Английская Википедия:Helmholtz equation
Шаблон:Short description In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the linear partial differential equation: <math display="block">\nabla^2 f = -k^2 f,</math> where Шаблон:Math is the Laplace operator, Шаблон:Math is the eigenvalue, and Шаблон:Mvar is the (eigen)function. When the equation is applied to waves, Шаблон:Mvar is known as the wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle.
In optics, the Helmholtz equation is the wave equation for the electric field.[1]
The equation is named after Hermann von Helmholtz, who studied it in 1860.[2]
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 a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis.
For example, consider the wave equation <math display="block">\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 Шаблон:Math is in fact separable: <math display="block">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 display="block">\frac{\nabla^2 A}{A} = \frac{1}{c^2 T} \frac{\mathrm{d}^2 T}{\mathrm{d} t^2}.</math>
Notice that the expression on the left side depends only on Шаблон:Math, whereas the right expression depends only on Шаблон:Mvar. As a result, this equation is valid in the general case if and only if both sides of the equation are equal to the same constant value. This argument is key in the technique of solving linear partial differential equations by separation of variables. From this observation, we obtain two equations, one for Шаблон:Math, the other for Шаблон:Math <math display="block">\frac{\nabla^2 A}{A} = -k^2</math> <math display="block">\frac{1}{c^2 T} \frac{\mathrm{d}^2 T}{\mathrm{d}t^2} = -k^2,</math>
where we have chosen, without loss of generality, the expression Шаблон:Math for the value of the constant. (It is equally valid to use any constant Шаблон:Mvar as the separation constant; Шаблон:Math is chosen only for convenience in the resulting solutions.)
Rearranging the first equation, we obtain the Helmholtz equation: <math display="block">\nabla^2 A + k^2 A = (\nabla^2 + k^2) A = 0.</math>
Likewise, after making the substitution Шаблон:Math, where Шаблон:Mvar is the wave number, and Шаблон:Mvar is the angular frequency (assuming a monochromatic field), the second equation becomes
<math display="block">\frac{\mathrm{d}^2 T}{\mathrm{d}t^2} + \omega^2T = \left( \frac{\mathrm{d}^2}{\mathrm{d}t^2} + \omega^2 \right) T = 0.</math>
We now have Helmholtz's equation for the spatial variable Шаблон:Math and a second-order ordinary differential equation in time. The solution in time will be a linear combination of sine and cosine functions, whose exact form is determined by initial conditions, 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 solution to the spatial Helmholtz equation: <math display="block"> \nabla^2 A = -k^2 A </math> can be obtained for simple geometries 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 Mathieu, leading to Mathieu's differential 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).
If the domain is a circle of radius Шаблон:Mvar, then it is appropriate to introduce polar coordinates Шаблон:Mvar and Шаблон:Mvar. The Helmholtz equation takes the form <math display="block">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 Шаблон:Mvar vanishes if Шаблон:Math; thus <math display="block">A(a,\theta) = 0.</math>
the method of separation of variables leads to trial solutions of the form <math display="block">A(r,\theta) = R(r)\Theta(\theta),</math> where Шаблон:Math must be periodic of period Шаблон:Math. This leads to
<math display="block">\Theta +n^2 \Theta =0,</math> <math display="block"> r^2 R + r R' + r^2 k^2 R - n^2 R=0.</math>
It follows from the periodicity condition that <math display="block"> \Theta = \alpha \cos n\theta + \beta \sin n\theta,</math> and that Шаблон:Mvar must be an integer. The radial component Шаблон:Mvar has the form <math display="block"> R(r) = \gamma J_n(\rho), </math> where the Bessel function Шаблон:Math satisfies Bessel's equation <math display="block"> \rho^2 J_n + \rho J_n' +(\rho^2 - n^2)J_n =0, </math> and Шаблон:Math. The radial function Шаблон:Math has infinitely many roots for each value of Шаблон:Mvar, denoted by Шаблон:Math. The boundary condition that Шаблон:Mvar vanishes where Шаблон:Math will be satisfied if the corresponding wavenumbers are given by <math display="block">k_{m,n} = \frac{1}{a} \rho_{m,n}.</math>
The general solution Шаблон:Mvar then takes the form of a generalized Fourier series of terms involving products of Шаблон:Math and the sine (or cosine) of Шаблон:Math. These solutions are the modes of vibration of a circular drumhead.
