paraxial wave equation

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11: 30.10 Series and Integrals

… ►Integrals and integral equations for ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). …

12: 29.11 Lamé Wave Equation

§29.11 Lamé Wave Equation

►The Lamé (or ellipsoidal) wave equation is given by …In the case $\omega =0$, (29.11.1) reduces to Lamé’s equation (29.2.1). …

13: 13.28 Physical Applications

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§13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation ${\nabla }^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta }\cos \varphi$, $y=2\sqrt{\xi \eta }\sin \varphi$, $z=\xi -\eta$, can be solved via separation of variables $w=f_1(\xi )f_2(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi }$, where …and $V_{\kappa ,\mu }^{(j)}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). …

14: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). …The agreement of these solutions with two-dimensional surface water waves in shallow water was considered in Hammack et al. (1989, 1995).

15: 30.6 Functions of Complex Argument

§30.6 Functions of Complex Argument

►The solutions ►

${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),$

… ►For results for Equation (30.2.1) with complex parameters see Meixner and Schäfke (1954).

16: 30.17 Tables

§30.17 Tables

… ►

Flammer (1957) includes 18 tables of eigenvalues, expansion coefficients, spheroidal wave functions, and other related quantities. Precision varies between 4S and 10S.

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Hanish et al. (1970) gives ${\lambda }_n^m\left({\gamma }^2\right)$ and $S_n^{m(j)}(z,\gamma )$, $j=1,2$, and their first derivatives, for $0\le m\le 2$, $m\le n\le m+49$, $-1600\le {\gamma }^2\le 1600$. The range of $z$ is given by $1\le z\le 10$ if ${\gamma }^2>0$, or $z=-\mathrm{i}\xi$, $0\le \xi \le 2$ if . Precision is 18S.

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EraŠevskaja et al. (1973, 1976) gives $S^{m(j)}(\mathrm{i}y,-\mathrm{i}c)$, $S^{m(j)}(z,\gamma )$ and their first derivatives for $j=1,2$, $0.5\le c\le 8$, $y=0,0.5,1,1.5$, $0.5\le \gamma \le 8$, $z=1.01,1.1,1.4,1.8$. Precision is 15S.

… ►

Zhang and Jin (1996) includes 24 tables of eigenvalues, spheroidal wave functions and their derivatives. Precision varies between 6S and 8S.

…

17: 21.9 Integrable Equations

… ►Typical examples of such equations are the Korteweg–de Vries equation …and the nonlinear Schrödinger equations … ►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)): …Here $x$ and $y$ are spatial variables, $t$ is time, and $u(x,y,t)$ is the elevation of the surface wave. … …

18: 30.4 Functions of the First Kind

§30.4 Functions of the First Kind

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§30.4(i) Definitions

… ►If $\gamma =0$, ${\mathrm{𝖯𝗌}}_n^m(x,0)$ reduces to the Ferrers function ${𝖯}_n^m\left(x\right)$: … ►

§30.4(ii) Elementary Properties

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§30.4(iv) Orthogonality

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19: 9.16 Physical Applications

… ►The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. …Within classical physics, they appear prominently in physical optics, electromagnetism, radiative transfer, fluid mechanics, and nonlinear wave propagation. Examples dealing with the propagation of light and with radiation of electromagnetic waves are given in Landau and Lifshitz (1962). … ►Airy functions play a prominent role in problems defined by nonlinear wave equations. These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …

20: Bibliography U

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On the transformation group of the second Painlevé equation. Nagoya Math. J. 157, pp. 15–46.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.7(viii).

Encodings:

BibTeX

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On Kelvin’s ship-wave pattern. J. Fluid Mech. 8 (3), pp. 418–431.

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External Links:

Document, MathReview (R. C. MacCamy), ZentralBlatt

Referenced by:

§36.13.

Encodings:

BibTeX

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Ship Hydrodynamics, Water Waves and Asymptotics. Collected works of F. Ursell, 1946-1992, Vol. 2, World Scientific, Singapore.

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External Links:

ZentralBlatt

Referenced by:

§36.13.

Encodings:

BibTeX

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Integral equations for paraboloidal wave functions. I. Quart. J. Math. Oxford Ser. (2) 15, pp. 309–315.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.31(iii), §28.31(iii).

Encodings:

BibTeX

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Integral equations for the paraboloidal wave functions. II. Quart. J. Math. Oxford Ser. (2) 16, pp. 257–262.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.31(iii), §28.31(iii).

Encodings:

BibTeX

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