Types of Wave Equations and Solutions  

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

2φv–2t2φ + a · ∇φ + btφ + cφ = f(x, t)   or   ∇2φv–2t2φ = 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:

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

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

Special Types of Solutions > s.a. computational physics; spectral geometry; special relativity [wave fronts]; wave phenomena [tails, negative frequencies].
* Plane waves: Characterized by one wave vector k and frequency ω, of the form φ(x, t) = (Re) A exp{i(k·xωt)}.
* Progressing / traveling waves: The ones that move without changing shape, of the form φ(x, t) = F(xvt).
@ Solutions: Varlamov IJTP(03)mp/02; Bičák & 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]; Khater et al RPMP(10) [using the mapping method and the extended F-expansion method]; Alekseev CQG(15)-a1411 [expanding spatially homogeneous spacetimes]; > s.a. Gross-Pitaevskii Equation.
@ Finite-order progressing waves: Couch & Torrence PLA(86); Torrence JPA(90) [acoustic equations]; Bombelli & Sonego JPA(94)mp/00.
@ Evanescent waves: Kleckner & Ron qp/98; Voigt et al PRA(00)qp/99 [radiation pressure]; Papathanassoglou & Vohnsen AJP(03)jul; Thio AS(06) [and data storage]; Wang & Xiong PRA(07) [superluminality]; Nimtz & Stahlhofen NS-a0708, comment Winful a0709 [as classical analogs of virtual particles, and Lorentz violation]; > s.a. Virtual Particles.
@ Non-radiating: Friedlander PLMS(73); Kim & Wolf OC(86); Gamliel et al JOSA(89); Berry et al AJP(98)feb.
@ Rogue waves: Akhmediev et al PLA(09); Onorato et al PRL(11) + news pw(11)oct; Bayindir & Ozaydin a1701 [and quantum Zeno dynamics].
@ Related topics: Couch & Torrence JPA(93) [equations with exact spreading solutions]; Hasse et al CQG(96) [caustics, in general relativity]; Kempf JMP(00) [superoscillations]; Kaiser in(05)mp/04 [eigenwavelets]; Okninski a1704 [non-standard solutions, involving higher-order spinors and describing decaying states]; > s.a. Shock Waves.

Types of Wave Equations > s.a. analysis [fractional]; green functions; Gross-Pitaevskii Equation; Helmholtz Equation; light; quaternions; sound.
* Acoustic equation: Often written as c2(x, t) wxx = wtt, with c(x, t) the speed of sound; The general form is

ρ φ, tt = ∇·(pφ) – qφ + F(x, t) ,

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

2φμεt2φμσ ∂tφ = –ρ/ε .

@ Non-linear: Foursov & Vorob'ev JPA(96) [utt = (uux)x]; Kneubühl 98; Infeld & Rowlands 00 [solitons, chaos]; Chugainova TMP(06) [with dispersion and dissipation]; Grochowski et al a1611 [bifurcations and non-linear spectral problem].
@ 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]; Bizoń et al Nonlin(10)-a0905 [cubic].
@ Generalizations: Barci et al IJMPA(95) [4th-order, tachyons].


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