paraxial wave equation

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21: Sidebar 21.SB1: Periodic Surface Waves

Sidebar 21.SB1: Periodic Surface Waves

… ►Two-dimensional periodic waves in a shallow water wave tank. Taken from Joe Hammack, Daryl McCallister, Norman Scheffner and Harvey Segur, “Two-dimensional periodic waves in shallow water. …Asymmetric waves”, J. …The caption reads “Mosaic of two overhead photographs, showing surface patterns of waves in shallow water”. …

22: 30.7 Graphics

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§30.7(i) Eigenvalues

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Figure 30.7.4: Eigenvalues ${\lambda }_n^{10}\left({\gamma }^2\right)$, $n=10,11,12,13$, $-50\le {\gamma }^2\le 150$. Magnify

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§30.7(ii) Functions of the First Kind

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Figure 30.7.5: ${\mathrm{𝖯𝗌}}_n^0(x,4)$, $n=0,1,2,3$, $-1\le x\le 1$. Magnify

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Figure 30.7.21: $|{\mathrm{𝑄𝑠}}_0^0(x+\mathrm{i}y,-4)|$, $-1.8\le x\le 1.8$, $-2\le y\le 2$. Magnify 3D Help

23: 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. …

24: 31.12 Confluent Forms of Heun’s Equation

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Confluent Heun Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. ►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. … ►

Biconfluent Heun Equation

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Triconfluent Heun Equation

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25: Mark J. Ablowitz

… ►Ablowitz is an applied mathematician who is interested in solutions of nonlinear wave equations. Certain nonlinear equations are special; e. …ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. Some of the relationships between IST and Painlevé equations are discussed in two books: Solitons and the Inverse Scattering Transform and Solitons, Nonlinear Evolution Equations and Inverse Scattering. Widespread interest in Painlevé equations re-emerged in the 1970s and thereafter partially due to the connection with IST and integrable systems. …

26: 28.33 Physical Applications

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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 … ►

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.

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Meixner and Schäfke (1954, §§4.3, 4.4) for elliptic membranes and electromagnetic waves.

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Alhargan and Judah (1992), Germey (1964), Ragheb et al. (1991), and Sips (1967) for electromagnetic waves.

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27: 30.12 Generalized and Coulomb Spheroidal Functions

§30.12 Generalized and Coulomb Spheroidal Functions

►Generalized spheroidal wave functions and Coulomb spheroidal functions are solutions of the differential equation …Equation (30.12.1) appears in astrophysics and molecular physics. … ►Another generalization is provided by the differential equation …

28: 30.14 Wave Equation in Oblate Spheroidal Coordinates

§30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(i) Oblate Spheroidal Coordinates

… ►The wave equation (30.13.7), transformed to oblate spheroidal coordinates $(\xi ,\eta ,\varphi )$, admits solutions of the form (30.13.8), where $w_1$ satisfies the differential equation … ►

§30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids

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29: Bernard Deconinck

… ►Deconinck is interested in nonlinear waves. 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. …

30: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

… ►Jacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena. …