Helmholtz equation

Failed to parse (Missing texvc executable; please see math/README to configure.): A
is the solution to the inhomogeneous Helmholtz equation Failed to parse (Missing texvc executable; please see math/README to configure.): (\nabla^2 + k^2) A = -f.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): (\nabla^2 + k^2) A = 0

where Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla^2

is the Laplacian, Failed to parse (Missing texvc executable; please see math/README to configure.): k
is a constant, and the unknown function Failed to parse (Missing texvc executable; please see math/README to configure.): A=A(x, y, z)
is defined on n-dimensional Euclidean space Rn (typically n=1, 2, or 3, when the solution to this equation makes physical sense).

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 the original equation, results from applying the technique of separation of variables to reduce the complexity of the analysis.

For example, consider the wave equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): u(\mathbf{r},t)=A (\mathbf{r}) \cdot T(t)

and

Failed to parse (Missing texvc executable; please see math/README to configure.): T(t) = e^{i\omega t}. \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla^2 A + \frac{\omega^2}{c^2} A = ( \nabla^2 + k^2) A = 0

and Failed to parse (Missing texvc executable; please see math/README to configure.): \omega \ \stackrel{\mathrm{def}}{=}\ kc

where k is the wave vector and ω is the angular frequency.  So we can write this as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla^2 A + k^2 A = ( \nabla^2 + k^2) A = 0.

Note that by the nature of our ansatz for Failed to parse (Missing texvc executable; please see math/README to configure.): T,

Failed to parse (Missing texvc executable; please see math/README to configure.): T
 satisfies:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d^2{T}}{d{t}^2} + \omega^2T = \left( { d^2 \over dt^2 } + \omega^2 \right) T = 0,

for Failed to parse (Missing texvc executable; please see math/README to configure.): A = e^{-i\mathbf{k}\cdot\mathbf{r}}.

We now have Helmholtz's equation for the spatial variable Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): ( \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 whose 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 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).

An interesting situation happens with a shape where about half of the solutions are integrable, but the remainder are not. A simple shape where this happens is with the regular hexagon. If the wavepacket describing a quantum billiard ball is made up of only the closed-form solutions, its motion will not be chaotic, but if any amount of non-closed-form solutions are included, the quantum billiard motion becomes chaotic. Another simple shape where this happens is with an "L" shape made by reflecting a square down, then to the right.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): A(a,\theta) = 0. \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): A(r,\theta) = R(r)\Theta(\theta), \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \Theta'' +n^2 \Theta =0, \,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): r^2 R'' + r R' + r^2 k^2 R - n^2 R=0. \,

It follows from the periodicity condition that

Failed to parse (Missing texvc executable; please see math/README to configure.): \Theta = \alpha \cos n\theta + \beta \sin n\theta, \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): R(r) = \gamma J_n(\rho), \,

where the Bessel function J_n(ρ) satisfies Bessel's equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 coordinates, the solution is:

Failed to parse (Missing texvc executable; please see math/README to configure.): A (r, \theta, \phi)= \sum_{n=0}^\infty \sum_{l=0}^\infty \sum_{m=-l}^l ( a_{n l m} j_n ( k r ) + b_{n l m} y_n ( k r ) ) Y ^ m_l ( { \theta,\phi} ) .

This solution arises from the spatial solution of the wave equation and diffusion equation. Here Failed to parse (Missing texvc executable; please see math/README to configure.): j_n ( k r )

and Failed to parse (Missing texvc executable; please see math/README to configure.):  y_n ( k r )
are the spherical Bessel functions, and 

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla_T^2 A - j 2k { \partial A \over \partial z } = 0

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 complex magnitude of the electric field E becomes

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \bigg| { \partial A \over \partial z } \bigg| \ll | kA |

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \bigg| { \partial^2 A \over \partial z^2 } \bigg| \ll | 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(\theta) \approx \theta \qquad \mathrm{and} \qquad \tan(\theta) \approx \theta

The paraxial form of the Helmholtz equation is found by substituting the above-stated complex magnitude of the electric field into the general form of the Helmholtz equation as follows.

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla^{2}(A\left( x,y,z \right) e^{-jkz}) + k^2 (A\left( x,y,z \right) e^{-jkz}) = 0

Expansion and cancellation yields the following:

Failed to parse (Missing texvc executable; please see math/README to configure.): \left( {\frac {\partial ^{2}}{\partial {x}^{2}}} + {\frac {\partial ^{2}}{\partial {y}^{2}}} \right)(A\left( x,y,z \right) e^{-jkz}) + \left( {\frac {\partial ^{2}}{\partial {z}^{2}}}A \left( x,y,z \right) \right) {e^{-jkz}}-2\, \left( {\frac {\partial }{\partial z}}A \left( x,y,z \right) \right) jk{e^{-jkz}}=0

Because of the paraxial inequalities stated above, the ∂²A/∂z² factor is neglected in comparison with the ∂A/∂z factor. The yields the Paraxial Helmholtz equation.

Inhomogeneous Helmholtz equation

The inhomogeneous Helmholtz equation is the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla^2 A + k^2 A = -f \mbox { in } \mathbb R^n

where Failed to parse (Missing texvc executable; please see math/README to configure.): f:\mathbb R^n\to \mathbb C

is a given function with compact support, and Failed to parse (Missing texvc executable; please see math/README to configure.): n=1, 2, 3.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{r \to \infty} r^{\frac{n-1}{2}} \left( \frac{\partial}{\partial r} - ik \right) A(r \hat {x}) = 0

uniformly in Failed to parse (Missing texvc executable; please see math/README to configure.): \hat {x}

with Failed to parse (Missing texvc executable; please see math/README to configure.): |\hat {x}|=1

, where the vertical bars denote the Euclidean norm.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): f

has compact support). Here, Failed to parse (Missing texvc executable; please see math/README to configure.): G
is the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with Failed to parse (Missing texvc executable; please see math/README to configure.): f
equaling the Dirac delta function, so Failed to parse (Missing texvc executable; please see math/README to configure.): G
satisfies

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla^2 G + k^2 G = -\delta \mbox { in } \mathbb R^n.

The expression for the Green's function depends on the dimension of the space. One has

Failed to parse (Missing texvc executable; please see math/README to configure.): G(x) = \frac{ie^{ik|x|}}{2k}

for Failed to parse (Missing texvc executable; please see math/README to configure.): n=1,

Failed to parse (Missing texvc executable; please see math/README to configure.): G(x) = \frac{i}{4}H^{(1)}_0(k|x|)

for Failed to parse (Missing texvc executable; please see math/README to configure.): n=2 , where Failed to parse (Missing texvc executable; please see math/README to configure.): H^{(1)}_0

is a Hankel function, and

Failed to parse (Missing texvc executable; please see math/README to configure.): G(x) = \frac{e^{ik|x|}}{4\pi |x|}

for Failed to parse (Missing texvc executable; please see math/README to configure.): n=3.

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.

• Bahaa E. A. Saleh and Malvin Carl Teich (1991). Fundamentals of Photonics. New York: John Wiley & Sons. ISBN 0-471-83965-5.  Chapter 3, "Beam Optics," pp. 80–107.

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

• Howe, M. S. (1998). Acoustics of fluid-structure interactions. Cambridge; New York: Cambridge University Press. ISBN 0-521-63320-6. 

External links

• Helmholtz Equation at EqWorld: The World of Mathematical Equations.de:Helmholtz-Gleichung

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