wave equation

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1: 29.11 Lamé Wave Equation

§29.11 Lamé Wave Equation

►The Lamé (or ellipsoidal) wave equation is given by …

2: 13.28 Physical Applications

… ►

§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\phi

y=2\sqrt{\xi\eta}\sin\phi

z=\xi-\eta

, can be solved via separation of variables

w=f_{1}(\xi)f_{2}(\eta)e^{\mathrm{i}p\phi}

, where …

3: 10.73 Physical Applications

… ►The Helmholtz equation,

(\nabla^{2}+k^{2})\psi=0

, follows from the wave equation …This equation governs problems in acoustic and electromagnetic wave propagation. … … ►

§10.73(ii) Spherical Bessel Functions

… ►In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …

4: 30.10 Series and Integrals

… ►Integrals and integral equations for

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)

are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). …

… ►He has worked on integrable systems, algorithms for computations with Riemann surfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations. …

6: 29.18 Mathematical Applications

… ►

§29.18(i) Sphero-Conal Coordinates

►The wave equation … ►

29.18.7

\frac{{\mathrm{d}}^{2}u_{3}}{{\mathrm{d}\gamma}^{2}}+(h-\nu(\nu+1)k^{2}{% \operatorname{sn}}^{2}\left(\gamma,k\right))u_{3}=0,

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter, $\gamma$ : sphero-conal coordinate and $u_{j}$ : solutions
Referenced by:: §29.18(i)
Permalink:: http://dlmf.nist.gov/29.18.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(i), §29.18 and Ch.29

… ►The wave equation (29.18.1), when transformed to ellipsoidal coordinates

\alpha,\beta,\gamma

: …where

u_{1}

u_{2}

u_{3}

each satisfy the Lamé wave equation (29.11.1). …

7: Mark J. Ablowitz

… ►Ablowitz is an applied mathematician who is interested in solutions of nonlinear wave equations. …

8: 30.14 Wave Equation in Oblate Spheroidal Coordinates

§30.14 Wave Equation in Oblate Spheroidal Coordinates

►

§30.14(i) Oblate Spheroidal Coordinates

… ►The wave equation (30.13.7), transformed to oblate spheroidal coordinates

(\xi,\eta,\phi)

, admits solutions of the form (30.13.8), where

w_{1}

satisfies the differential equation … ►

9: 28.33 Physical Applications

… ►

•

Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

… ►The wave equation ►

28.33.1

\frac{{\partial}^{2}W}{{\partial x}^{2}}+\frac{{\partial}^{2}W}{{\partial y}^{% 2}}-\frac{\rho}{\tau}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0,

ⓘ

Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $x$ : real variable, $y$ : real variable, $\rho$ : mass, $\tau$ : radial tension and $W(x,y,t)$ : solution
Permalink:: http://dlmf.nist.gov/28.33.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.33(ii), §28.33 and Ch.28

… ►

•

McLachlan (1947, Chapters XVI–XIX) for applications of the wave equation to vibrational systems, electrical and thermal diffusion, electromagnetic wave guides, elliptical cylinders in viscous fluids, and diffraction of sound and electromagnetic waves.

…

10: 30.13 Wave Equation in Prolate Spheroidal Coordinates

§30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►The wave equation … ►