Helmholtz equation

Last updated Two sources of radiation in the plane, given mathematically by a function ƒ, which is zero in the blue region The real part of the resulting field A, A is the solution to the inhomogeneous Helmholtz equation (∇2+k2)A=−f{\displaystyle (\nabla ^{2}+k^{2})A=-f}

In mathematics and physics, the Helmholtz equation, named for Hermann von Helmholtz, is the linear partial differential equation Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions in several scientific fields. The largest German association of research institutions, the Helmholtz Association, is named after him. In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

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$(\nabla ^{2}+k^{2})A=0,$ where $\nabla ^{2}$ is the Laplacian, $k$ is the wave number, and $A$ is the amplitude. This is also an eigenvalue equation.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, 2. The Laplacian ∇·∇f(p) of a function f at a point p, is the rate at which the average value of f over spheres centered at p deviates from f(p) as the radius of the sphere shrinks towards 0. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems such as cylindrical and spherical coordinates, the Laplacian also has a useful form. In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

The amplitude of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the variable's extreme values. In older texts the phase is sometimes called the amplitude.

Motivation and uses

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. The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics. In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

For example, consider the wave equation

$\left(\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)u(\mathbf {r} ,t)=0.$ Separation of variables begins by assuming that the wave function $u(\mathbf {r} ,t)$ is in fact separable:

$u(\mathbf {r} ,t)=A(\mathbf {r} )T(t).$ Substituting this form into the wave equation and then simplifying, we obtain the following equation:

${\frac {\nabla ^{2}A}{A}}={\frac {1}{c^{2}T}}{\frac {d^{2}T}{dt^{2}}}.$ Notice that the expression on the left side depends only on $\mathbf {r}$ , whereas the right expression depends only on $t$ . As a result, this equation is valid in the general case if and only if both sides of the equation are equal to a 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 $A(\mathbf {r} )$ , the other for $T(t)$ :

${\frac {\nabla ^{2}A}{A}}=-k^{2}$ and

${\frac {1}{c^{2}T}}{\frac {d^{2}T}{dt^{2}}}=-k^{2},$ where we have chosen, without loss of generality, the expression $-k^{2}$ for the value of the constant. (It is equally valid to use any constant $k$ as the separation constant; $-k^{2}$ is chosen only for convenience in the resulting solutions.)

Rearranging the first equation, we obtain the Helmholtz equation:

$\nabla ^{2}A+k^{2}A=(\nabla ^{2}+k^{2})A=0.$ Likewise, after making the substitution $\omega =kc$ , where $k$ is the wave number, and $\omega$ is the angular frequency, the second equation becomes In physics, angular frequencyω is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

${\frac {d^{2}T}{dt^{2}}}+\omega ^{2}T=\left({\frac {d^{2}}{dt^{2}}}+\omega ^{2}\right)T=0,$ We now have Helmholtz's equation for the spatial variable $\mathbf {r}$ 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.

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results. The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article. In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle.

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. Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves. In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays. Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake environmental effects such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic, atmospheric, and artificial processes such as explosions. A related field that uses geology to infer information regarding past earthquakes is paleoseismology. A recording of earth motion as a function of time is called a seismogram. A seismologist is a scientist who does research in seismology.

Solving the Helmholtz equation using separation of variables

The solution to the spatial Helmholtz equation

$(\nabla ^{2}+k^{2})A=0$ 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

$A_{rr}+{\frac {1}{r}}A_{r}+{\frac {1}{r^{2}}}A_{\theta \theta }+k^{2}A=0.$ We may impose the boundary condition that A vanish if r = a; thus

$A(a,\theta )=0.\,$ The method of separation of variables leads to trial solutions of the form

$A(r,\theta )=R(r)\Theta (\theta ),\,$ where Θ must be periodic of period 2π. This leads to

$\Theta ''+n^{2}\Theta =0,\,$ and

$r^{2}R''+rR'+r^{2}k^{2}R-n^{2}R=0.\,$ It follows from the periodicity condition that

$\Theta =\alpha \cos n\theta +\beta \sin n\theta ,\,$ and that n must be an integer. The radial component R has the form

$R(r)=\gamma J_{n}(\rho ),\,$ where the Bessel function Jn(ρ) satisfies Bessel's equation

$\rho ^{2}J_{n}''+\rho J_{n}'+(\rho ^{2}-n^{2})J_{n}=0,$ and ρ = kr. The radial function Jn has infinitely many roots for each value of n, denoted by ρm,n. The boundary condition that A vanishes where r = a will be satisfied if the corresponding wavenumbers are given by

$k_{m,n}={\frac {1}{a}}\rho _{m,n}.\,$ The general solution A then takes the form of a generalized Fourier series of terms involving products of

$\sin(n\theta ){\text{ or }}\cos(n\theta ),{\text{ and }}J_{n}(k_{m,n}r).$ These solutions are the modes of vibration of a circular drumhead.

