|
|
| Line 1: |
Line 1: |
| [[Image:Helmholtz source.png|right|thumb|Two sources of radiation in the plane, given mathematically by a function ''ƒ'' which is zero in the blue region.]]
| | I am Oscar and I totally dig that title. To gather coins is what his family members and him enjoy. For years I've been operating as a payroll clerk. North Dakota is exactly where me and my spouse reside.<br><br>my webpage :: [http://www.gradbible.com/members/astrimulquin/activity/128883/ www.gradbible.com] |
| [[Image:Helmholtz solution.png|right|thumb|The [[real part]] of the resulting field ''A'', ''A'' is the solution to the inhomogeneous Helmholtz equation <math>(\nabla^2 + k^2) A = -f.</math>]]
| |
| The '''Helmholtz equation''', named for [[Hermann von Helmholtz]], is the [[partial differential equation]]
| |
| :<math>\nabla^2 A + k^2 A = 0</math>
| |
| where ∇<sup>2</sup> is the [[Laplace operator|Laplacian]], ''k'' is the [[wavenumber]], and ''A'' is the [[amplitude]].
| |
| | |
| ==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 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]]
| |
| :<math>\left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial{t}^2}\right)u(\mathbf{r},t)=0.</math>
| |
| | |
| Separation of variables begins by assuming that the wave function ''u''('''r''', ''t'') is in fact separable:
| |
| :<math>u(\mathbf{r},t)=A (\mathbf{r}) T(t).</math>
| |
| | |
| Substituting this form into the wave equation, and then simplifying, we obtain the following equation:
| |
| :<math>{\nabla^2 A \over A } = {1 \over c^2 T } { d^2 T \over d t^2 }.</math>
| |
| | |
| Notice the expression on the left-hand side depends only on '''r''', whereas the right-hand 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 a constant value. From this observation, we obtain two equations, one for ''A''('''r'''), the other for ''T''(''t''):
| |
| :<math>{\nabla^2 A \over A } = -k^2 </math>
| |
| and
| |
| :<math> {1 \over c^2 T } { d^2 T \over dt^2 } = -k^2</math>
| |
| where we have chosen, without loss of generality, the expression −''k''<sup>2</sup> for the value of the constant. (It is equally valid to use any constant ''k'' as the separation constant; −''k''<sup>2</sup> is chosen only for convenience in the resulting solutions.)
| |
| | |
| Rearranging the first equation, we obtain the Helmholtz equation:
| |
| :<math>\nabla^2 A + k^2 A = ( \nabla^2 + k^2) A = 0. </math>
| |
| | |
| Likewise, after making the substitution
| |
| :<math> \omega \stackrel{\mathrm{def}}{=} kc </math>
| |
| the second equation becomes
| |
| :<math>\frac{d^2{T}}{d{t}^2} + \omega^2T = \left( { d^2 \over dt^2 } + \omega^2 \right) T = 0,</math>
| |
| where ''k'' is the [[wave vector]] and ''ω'' is the [[angular frequency]].
| |
| | |
| We now have Helmholtz's equation for the spatial variable '''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 transform|Laplace]] or [[Fourier transform]], are often used to transform a [[Hyperbolic partial differential equation|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
| |
| | |
| :<math> ( \nabla^2 + k^2 ) A = 0 </math>
| |
| | |
| 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 [[Émile Léonard Mathieu|Émile Mathieu]], leading to [[Mathieu's differential equation]]. The solvable shapes all correspond to shapes whose [[dynamical billiard table]] is [[Integrable system|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]] (see [[Schrödinger equation]]).
| |
| | |
| 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
| |
| | |
| :<math> A_{rr} + \frac{1}{r} A_r + \frac{1}{r^2}A_{\theta\theta} + k^2 A = 0. </math>
| |
| | |
| We may impose the boundary condition that ''A'' vanish if ''r'' = ''a''; thus
| |
| | |
| :<math> A(a,\theta) = 0. \,</math>
| |
| | |
| The method of separation of variables leads to trial solutions of the form
| |
| | |
| :<math> A(r,\theta) = R(r)\Theta(\theta), \,</math>
| |
| | |
| where Θ must be periodic of period 2π. This leads to | |
| | |
| :<math> \Theta'' +n^2 \Theta =0, \,</math>
| |
| | |
| and | |
| :<math> r^2 R'' + r R' + r^2 k^2 R - n^2 R=0. \,</math>
| |
| | |
| It follows from the periodicity condition that
| |
| | |
| :<math> \Theta = \alpha \cos n\theta + \beta \sin n\theta, \,</math>
| |
| | |
| and that ''n'' must be an integer. The radial component ''R'' has the form
| |
| | |
| :<math> R(r) = \gamma J_n(\rho), \,</math>
| |
| | |
| where the [[Bessel function]] ''J<sub>n</sub>''(ρ) satisfies Bessel's equation
| |
| | |
| :<math> \rho^2 J_n'' + \rho J_n' +(\rho^2 - n^2)J_n =0, </math>
| |
| | |
| and ''ρ'' = ''kr''. The radial function ''J''<sub>''n''</sub> has infinitely many roots for each value of ''n'', denoted by ''ρ''<sub>''m'',''n''</sub>. The boundary condition that ''A'' vanishes where ''r'' = ''a'' will be satisfied if the corresponding wavenumbers are given by
| |
| | |
| :<math> k_{m,n} = \frac{1}{a} \rho_{m,n}. \,</math>
| |
| | |
| The general solution ''A'' then takes the form of a doubly infinite sum of terms involving products of
| |
| | |
| :<math> \sin(n\theta) \, \hbox{or} \, \cos(n\theta), \text{ and } J_n(k_{m,n}r). </math>
| |
| | |
| These solutions are the modes of [[Vibrations of a circular drum|vibration of a circular drumhead]].
