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.
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- ^ 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
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