Three-dimensional solutions

In spherical coordinates, the solution is:

$A(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left(a_{\ell m}j_{\ell }(kr)+b_{\ell m}y_{\ell }(kr)\right)Y_{\ell }^{m}(\theta ,\varphi ).$ This solution arises from the spatial solution of the wave equation and diffusion equation. Here $j_{\ell }(kr)$ and $y_{\ell }(kr)$ are the spherical Bessel functions, and

$Y_{\ell }^{m}(\theta ,\varphi )$ 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 $\mathbf {r_{0}} =(x,y,z)$ function $A(r_{0})$ has asymptotics

$A(r_{0})={\frac {e^{ikr_{0}}}{r_{0}}}f\left({\frac {\mathbf {r} _{0}}{r_{0}}},k,u_{0}\right)+o\left({\frac {1}{r_{0}}}\right){\text{ as }}r_{0}\to \infty$ where function f is called scattering amplitude and $u_{0}(r_{0})$ is the value of A at each boundary point $r_{0}$ .

Paraxial approximation

In the paraxial approximation of the Helmholtz equation,  the complex amplitude A is expressed as

$A(\mathbf {r} )=u(\mathbf {r} )e^{ikz}$ 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

$\nabla _{\perp }^{2}u+2ik{\frac {\partial u}{\partial z}}=0,$ where $\textstyle \nabla _{\perp }^{2}{\stackrel {\mathrm {def} }{=}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}$ 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 z derivative of the amplitude function u is a slowly-varying function of z:

${\bigg |}{\partial ^{2}u \over \partial z^{2}}{\bigg |}\ll {\bigg |}{k{\partial u \over \partial z}}{\bigg |}.$ This condition is equivalent to saying that the angle θ between the wave vector k and the optical axis z is small: $\theta \ll 1$ .

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:

$\nabla ^{2}(u\left(x,y,z\right)e^{ikz})+k^{2}u\left(x,y,z\right)e^{ikz}=0.$ Expansion and cancellation yields the following:

$\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)u(x,y,z)e^{ikz}+\left({\frac {\partial ^{2}}{\partial z^{2}}}u(x,y,z)\right)e^{ikz}+2\left({\frac {\partial }{\partial z}}u(x,y,z)\right)ik{e^{ikz}}=0.$ Because of the paraxial inequalitiy stated above, the ∂2u/∂z2 term is neglected in comparison with the k·∂u/∂z term. This yields the paraxial Helmholtz equation. Substituting $u(\mathbf {r} )=A(\mathbf {r} )e^{-ikz}$ then gives the paraxial equation for the original complex amplitude A:

$\nabla _{\perp }^{2}A+2ik{\frac {\partial A}{\partial z}}+2k^{2}A=0.$ The Fresnel diffraction integral is an exact solution to the paraxial Helmholtz equation. 

There is even a subject named "Helmholtz optics" based on the equation, named in honour of Helmholtz.   

Inhomogeneous Helmholtz equation

The inhomogeneous Helmholtz equation is the equation

$\nabla ^{2}A(x)+k^{2}A(x)=-f(x)\ {\text{ in }}\mathbb {R} ^{n},$ 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) is switched to a minus sign.

In order to solve this equation uniquely, one needs to specify a boundary condition at infinity, which is typically the Sommerfeld radiation condition

$\lim _{r\to \infty }r^{\frac {n-1}{2}}\left({\frac {\partial }{\partial r}}-ik\right)A(r{\hat {x}})=0$ 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

$A(x)=(G*f)(x)=\int \limits _{\mathbb {R} ^{n}}\!G(x-y)f(y)\,dy$ (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 ƒ equaling the Dirac delta function, so G satisfies

$\nabla ^{2}G(x)+k^{2}G(x)=-\delta (x){\text{ in }}\mathbb {R} ^{n}.\,$ The expression for the Green's function depends on the dimension $n$ of the space. One has

$G(x)={\frac {ie^{ik|x|}}{2k}}$ 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. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for $|x|\to \infty$ .