Types of Wave Equations and Solutions| Types of Wave Equations and Solutions |
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
∇2φ − v−2 ∂t2φ + a · ∇φ + b ∂tφ + cφ = f(x, t) or ∇2φ − v−2 ∂t2φ = 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.
> Generalized backgrounds: see
waves [in curved spacetimes]; dynamics of causal sets.
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·x − ωt)}.
* Progressing / traveling waves:
The ones that move without changing shape, of the form φ(x, t)
= F(x−vt).
@ 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;
Bayındır & 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 = ∇·(p∇φ) − 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
∇2φ − με ∂t2φ − μσ ∂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].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 3 jan 2021