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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 2 w = k 2 w in paraboloidal coordinates, x = 2 ξ η cos ϕ , y = 2 ξ η sin ϕ , z = ξ - η , can be solved via separation of variables w = f 1 ( ξ ) f 2 ( η ) e i p ϕ , where …
3: 10.73 Physical Applications
The Helmholtz equation, ( 2 + k 2 ) ψ = 0 , follows from the wave equation
10.73.2 2 ψ = 1 c 2 2 ψ t 2 ,
§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 Ps n m ( x , γ 2 ) are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). …
5: Bernard Deconinck
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 d 2 u 3 d γ 2 + ( h - ν ( ν + 1 ) k 2 sn 2 ( γ , k ) ) u 3 = 0 ,
The wave equation (29.18.1), when transformed to ellipsoidal coordinates α , β , γ : …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 ( ξ , η , ϕ ) , 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 2 W x 2 + 2 W y 2 - ρ τ 2 W t 2 = 0 ,
  • 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