Helmholtz equation

The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation

(\nabla^2 + k^2) A = 0

where $\nabla^2$ 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 equations (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(\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(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:

\nabla^2 A + k^2 A  =  ( \nabla^2 + k^2)  A  =  0

and

\frac{d^2{T}}{d{t}^2} + \omega^2T  =  \left( { d^2 \over dt^2 } + \omega^2 \right) T  =  0,

where

k

is the wave vector and

\omega \equiv kc

is the angular frequency.

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 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.

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

( \nabla^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, 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 Emile Mathieu, leading to Mathieu's differential equation. The solvable shapes all correspond to shapes whos dynamical billiard table is integrable, that is, not chaotic. When the motion on a correspondingly-shaped billiard table is chaotic, then no closed form solutions to the Helmholtz equation are known. The study of such systems is known as quantum chaos, as the Helmholtz equation and similar equations occur in quantum mechanics.

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'' + r R' + 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 J_n(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 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 frequencies are given by

k_{m,n} = \frac{1}{a} \rho_{m,n}. \,

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

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

These solutions are the modes of vibration of a circular drumhead.

Three-dimensional solutions

In spherical polar coordinates, the solution is:

A (r, \theta, \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 ( { \theta,\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 functions, and

Y^m_l ( {\theta,\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:

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

where

\nabla_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 beams. Most lasers 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:

\bigg| { \partial A \over \partial z } \bigg|  <<   | kA |

and

\bigg| { \partial^2 A \over \partial z^2 } \bigg|  <<   | k^2 A |

These conditions are equivalent to saying that the angle θ between the wave vector k and the optical axis z must be small enough so that

\sin(\theta) \approx \theta \qquad \mathrm{and} \qquad \tan(\theta) \approx \theta

References

M. Abramowitz and I. Stegun eds., Handbook of Mathematical functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards. Washington, D. C., 1964.

Riley, K.F., Hobson, M.P., and Bence, S.J. (2002). Mathematical methods for physics and engineering, Cambridge University Press, ch. 19. ISBN 0-521-89067-5.

McQuarrie, Donald A. (2003). Mathematical Methods for Scientists and Engineers, University Science Books: Sausalito, California, Ch. 16. ISBN 1-891389-24-6.

Chapter 3, "Beam Optics," pp. 80–107.

A. Sommerfeld, Partial Differential Equations in Physics, Academic Press, New York, New York, 1949.

External link

Helmholtz Equation at EqWorld: The World of Mathematical Equations.

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