Three-dimensional solutions
In spherical coordinates, the solution is:
<math display="block"> A (r, \theta, \varphi)= \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell \left( a_{\ell m} j_\ell ( k r ) + b_{\ell m} y_\ell(kr) \right) Y^m_\ell (\theta,\varphi) .</math>
This solution arises from the spatial solution of the wave equation and diffusion equation. Here Шаблон:Math and Шаблон:Math are the spherical Bessel functions, and Шаблон: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).
Writing Шаблон:Math function Шаблон:Math has asymptotics <math display="block">A(r_0)=\frac{e^{i k r_0}}{r_0} f\left(\frac{\mathbf{r}_0}{r_0},k,u_0\right) + o\left(\frac 1 {r_0}\right)\text{ as } r_0\to\infty</math>
where function Шаблон:Mvar is called scattering amplitude and Шаблон:Math is the value of Шаблон:Mvar at each boundary point Шаблон:Math
Three-dimensional solutions given the function on a 2-dimensional plane
Given a 2-dimensional plane where A is known, the solution to the Helmholtz equation is given by:[3] <math display=block> A(x, y, z) = -\frac{1}{2 \pi} \iint_{-\infty}^{+\infty} A'(x', y') \frac{e^{ikr}}{r} \frac{z}{r} \left(ik-\frac{1}{r}\right) \,dx'dy', </math>
where
- <math>A'(x', y')</math> is the solution at the 2-dimensional plane,
- <math>r = \sqrt{(x - x')^2 + (y - y')^2 + z^2},</math>
As z approaches zero, all contributions from the integral vanish except for r=0. Thus <math>A(x, y, 0)=A'(x,y)</math> up to a numerical factor, which can be verified to be 1 by transforming the integral to polar coordinates <math>(\rho, \theta)</math>.
This solution is important in diffraction theory, e.g. in deriving Fresnel diffraction.
Paraxial approximation
Шаблон:Further In the paraxial approximation of the Helmholtz equation,[4] the complex amplitude Шаблон:Mvar is expressed as <math display="block">A(\mathbf{r}) = u(\mathbf{r}) e^{ikz} </math> where Шаблон:Mvar represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Then under a suitable assumption, Шаблон:Mvar approximately solves <math display="block">\nabla_{\perp}^2 u + 2ik\frac{\partial u}{\partial z} = 0,</math> where <math display="inline">\nabla_\perp^2 \overset{\text{ def }}{=} \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}</math> is the transverse part 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.
The assumption under which the paraxial approximation is valid is that the Шаблон:Mvar derivative of the amplitude function Шаблон:Mvar is a slowly varying function of Шаблон:Mvar:
<math display="block"> \left| \frac{ \partial^2 u }{ \partial z^2 } \right| \ll \left| k \frac{\partial u}{\partial z} \right| .</math>
This condition is equivalent to saying that the angle Шаблон:Mvar between the wave vector Шаблон:Math and the optical axis Шаблон:Mvar is small: Шаблон:Math.
