View source for Parabolic-equation method

A method for the approximate solution of a high-frequency diffraction problem (see [[Diffraction, mathematical theory of|Diffraction, mathematical theory of]]). As a rule it is necessary to resort to the parabolic-equation method for the determination of a wave field in those domains where the [[Ray method|ray method]] cannot be employed because the field of rays suffers from a singularity in one sense or another. Consider, for example, the incidence of a plane wave on a perfectly reflecting convex body. The wave phenomenon is described by the Helmholtz equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p0712001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

Here the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p0712002.png" /> is located outside the bounded convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p0712003.png" />, on the boundary of which the Dirichlet boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p0712004.png" /> is satisfied. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p0712005.png" /> has everywhere positive curvature. The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p0712006.png" /> can be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p0712007.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p0712008.png" /> satisfies the [[Radiation conditions|radiation conditions]]. The solution of such a problem exists and is unique.

In the high-frequency range (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p0712009.png" />  "large" ) it is important to construct a formal small wavelength solution to this problem (i.e. roughly speaking, an expansion that formally satisfies all the conditions of the problem, while successive terms are of increasingly high order of smallness as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120010.png" />). It is possible to show that, in the case under consideration, a formal solution will be an asymptotic expansion of the classical solution.

The ray method allows one to construct the unknown small wavelength expansion everywhere, except in the shadow zone (see Fig.). Rays of the incident wave shadow penumbra region

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071200a.gif" />

Figure: p071200a

The expression for the wave field constructed with the aid of the ray method loses smoothness on the boundary between shadow zone and illuminated zone (the half-lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120012.png" /> in the figure). In a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120013.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120014.png" />) the small wavelength asymptotic expansion of the wavefield is no longer given by the ray formulas. A neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120015.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120016.png" />) is usually called a penumbra region.

Of key importance for the construction of a formal solution to the problem given above is an examination of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120018.png" /> where the rays of the incident wave are the boundary curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120019.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120020.png" /> is taken as the origin of coordinates, and the positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120021.png" />-axis separates the shadow zone and the illuminated zone.

One introduces new coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120023.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120024.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120025.png" /> is determined by its distance along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120026.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120027.png" />. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120028.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120029.png" />) in the shadow zone (respectively, in the illuminated zone). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120030.png" />, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120031.png" /> is characterized by its distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120032.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120033.png" /> and the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120034.png" /> of the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120036.png" />. With respect to the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120038.png" />, the Helmholtz equation takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120040.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120041.png" /> is the radius of curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120042.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120043.png" />). Through the MacLaurin series for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120044.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120045.png" /></td> </tr></table>

it is possible to replace all the coefficients of equation (2) by their formal expansions in powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120047.png" />. Substituting the coordinates

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120048.png" /></td> </tr></table>

in equations (2), one arrives at

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

By equating to zero the coefficients of the successive powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120050.png" />, one obtains the following sequence of recurrence relations (which is typical for the method of boundary layers, cf. also [[Boundary-layer theory|Boundary-layer theory]]):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120051.png" /></td> </tr></table>

Here the first equation is a  "parabolic"  equation, leading to the name parabolic-equation method:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

Basically, equation (4) is an equation of Schrödinger type. The coefficients in the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120053.png" /> are polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120055.png" />. For the formal satisfaction of the boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120056.png" /> it is sufficient to require that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120057.png" />. The other boundary conditions require that for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120059.png" />, the sequence (3) formally becomes the expansion of the ray method. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120060.png" /> it is possible to derive an explicit formula (the so-called Fock formula), having the form of a Fourier integral:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120061.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071200/p07120062.png" /> is expressed comparatively simply in terms of [[Airy functions|Airy functions]]. Methods of matching asymptotics enable one to obtain formulas for the wave field in all shadow and penumbra regions.

In the case of creeping waves and of whispering gallery modes, corresponding  "parabolic"  equations have been deduced and their solutions are expressed in terms of Airy functions. The development of the theory of lasers made it necessary to consider waves concentrated in a neighbourhood of isolated rays. Selecting the corresponding phase coefficient and carrying out further constructions analogous to the constructions of the boundary-layer method, a  "parabolic"  equation is obtained, in terms of the solution of which the wave field is expressed in a first approximation. In this case the  "parabolic"  equation will be the Schrödinger equation with a quadratic potential. The parabolic-equation method finds application also in the calculation of the wave field in shallow water, in statistical non-homogeneous media and in many other problems. Analogues of the parabolic-equation method are used in the theory of non-linear waves.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. [V.A. Fok] Fock,   "Electromagnetic diffraction and propagation problems" , Pergamon  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.M. Babich,   V.S. Buldyrev,   "Asymptotic methods in the diffraction of short waves" , Moscow  (1972)  (In Russian)  (Translation forthcoming: Springer)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.M. [V.M. Babich] Babič,   N.Y. [N.Ya. Kirpichnikova] Kirpičnikova,   "The boundary-layer method in diffraction problems" , Springer  (1979)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Maruvitz,   "Radiation and scattering of waves" , Prentice-Hall  (1973)</TD></TR></table>