In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator.
[1] The equation is named after Hermann von Helmholtz, who studied it in 1860.
[2] 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.
Notice that the expression on the left side depends only on 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 the same constant value.
This argument is key in the technique of solving linear partial differential equations by separation of variables.
Likewise, after making the substitution ω = kc, where k is the wave number, and ω is the angular frequency (assuming a monochromatic field), the second equation becomes
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.
[3] 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.
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.
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 method of separation of variables leads to trial solutions of the form
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
The general solution A then takes the form of a generalized Fourier series of terms involving products of Jn(km,nr) and the sine (or cosine) of n θ .
Here jℓ(kr) and yℓ(kr) are the spherical Bessel functions, and Ymℓ(θ, φ) 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).
where function f is called scattering amplitude and u0(r0) is the value of A at each boundary point r0.
up to a numerical factor, which can be verified to be 1 by transforming the integral to polar coordinates
In the paraxial approximation of the Helmholtz equation,[5] the complex amplitude A is expressed as
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.
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:
Substituting u(r) = A(r) e−ikz then gives the paraxial equation for the original complex amplitude A:
The Fresnel diffraction integral is an exact solution to the paraxial Helmholtz equation.
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.
(notice this integral is actually over a finite region, since f has compact support).
The expression for the Green's function depends on the dimension n of the space.
Note that we have chosen the boundary condition that the Green's function is an outgoing wave for |x| → ∞.