Helmholtz equation - Citizendia

Two sources of radiation in the plane, given mathematically by a function

f

which is zero in the blue region.

The real part of the resulting field

A,

A

is the solution to the inhomogeneous Helmholtz equation $(\nabla^2 + k^2) A = -f.$

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 n-dimensional Euclidean space Rⁿ (typically n=1, 2, or 3, when the solution to this equation makes physical sense). In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z In Mathematics, an elliptic operator is one of the major types of Differential operator. In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

1 Motivation and uses
2 Solving the Helmholtz equation using separation of variables
- 2.1 Vibrating membrane
- 2.2 Three-dimensional solutions
3 Paraxial form
4 Inhomogeneous Helmholtz equation
5 References
6 External links

Motivation and uses

The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i 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. In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an

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(r,t) is in fact separable:

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

and

$T(t) = e^{i\omega t}. \,$

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

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

and $\omega \ \stackrel{\mathrm{def}}{=}\ kc$ where k is the wave vector and ω is the angular frequency. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves A wave vector is a vector representation of a Wave. The wave vector has magnitude indicating Wavenumber (reciprocal of Wavelength) and the Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency So we can write this as:

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

Note that by the nature of our ansatz for $T,$ $T$ satisfies:

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

for $A = e^{-i\mathbf{k}\cdot\mathbf{r}}.$

We now have Helmholtz's equation for the spatial variable $\mathbf{r}$ and a second-order ordinary differential equation in time. In physics and mathematics an ansatz ( Ger, "anset onset" today "setup" plural Ansätze) is an educated guess that is In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its 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. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints 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 integral transform is any transform T of the following form (Tf(u = \int_{t_1}^{t_2} K(t u\ f(t\ dt In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and In Mathematics, a hyperbolic partial differential equation is usually a second-order Partial differential equation (PDE of the form A u_{xx}

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 (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. Seismology (from Greek grc σεισμός seismos, "earthquake" and grc -λογία -logia) is the scientific study of Earthquakes Acoustics is the interdisciplinary science that deals with the study of Sound, Ultrasound and Infrasound (all mechanical waves in gases liquids and solids

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. In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an

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. Siméon-Denis Poisson (21 June 1781 &ndash 25 April 1840 was a French Mathematician, Geometer, and Physicist. Gabriel Lamé ( July 22, 1795 - May 1, 1870) was a French Mathematician. Rudolf Friedrich Alfred Clebsch ( 19 January 1833 – 7 November 1872) was a German Mathematician who made important contributions The elliptical drumhead was studied by Emile Mathieu, leading to Mathieu's differential equation. Émile Léonard Mathieu ( May 15, 1835, Metz – October 19, 1890, Nancy) was a French Mathematician In Mathematics, the Mathieu functions are certain Special functions useful for treating a variety of interesting problems in applied mathematics including The solvable shapes all correspond to shapes whose dynamical billiard table is integrable, that is, not chaotic. A billiard is a Dynamical system in which a particle alternates between motion in a straight line and Specular reflections off of a boundary 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. Quantum chaos is a branch of Physics which studies how chaotic classical systems (see Dynamical systems and Chaos theory) can be shown to be limits of Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

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

$A_{rr} + \frac{1}{r} A_r + \frac{1}{r^2}A_{\theta\theta} + k^2 A = 0.$

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

ρ 2 J n'' + ρ J n' + (ρ 2 - n 2) J n = 0,

and ρ=kr. In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical 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 coordinates, the solution is:

$A (r, \theta, \phi)= \sum_{n=0}^\infty \sum_{l=0}^\infty \sum_{m=-l}^l ( a_{n l m} j_n ( k r ) + b_{n l m} y_n ( k r ) ) Y ^ m_l ( { \theta,\phi} ) .$

This solution arises from the spatial solution of the wave equation and diffusion equation. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves The diffusion equation is a Partial differential equation which describes density fluctuations in a material undergoing Diffusion. Here $j n (k r)$ and $y n (k r)$ are the spherical Bessel functions, and

$Y^m_l ( {\theta,\phi} )$

are the spherical harmonics (Abramowitz and Stegun, 1964). In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of Note that these forms are general solutions, and require boundary conditions to be specified to be used in any specific case. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints For infinite exterior domains, a radiation condition may also be required (Sommerfeld, 1949). Arnold Sommerfeld defined the condition of radiation for a scalar field satisfying the Helmholtz equation as "the sources must be sources not sinks of

For $a = (x, y, z)$ function $A (a)$ has asymptotics

$A(a)=\frac{e^{i k |a|}}{|a|} f(a/|a|,k,u_0) + o(1/|a|)$ when $a\to\infty$

where function f is called scattering amplitude and $u 0 (a)$ is the value of $A$ at each boundary point $a$ .

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. In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after

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. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Optics, a Gaussian beam is a Beam of Electromagnetic radiation whose transverse Electric field and Intensity ( Irradiance Most lasers emit beams that take this form. A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission.

In the paraxial approximation, the complex magnitude of the electric field 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. In Geometric optics, the paraxial approximation is an Approximation used in ray tracing of light through an optical system (such as a lens) Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can

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

$\bigg| { \partial A \over \partial z } \bigg| \ll | kA |$

and

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

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

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

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. A wave vector is a vector representation of a Wave. The wave vector has magnitude indicating Wavenumber (reciprocal of Wavelength) and the

$\nabla^{2}(A\left( x,y,z \right) e^{-jkz}) + k^2 (A\left( x,y,z \right) e^{-jkz}) = 0$

Expansion and cancellation yields the following:

$\left( {\frac {\partial ^{2}}{\partial {x}^{2}}} + {\frac {\partial ^{2}}{\partial {y}^{2}}} \right)(A\left( x,y,z \right) e^{-jkz}) + \left( {\frac {\partial ^{2}}{\partial {z}^{2}}}A \left( x,y,z \right) \right) {e^{-jkz}}-2\, \left( {\frac {\partial }{\partial z}}A \left( x,y,z \right) \right) jk{e^{-jkz}}=0.$

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

$\nabla^2 A + k^2 A = -f \mbox { in } \mathbb R^n$

where $f:\mathbb R^n\to \mathbb C$ is a given function with compact support, and $n = 1,2,3. In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set$

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. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints Arnold Sommerfeld defined the condition of radiation for a scalar field satisfying the Helmholtz equation as "the sources must be sources not sinks of In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

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). In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and Here, $G$ is the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with $f$ equaling the Dirac delta function, so $G$ satisfies

$\nabla^2 G + k^2 G = -\delta \mbox { in } \mathbb R^n.$

The expression for the Green's function depends on the dimension of the space. In Mathematics, Green's function is a type of function used to solve inhomogeneous Differential equations subject to boundary conditions The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. One has

$G(x) = \frac{ie^{ik|x|}}{2k}$

for $n = 1,$

$G(x) = \frac{i}{4}H^{(1)}_0(k|x|)$

for $n = 2$ , where $H^{(1)}_0$ is a Hankel function, and

$G(x) = \frac{e^{ik|x|}}{4\pi |x|}$

for $n = 3. In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical$

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.
Vibrating Circular Membrane by Sam Blake, The Wolfram Demonstrations Project.

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