The paraxial form of the Helmholtz equation is found by substituting the above-stated expression for the complex amplitude into the general form of the Helmholtz equation as follows:
<math display="block">\nabla^{2}(u\left( x,y,z \right) e^{ikz}) + k^2 u\left( x,y,z \right) e^{ikz} = 0.</math>
Expansion and cancellation yields the following:
<math display="block">\left( \frac {\partial^2}{\partial x^2} + \frac {\partial^2}{\partial y^2} \right) u(x,y,z) e^{ikz} + \left( \frac {\partial^2}{\partial z^2} u (x,y,z) \right) e^{ikz} + 2 \left( \frac \partial {\partial z} u(x,y,z) \right) ik{e^{ikz}}=0.</math>
Because of the paraxial inequality stated above, the Шаблон:Math term is neglected in comparison with the Шаблон:Math term. This yields the paraxial Helmholtz equation. Substituting Шаблон:Math then gives the paraxial equation for the original complex amplitude Шаблон:Mvar:
<math display="block">\nabla_{\perp}^2 A + 2ik\frac{\partial A}{\partial z} + 2k^2A = 0.</math>
The Fresnel diffraction integral is an exact solution to the paraxial Helmholtz equation.[5]
Inhomogeneous Helmholtz equation
Шаблон:Multiple image The inhomogeneous Helmholtz equation is the equation <math display="block">\nabla^2 A(\mathbf{x}) + k^2 A(\mathbf{x}) = -f(\mathbf{x}) \ \text { in } \R^n,</math> where Шаблон:Math is a function with compact support, and Шаблон:Math This equation is very similar to the screened Poisson equation, and would be identical if the plus sign (in front of the Шаблон:Mvar term) were 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 display="block">\lim_{r \to \infty} r^{\frac{n-1}{2}} \left( \frac{\partial}{\partial r} - ik \right) A(\mathbf{x}) = 0</math>
in <math>n</math> spatial dimensions, for all angles (i.e. any value of <math>\theta, \phi</math>). Here <math>r = \sqrt{\sum_{i=1}^n x_i^2} </math> where <math>x_i</math> are the coordinates of the vector <math>\mathbf{x}</math>.
With this condition, the solution to the inhomogeneous Helmholtz equation is
<math display="block">A(\mathbf{x})=\int_{\R^n}\! G(\mathbf{x},\mathbf{x'})f(\mathbf{x'})\,\mathrm{d}\mathbf{x'}</math>
(notice this integral is actually over a finite region, since Шаблон:Mvar has compact support). Here, Шаблон:Mvar is the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with Шаблон:Math equaling the Dirac delta function, so Шаблон:Mvar satisfies
<math display="block">\nabla^2 G(\mathbf{x},\mathbf{x'}) + k^2 G(\mathbf{x},\mathbf{x'}) = -\delta(\mathbf{x},\mathbf{x'}) \in \R^n. </math>
The expression for the Green's function depends on the dimension Шаблон:Mvar of the space. One has <math display="block">G(x,x') = \frac{ie^{ik|x - x'|}}{2k}</math> for Шаблон:Math,
<math display="block">G(\mathbf{x},\mathbf{x'}) = \frac{i}{4}H^{(1)}_0(k|\mathbf{x}-\mathbf{x'}|)</math> for Шаблон:Math, where Шаблон:Math is a Hankel function, and <math display="block">G(\mathbf{x},\mathbf{x'}) = \frac{e^{ik|\mathbf{x}-\mathbf{x'}|}}{4\pi |\mathbf{x}-\mathbf{x'}|}</math> for Шаблон:Math. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for Шаблон:Math.
Finally, for general n,
<math display="block">G(\mathbf{x},\mathbf{x'}) = c_d k^p \frac{H_p^{(1)}(k|\mathbf{x}-\mathbf{x'}|)}{|\mathbf{x}-\mathbf{x'}|^p}</math>
where <math> p = \frac{n - 2}{2} </math> and <math>c_d = \frac{1}{2i(2\pi)^p} </math>.[6]
See also
- Laplace's equation (a particular case of the Helmholtz equation)
- Weyl expansion
Notes
References
External links
- Helmholtz Equation at EqWorld: The World of Mathematical Equations.
- Шаблон:Springer
- Vibrating Circular Membrane by Sam Blake, The Wolfram Demonstrations Project.
- Green's functions for the wave, Helmholtz and Poisson equations in a two-dimensional boundless domain
- ↑ Шаблон:Cite book
- ↑ Helmholtz Equation, from the Encyclopedia of Mathematics.
- ↑ Mehrabkhani, S., & Schneider, T. (2017). Is the Rayleigh-Sommerfeld diffraction always an exact reference for high speed diffraction algorithms?. Optics express, 25(24), 30229-30240.
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
