Wave Equations  

General Concepts > s.a. Coherence; green functions; partial differential equations; velocity; wave phenomena.
* Radiation vs waves: Can usually be identified, physically, but conceptually radiation involves energy transport.
* Frequency: Measured by an observer with 4-velocity ua, = –ka ua.
* Wave number / vector:
* Power: If I = p u is the intensity vector, the power is P = S I · dA.
@ General references: Havelock 14 [dispersive media]; Georgi 93; Jonsson & Yngvason 95 [math]; Nettel 95; Kneubühl 98 [including non-linear]; AJP(01)aug [book reviews].
@ Related topics: Abraham-Shrauner et al JPA(06) [type-II hidden symmetries]

Forms of the Wave Equation > s.a. Fermat's Principle; huygens principle; quantum field theory in curved spacetime.
* General form: In its most general form, a linear wave equation can be written as

2 – v–2 t2 + a · + b t + c = f(x, t)   or   2 – v–2 t2 = 0 ,

where v (speed), a, b (diffusion), and c (mass) depend on the medium, and f is a source; The second form is for a homogeneous, non-conducting medium without sources.
* Remark: It can always be put in normal form.
* Exactly solvable: One whose general solution is a finite sum of progressing waves of finite order:

= n=0infty hn(u,v) dna(z)/dzn ,   z = u or v .

* Boundary conditions: Solid wall (vperp = 0); Free surface p = 0; > s.a. boundaries in field theory.
@ General references: Greiner 90; Kitano PRA(95)qp [1D propagation]; Pol'shin gq/98, gq/98, gq/98 [on curved spacetime, and group theory].
@ Related topics: Torre JMP(03)mp, JMP(06)mp [helically reduced]; Vasy AM(08) [singularities on manifolds with corners]; > s.a. toda lattice.

Special Techniques and Solutions > s.a. computational physics; spectral geometry; special relativity [wave fronts].
* Plane waves: Characterized by one wave vector k and frequency , of the form (x, t) = (Re) A exp{i(k·x – t)}.
* Progressing / traveling waves: The ones that move without changing shape, of the form (x, t) = F(x–vt).
@ Solutions: Varlamov IJTP(03)mp/02; Bicák & Schmidt PRD(07)-a0803 [wth helical symmetry].
@ Traveling waves: Ward CQG(87); Rodríguez et al JPA(90) [1+1, stability]; Rodrigues & Lu FP(97)ht/96 [existence]; Hu PLA(04) and PLA(04) [coupled non-linear differential equations]; Sirendaoreji PLA(06) [non-linear equations]; Bazeia et al AP(08) [solution-generating method]; Fernández a0902 [non-linear equations]; > s.a. Gross-Pitaevskii Equation.
@ Finite-order progressing waves: Couch & Torrence PLA(86); Torrence JPA(90) [acoustic equations]; Bombelli & Sonego JPA(94)mp/00.
@ Characteristic problem: Frittelli JPA(04) [first-order reduction].
@ Related topics: Couch & Torrence JPA(93) [equations with exact spreading solutions]; Kaiser in(05)mp/04 [eigenwavelets]; > s.a. decomposition of functions, hamiltonian systems, Shock Waves.

Types of Wave Equations > s.a. green functions; perturbations in general relativity; Gross-Pitaevskii Equation; light; Quaternions; sound.
* In a fluid: Acoustic waves; Surface waves; Buoyancy waves; Lamb waves (acoustic in nature, different propagation).
* Acoustic equation: Often written as c2(x, t) wxx = wtt, with c(x, t) the speed of sound; The general form is

 , tt = ·(p) – q + F(x, t) ,

where the functions , p and q depend on the medium and F(x, t) is an external perturbation.
* Electromagnetic: If = magnetic permeability, = dielectric constant, and = conductivity, the scalar potential obeys

2 – t2 –  t = –/ .

* Helmholtz equation: The equation obtained from the wave equation for a homogeneous medium, with an oscillating disturbance of frequency and amplitude a2 f(x), and when one looks for a solution u of the same frequency,

2u + k2u = –f(x) ,   with   k:= 2/a2 .

@ Non-linear: Foursov & Vorob'ev JPA(96) [utt = (uux)x]; Infeld & Rowlands 00 [solitons, chaos]; Chugainova TMP(06) [with dispersion and dissipation].
@ Solvable: Friedlander PCPS(47); Degasperis & Tinebra JMP(93), Barashenkov et al JMP(93), JMP(93) [relativistic].
@ Other types: Couch & Torrence JPA(95) [2D, wave-splitting approach]; Bizon et al a0905 [cubic].
@ In curved spacetime: Grant et al CMP(09)-a0710 [singular space-times]; Ionescu & Klainerman CMP(09) [ill-posed problems, uniqueness results]; Galstian et al a0908 [Einstein and de Sitter spacetimes]; > s.a. FRW spacetimes, gödel solution, schwarzschild metric.

Generalizations > s.a. differential equations; non-commutative theories.
@ References: Barci et al IJMPA(95) [4th-order, tachyons]; Das a0811 [discretized, covariant phase space].

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