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:
(
abla^2 + k^2) A = 0


where
abla^2 is the Laplacian , k is a constant, and the unknown Function A=A(x, y, z) is defined on three-dimensional Euclidean Space R3.


MOTIVATION AND USES


The Helmholtz equation often arises in the study of physical problems involving Partial Differential Equation s (PDEs) in both space and time. The Helmholtz equation, which represents the time-independent form of original equation, results from applying the technique of Separation Of Variables to reduce the complexity of the analysis.

For example, consider the Wave Equation :

:
\left(
abla^2- rac{1}{c^2} rac{\partial^2}{\partial{t}^2} ight)u(\mathbf{r},t)=0


Separation of variables begins by assuming that the wave function ''u''(''t'') is in fact separable:

:u(\mathbf{r},t)=A (\mathbf{r}) \cdot T(t)

Substituting this form into the wave equation, and then simplifying, we obtain two differential equations:

:
abla^2 A + k^2 A = (
abla^2 + k^2) A = 0
and
: rac{d^2{T}}{d{t}^2} + \omega^2T = \left( { d^2 \over dt^2 } + \omega^2 ight) T = 0,
where
:k \, is the separation constant and
: \omega \equiv kc is defined for convenience (the physical interpretation of both parameters will become apparent shortly).

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, with Angular Frequency of ω, while the form of the solution in space will depend on the Boundary Condition s. Alternatively, Integral Transform s, such as the Laplace or Fourier Transform , are often used to transform a hyperbolic PDE into a form of the Helmholtz equation.

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 VARIABLES


The general solution to the spatial Helmholtz equation

: (
abla^2 + k^2 ) A = 0

can be obtained using Separation Of Variables .


Vibrating membrane


The two-dimensional analogue of the vibrating string is the vibrating membrane. 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} + rac{1}{r} A_r + rac{1}{r^2}A_{ heta heta} + k^2 A=0. \,

We may impose the boundary condition that ''A'' vanish if ''r''=''a''; thus

: A(a, heta) = 0. \,

The method of separation of variables leads to trial solutions of the form

: A(r, heta) = R(r)\Theta( heta), \,

where Θ must be periodic of period 2π. This leads to

: \Theta'' +n^2 \Theta =0, \,

and
: r^2 R'' + r R' + r^2 k^2 R + n^2 R=0. \,

It follows from the periodicity condition that

: \Theta = \alpha \cos n heta + \beta \sin n heta, \,

and that ''n'' must be an integer. The radial component ''R'' has the form

: R(r) = \gamma J_n( ho), \,

where the Bessel Function ''Jn''(ρ) satisfies Bessel's equation

: ho^2 J_n'' + ho J_n' +( ho^2 + n^2)J_n =0, \,

and ρ=''kr''. The radial function ''Jn''
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 frequencies are given by

: k_{m,n} = rac{1}{a} ho_{m,n}. \,

The general solution ''A'' then takes the form of a doubly infinite sum of terms involving products of

: \sin(n heta) \, \hbox{or} \, \cos(n heta), \, \hbox{and} \, J_n(k_{m,n}r).

These solutions are the modes of vibration of a drumhead.


Three-dimensional solutions


In spherical polar coordinates, the solution is:

: A (r, heta, \phi)= \sum_{k} \sum_{l=0}^\infty \sum_{m=-l}^l ( a_{l m} j_l ( k r ) + b_{l m} n_l ( k r ) ) Y ^ m_l ( { heta,\phi} )

This solution arises from the spatial solution of the Wave Equation and Diffusion Equation . Here j_l ( k r ) and n_l ( k r ) are the Spherical Bessel Function s, and

: Y^m_l ( { heta,\phi} )

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).


PARAXIAL FORM


The paraxial form of the Helmholtz equation is:

:
abla_T^2 A - j 2k { \partial A \over \partial z } = 0

where

:
abla_T^2 = { \partial^2 \over \partial x^2 } + { \partial^2 \over \partial y^2 }

is the transverse form 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 Beam s. Most Laser s emit beams that take this form.

In the Paraxial Approximation , the Electric Field Complex Magnitude ''E'' becomes

:E(\mathbf{r}) = A(\mathbf{r}) e^{-jkz}

where ''A'' represents the complex-valued amplitude of the electric field, which modulates the sinusoidal plane wave represented by the exponential factor.

The paraxial approximation places certain upper limits on the variation of the amplitude function ''A'' with respect to longitudinal distance ''z''. Specifically: