For a burst of waves seen in quantum mechanics, see Wave packet.
In mathematics, a periodic travelling wave (or wavetrain) is a periodic function of one-dimensional space that moves with constant speed. Consequently, it is a special type of spatiotemporal oscillation that is a periodic function of both space and time.
Periodic travelling waves play a fundamental role in many mathematical equations, including self-oscillatory systems,[1][2]
excitable systems[3] and
reaction–diffusion–advection systems.[4]
Equations of these types are widely used as mathematical models of biology, chemistry and physics, and many examples in phenomena resembling periodic travelling waves have been found empirically.
The mathematical theory of periodic travelling waves is most fully developed for partial differential equations, but these solutions also occur in a number of other types of mathematical system, including integrodifferential equations,[5][6]
integrodifference equations,[7]
coupled map lattices[8]
and cellular automata[9][10]
As well as being important in their own right, periodic travelling waves are significant as the one-dimensional equivalent of spiral waves and target patterns in two-dimensional space, and of scroll waves in three-dimensional space.
^N. Kopell, L.N. Howard (1973) "Plane wave solutions to reaction–diffusion equations", Stud. Appl. Math. 52: 291–328.
^I. S. Aranson, L. Kramer (2002) "The world of the complex Ginzburg–Landau equation", Rev. Mod. Phys. 74: 99–143. DOI:10.1103/RevModPhys.74.99
^S. Coombes (2001) "From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves", Math. Biosci. 170: 155–172.
DOI:10.1016/S0025-5564(00)00070-5
^J.A. Sherratt, G. J. Lord (2007) "Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments", Theor. Popul. Biol. 71 (2007): 1–11.
DOI:10.1016/j.tpb.2006.07.009
^S. A. Gourley, N. F. Britton (1993) "Instability of traveling wave solutions of a population model with nonlocal effects", IMA J. Appl. Math. 51: 299–310.
DOI:10.1093/imamat/51.3.299
^P. Ashwin, M. V. Bartuccelli, T. J. Bridges, S. A. Gourley (2002) "Travelling fronts for the KPP equation with spatio-temporal delay", Z. Angew. Math. Phys. 53: 103–122.
DOI:0010-2571/02/010103-20
^M. Kot (1992) "Discrete-time travelling waves: ecological examples", J. Math. Biol. 30: 413-436. DOI:10.1007/BF00173295
^M. D. S. Herrera, J. S. Martin (2009) "An analytical study in coupled map lattices of synchronized states and traveling waves, and of their period-doubling cascades", Chaos, Solitons & Fractals 42: 901–910.DOI:10.1016/j.chaos.2009.02.040
^J. A. Sherratt (1996) "Periodic travelling waves in a family of deterministic cellular automata", Physica D 95: 319–335.
DOI:10.1016/0167-2789(96)00070-X
^M. Courbage (1997) "On the abundance of traveling waves in 1D infinite cellular automata", Physica D 103: 133–144.
DOI:10.1016/S0167-2789(96)00256-4
and 25 Related for: Periodic travelling wave information
In mathematics, a periodictravellingwave (or wavetrain) is a periodic function of one-dimensional space that moves with constant speed. Consequently...
A periodic function or cyclic function, also called a periodic waveform (or simply periodicwave), is a function that repeats its values at regular intervals...
be a travellingwave; by contrast, a pair of superimposed periodicwavestraveling in opposite directions makes a standing wave. In a standing wave, the...
A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodicwave whose waveform (shape) is the trigonometric sine function. In mechanics, as a...
mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance...
R=0} the wave becomes a travellingwave moving leftwards. A standing wave can be obtained by R = 0.5 {\displaystyle R=0.5} . In a travellingwave pressure...
An atmospheric wave is a periodic disturbance in the fields of atmospheric variables (like surface pressure or geopotential height, temperature, or wind...
In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has...
sharp-crested wave shape. After the wave breaks, it becomes a wave of translation and erosion of the ocean bottom intensifies. Cnoidal waves are exact periodic solutions...
sidelobe level or beam shape. Leaky-wave antennas can be divided into two important categories, uniform and periodic, depending on the type of guiding structure...
spatial analog to temporal period. For periodicwaves in nondispersive media (that is, media in which the wave speed is independent of frequency), frequency...
interference. One wave is travelling horizontally, and the other is travelling downwards at an angle θ to the first wave. Assuming that the two waves are in phase...
Extended periodic table Element 119 (Uue, marked here) in period 8 (row 8) marks the start of theorisations. An extended periodic table theorizes about...
(also called the celerity) of a surface gravity wave is—for pure periodicwave motion of small-amplitude waves—well approximated by c = g λ 2 π tanh ( 2...
matter waves as probes of materials or of fundamental quantum properties. In most cases these involve some method of producing travelling matter waves which...
rotates at twice the orbital rate, so the time-varying gravitational wave size, or 'periodic spacetime strain', exhibits a variation as shown in the animation...
surface electromagnetic waves in the periodic structure using a prism coupling technique. In quantum mechanics, the evanescent-wave solutions of the Schrödinger...
hydrogen-like wave functions for all atomic orbitals up to 7s, and therefore covers the occupied orbitals in the ground state of all elements in the periodic table...
Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large and unpredictable...
terahertz (THz). Some of the unit's most common uses are in the description of periodic waveforms and musical tones, particularly those used in radio- and audio-related...
ranges, the bands are periodic. The Bloch waves are periodic functions of space, with periods lx, ly, lz and the bands are periodic functions of wavenumber...
In fluid dynamics, a cnoidal wave is a nonlinear and exact periodicwave solution of the Korteweg–de Vries equation. These solutions are in terms of the...
Numerics for Wave Propagation, Basque Center for Applied Mathematics, 2012, Spain, p. 39 Brillouin, L. (1946). Wave propagation in Periodic Structures:...
Global Information