| |
| | |
| ===Three-dimensional solutions===
| |
| | |
| In spherical coordinates, the solution is:
| |
| | |
| : <math> A (r, \theta, \varphi)= \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell ( a_{\ell m} j_\ell ( k r ) + b_{\ell m} y_\ell ( k r ) ) Y ^ m_\ell ( { \theta,\varphi} ) .</math>
| |
| | |
| This solution arises from the spatial solution of the [[wave equation]] and [[diffusion equation]]. Here <math> j_\ell ( k r ) </math> and <math> y_\ell ( k r )</math> are the [[spherical Bessel function]]s, and
| |
| | |
| :<math> Y^m_\ell ( {\theta,\varphi} )</math>
| |
| | |
| 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).
| |
| | |
| For <math> \mathbf{r_0}=(x,y,z)</math> function <math>A(r_0)</math> has asymptotics
| |
| | |
| : <math>A(r_0)=\frac{e^{i k r_0}}{r_0} f(\mathbf{r}_0/r_0,k,u_0) + o(1/r_0)\text{ as } r_0\to\infty</math>
| |
| | |
| where function ƒ is called scattering amplitude and <math> u_0(r_0) </math> is the value of ''A'' at each boundary point <math> r_0 </math>.
| |
| | |
| ==Paraxial approximation==<!-- This section is linked from [[Gaussian beam]] -->
| |
| {{further|Slowly varying envelope approximation}}
| |
| The paraxial approximation of the Helmholtz equation is:<ref>{{cite book |title=Introduction to Fourier Optics |edition=2nd |author=J. W. Goodman |pages=61–62 }}</ref>
| |
| | |
| :<math>\nabla_{\perp}^2 A + 2ik\frac{\partial A}{\partial z} = 0,</math>
| |
| | |
| where <math>\textstyle \nabla_{\perp}^2 \stackrel{\mathrm {def}}{=} \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2 }</math> is the transverse part of the [[Laplace operator|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 [[parabola|paraboloidal]] waves or [[Gaussian beam]]s. Most [[laser]]s emit beams that take this form.
| |
| | |
| In the [[paraxial approximation]], the [[complex number|complex]] [[magnitude (mathematics)|magnitude]] of the [[electric field]] ''E'' becomes
| |
| | |
| :<math>E(\mathbf{r}) = A(\mathbf{r}) e^{ikz} </math>
| |
| | |
| 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:
| |
| | |
| :<math> \bigg| { \partial A \over \partial z } \bigg| \ll | kA | </math>
| |
| and
| |
| :<math> \bigg| { \partial^2 A \over \partial z^2 } \bigg| \ll | k^2 A |. </math>
| |
| | |
| 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
| |
| | |
| :<math>\sin(\theta) \approx \theta \qquad \mathrm{and} \qquad \tan(\theta) \approx \theta. </math>
| |
| | |
| 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.
| |
| | |
| :<math>\nabla^{2}(A\left( x,y,z \right) e^{ikz}) + k^2 (A\left( x,y,z \right) e^{ikz}) = 0</math>
| |
| | |
| Expansion and cancellation yields the following:
| |
| | |
| :<math>\left( {\frac {\partial ^{2}}{\partial {x}^{2}}} + {\frac {\partial ^{2}}{\partial {y}^{2}}} \right)(A\left( x,y,z \right) e^{ikz}) + \left( {\frac {\partial ^{2}}{\partial {z}^{2}}}A \left( x,y,z \right) \right) {e^{ikz}}+2\, \left( {\frac {\partial }{\partial z}}A \left( x,y,z \right) \right) ik{e^{ikz}}=0.</math> | |
| | |
| Because of the paraxial inequalities stated above, the ∂<sup>2</sup>A/∂z<sup>2</sup> factor is neglected in comparison with the ∂A/∂z factor. The yields the Paraxial Helmholtz equation.
| |
| | |
| There is even a topic by name "Helmholtz Optics" based on the equation named in his honour.
| |
| <ref>
| |
| [http://scholar.google.com/citations?user=hTKwGHoAAAAJ&hl=en Kurt Bernardo Wolf] and Evgenii V. Kurmyshev, | |
| [http://link.aps.org/doi/10.1103/PhysRevA.47.3365 Squeezed states in Helmholtz optics], [[Physical Review]] A 47, 3365–3370 (1993).
