e^{ikz} = 0.</math>

Expansion and cancellation yields the following:

<math display="block">\left( \frac {\partial^2}{\partial x^2} + \frac {\partial^2}{\partial y^2} \right) u(x,y,z) e^{ikz} + \left( \frac {\partial^2}{\partial z^2} u (x,y,z) \right) e^{ikz} + 2 \left( \frac \partial {\partial z} u(x,y,z) \right) ik{e^{ikz}}=0.</math>

Because of the paraxial inequality stated above, the Template:Math term is neglected in comparison with the Template:Math term. This yields the paraxial Helmholtz equation. Substituting Template:Math then gives the paraxial equation for the original complex amplitude Template:Mvar:

<math display="block">\nabla_{\perp}^2 A + 2ik\frac{\partial A}{\partial z} + 2k^2A = 0.</math>

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

Inhomogeneous Helmholtz equationEdit

Template:Multiple image The inhomogeneous Helmholtz equation is the equation <math display="block">\nabla^2 A(\mathbf{x}) + k^2 A(\mathbf{x}) = -f(\mathbf{x}), \quad \forall \mathbf{x} \in \mathbb{R}^n, </math> where Template:Math is a function with compact support, and Template:Math This equation is very similar to the screened Poisson equation, and would be identical if the plus sign (in front of the Template:Mvar term) were switched to a minus sign.

SolutionEdit

In order to solve this equation uniquely, one needs to specify a boundary condition at infinity, which is typically the Sommerfeld radiation condition <math display="block">\lim_{r \to \infty} r^{\frac{n-1}{2}} \left( \frac{\partial}{\partial r} - ik \right) A(\mathbf{x}) = 0,</math> in <math>n</math> spatial dimensions, for all angles (i.e. any value of <math>\theta, \phi</math>). Here <math display="block">r = \sqrt{\sum_{i=1}^n x_i^2} </math> where <math>x_i,</math> are the coordinates of the vector <math>\mathbf{x}</math>.

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

<math display="block">A(\mathbf{x})=\int_{\R^n}\! G(\mathbf{x},\mathbf{x'})f(\mathbf{x'})\,\mathrm{d}\mathbf{x'}</math>

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

<math display="block">\nabla^2 G(\mathbf{x},\mathbf{x'}) + k^2 G(\mathbf{x},\mathbf{x'}) = -\delta(\mathbf{x},\mathbf{x'}) \in \R^n. </math>

The expression for the Green's function depends on the dimension Template:Mvar of the space. One has <math display="block">G(x,x') = \frac{ie^{ik|x - x'|}}{2k}</math> for Template:Math,

<math display="block">G(\mathbf{x},\mathbf{x'}) = \frac{i}{4}H^{(1)}_0(k|\mathbf{x}-\mathbf{x'}|)</math> for Template:Math, where Template:Math is a Hankel function, and <math display="block">G(\mathbf{x},\mathbf{x'}) = \frac{e^{ik|\mathbf{x}-\mathbf{x'}|}}{4\pi |\mathbf{x}-\mathbf{x'}|}</math> for Template:Math. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for Template:Math.

Finally, for general n,

<math display="block">G(\mathbf{x},\mathbf{x'}) = c_d k^p \frac{H_p^{(1)}(k|\mathbf{x}-\mathbf{x'}|)}{|\mathbf{x}-\mathbf{x'}|^p}</math>

where <math> p = \frac{n - 2}{2} </math> and <math>c_d = \frac{i}{4(2\pi)^p} </math>.Template:Sfn