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
Separation of variables begins by assuming that the wave function u(r, t) is in fact separable:
Substituting this form into the wave equation and then simplifying, we obtain the following equation:
Notice that the expression on the left side depends only on r, whereas the right 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 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 A(r), the other for T(t):
where we have chosen, without loss of generality, the expression −k2 for the value of the constant. (It is equally valid to use any constant k as the separation constant; −k2 is chosen only for convenience in the resulting solutions.)
Rearranging the first equation, we obtain the (homogeneous) Helmholtz equation:
Likewise, after making the substitution ω = kc, where k is the wave number, and ω is the angular frequency (assuming a monochromatic field), the second equation becomes
Solving the Helmholtz equation using separation of variables
edit
The solution to the spatial Helmholtz equation:
can be obtained for simple geometries using separation of variables.
Vibrating membrane
edit
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 a, then it is appropriate to introduce polar coordinates r and θ. The Helmholtz equation takes the form
We may impose the boundary condition that A vanishes if r = a; thus
the method of separation of variables leads to trial solutions of the form
where Θ must be periodic of period 2 π . This leads to
It follows from the periodicity condition that
and that n must be an integer. The radial component R has the form
where the Bessel functionJn(ρ) satisfies Bessel's equation
and z = k r. The radial function Jn has infinitely many roots for each value of n , denoted by ρm,n. The boundary condition that A vanishes where r = a will be satisfied if the corresponding wavenumbers are given by
Writing r0 = (x, y, z) function A(r0) has asymptotics
where function f is called scattering amplitude and u0(r0) is the value of A at each boundary point r0.
Three-dimensional solutions given the function on a 2-dimensional plane
edit
Given a 2-dimensional plane where A is known, the solution to the Helmholtz equation is given by:[4]
where
is the solution at the 2-dimensional plane,
As z approaches zero, all contributions from the integral vanish except for r = 0 . Thus up to a numerical factor, which can be verified to be 1 by transforming the integral to polar coordinates
This solution is important in diffraction theory, e.g. in deriving Fresnel diffraction.
Paraxial approximation
edit
In the paraxial approximation of the Helmholtz equation,[5] the complex amplitudeA is expressed as
where u represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Then under a suitable assumption, u approximately solves
where 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 z derivative of the amplitude function u is a slowly varying function of z:
This condition is equivalent to saying that the angle θ between the wave vectork and the optical axis z is small: θ ≪ 1.
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:
Expansion and cancellation yields the following:
Because of the paraxial inequality stated above, the ∂2u/∂z2 term is neglected in comparison with the k·∂u/∂z term. This yields the paraxial Helmholtz equation. Substituting u(r) = A(r) e−ikz then gives the paraxial equation for the original complex amplitude A:
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 (∇2 + k2) A = −f.
The inhomogeneous Helmholtz equation is the equation
where ƒ : Rn → C is a 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) were switched to a minus sign.
in spatial dimensions, for all angles (i.e. any value of ). Here where are the coordinates of the vector .
With this condition, the solution to the inhomogeneous Helmholtz equation is
(notice this integral is actually over a finite region, since f has compact support). 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
The expression for the Green's function depends on the dimension n of the space. One has
for n = 1,
for n = 2, where H(1) 0 is a Hankel function, and
for n = 3. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for |x| → ∞.
Blanche, Pierre-Alexandre (2014). Field Guide to Holography. Bellingham, Washington USA: SPIE-International Society for Optical Engineering. ISBN 978-0-8194-9957-8.
Engquist, Björn; Zhao, Hongkai (2018). "Approximate Separability of the Green's Function of the Helmholtz Equation in the High Frequency Limit". Communications on Pure and Applied Mathematics. 71 (11): 2220–2274. doi:10.1002/cpa.21755. ISSN 0010-3640.
Goodman, Joseph W. (1996). Introduction to Fourier Optics. New York: McGraw-Hill Science, Engineering & Mathematics. ISBN 978-0-07-024254-8.
Grella, R (1982). "Fresnel propagation and diffraction and paraxial wave equation". Journal of Optics. 13 (6): 367–374. doi:10.1088/0150-536X/13/6/006. ISSN 0150-536X.
Mehrabkhani, Soheil; Schneider, Thomas (2017). "Is the Rayleigh-Sommerfeld diffraction always an exact reference for high speed diffraction algorithms?". Optics Express. 25 (24): 30229-30240. arXiv:1709.09727. doi:10.1364/OE.25.030229. ISSN 1094-4087.
Noble, Ben (1958). Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. New York, N.Y: Taylor & Francis US. ISBN 978-0-8284-0332-0.
Further reading
edit
Abramowitz, Milton; Stegun, Irene, eds. (1964). Handbook of Mathematical functions with Formulas, Graphs and Mathematical Tables. New York: Dover Publications. ISBN 978-0-486-61272-0.
Riley, K. F.; Hobson, M. P.; Bence, S. J. (2002). "Chapter 19". Mathematical methods for physics and engineering. New York: Cambridge University Press. ISBN 978-0-521-89067-0.
Riley, K. F. (2002). "Chapter 16". Mathematical Methods for Scientists and Engineers. Sausalito, California: University Science Books. ISBN 978-1-891389-24-5.
Saleh, Bahaa E. A.; Teich, Malvin Carl (1991). "Chapter 3". Fundamentals of Photonics. Wiley Series in Pure and Applied Optics. New York: John Wiley & Sons. pp. 80–107. ISBN 978-0-471-83965-1.
Sommerfeld, Arnold (1949). "Chapter 16". Partial Differential Equations in Physics. New York: Academic Press. ISBN 978-0126546569.
Howe, M. S. (1998). Acoustics of fluid-structure interactions. New York: Cambridge University Press. ISBN 978-0-521-63320-8.
External links
edit
Helmholtz Equation at EqWorld: The World of Mathematical Equations.