| |
| </ref>
| |
| <ref>
| |
| [http://inspirehep.net/author/S.A.Khan.5/ Sameen Ahmed Khan],
| |
| [http://dx.doi.org/10.1007/s10773-005-1488-0 Wavelength-dependent modifications in Helmholtz Optics],
| |
| [[International Journal of Theoretical Physics]], 44(1), 95http://www.maa.org/programs/maa-awards/writing-awards/can-one-hear-the-shape-of-a-drum125 (January 2005).
| |
| </ref>
| |
| <ref>
| |
| [http://scholar.google.com/citations?user=hZvL5eYAAAAJ&hl Sameen Ahmed Khan],
| |
| [http://www.osa-opn.org/Content/ViewFile.aspx?id=12977 A Profile of Hermann von Helmholtz],
| |
| [[Optics and Photonics News|Optics & Photonics News]], Vol. 21, No. 7, pp. 7 (July/August 2010).
| |
| </ref>
| |
| | |
| ==Inhomogeneous Helmholtz equation==
| |
| | |
| The '''inhomogeneous Helmholtz equation''' is the equation
| |
| | |
| : <math>\nabla^2 A(x) + k^2 A(x) = -f(x) \mbox { in } \mathbb R^n</math>
| |
| | |
| where ''ƒ'' : '''R'''<sup>''n''</sup> → '''C''' is a given function with [[compact support]], and ''n'' = 1, 2, 3. This equation is very similar to the [[screened Poisson equation]], and would be identical if the plus sign (in front of the ''k'' term) is switched to a minus sign.
| |
| | |
| In order to solve this equation uniquely, one needs to specify a [[boundary condition]] at infinity, which is typically the [[Sommerfeld radiation condition]]
| |
| | |
| : <math>\lim_{r \to \infty} r^{\frac{n-1}{2}} \left( \frac{\partial}{\partial r} - ik \right) A(r \hat {x}) = 0</math>
| |
| | |
| uniformly in <math>\hat {x}</math> with <math>|\hat {x}|=1</math>, where the vertical bars denote the [[Euclidean norm]].
| |
| | |
| With this condition, the solution to the inhomogeneous Helmholtz equation is the [[convolution]]
| |
| | |
| : <math>A(x)=(G*f)(x)=\int\limits_{\mathbb R^n}\! G(x-y)f(y)\,dy</math>
| |
| | |
| (notice this integral is actually over a finite region, since <math>f</math> has compact support). Here, <math>G</math> is the [[Green's function]] of this equation, that is, the solution to the inhomogeneous Helmholtz equation with ƒ equaling the [[Dirac delta function]], so ''G'' satisfies
| |
| | |
| : <math>\nabla^2 G(x) + k^2 G(x) = -\delta(x) \text{ in }\mathbb R^n. \, </math>
| |
| | |
| The expression for the Green's function depends on the dimension <math>n</math> of the space. One has
| |
| | |
| : <math>G(x) = \frac{ie^{ik|x|}}{2k}</math>
| |
| | |
| for ''n'' = 1,
| |
| | |
| : <math>G(x) = \frac{i}{4}H^{(1)}_0(k|x|)</math>
| |
| | |
| for ''n'' = 2, where <math>H^{(1)}_0</math> is a [[Bessel_function#Hankel_functions_:_H.CE.B1|Hankel function]], and
| |
| | |
| : <math>G(x) = \frac{e^{ik|x|}}{4\pi |x|}</math>
| |
| | |
| for ''n'' = 3. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for <math>|x| \to \infty </math>.
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==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.
| |
| *{{cite book | title = Fundamentals of Photonics | author = Bahaa E. A. Saleh and Malvin Carl Teich | publisher = John Wiley & Sons | location = New York | year = 1991 | 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.
| |
| *{{cite book
| |
| | last = Howe
| |
| | first = M. S.
| |
| | title = Acoustics of fluid-structure interactions
| |
| | publisher = Cambridge; New York: Cambridge University Press
| |
| | year = 1998
| |
| | pages =
| |
| | isbn = 0-521-63320-6
| |
| }}
| |
| | |
| ==External links==
| |
| * [http://eqworld.ipmnet.ru/en/solutions/lpde/lpde303.pdf Helmholtz Equation] at EqWorld: The World of Mathematical Equations.
| |
| * {{springer|title=Helmholtz equation|id=p/h046920}}
| |
| * [http://demonstrations.wolfram.com/VibratingCircularMembrane/ Vibrating Circular Membrane] by Sam Blake, [[The Wolfram Demonstrations Project]].
| |
| * [http://www.sbfisica.org.br/rbef/pdf/351304.pdf Green's functions for the wave, Helmholtz and Poisson equations in a two-dimensional boundless domain]
| |
| | |
| [[Category:Waves]]
| |
| [[Category:Elliptic partial differential equations]]
| |