Periodic travelling wave

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, excitable systems and reaction–diffusion–advection systems. 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, integrodifference equations, coupled map lattices and cellular automata 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. While periodic travelling waves have been known as solutions of the wave equation since the 18th century, their study in nonlinear systems began in the 1970s. A key early research paper was that of Nancy Kopell and Lou Howard which proved several fundamental results on periodic travelling waves in reaction–diffusion equations. This was followed by significant research activity during the 1970s and early 1980s. There was then a period of inactivity, before interest in periodic travelling waves was renewed by mathematical work on their generation, and by their detection in ecology, in spatiotemporal data sets on cyclic populations. Since the mid-2000s, research on periodic travelling waves has benefitted from new computational methods for studying their stability and absolute stability. The existence of periodic travelling waves usually depends on the parameter values in a mathematical equation.

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Related publications (1)

An Effective EMTR-Based High-Impedance Fault Location Method for Transmission Lines

Rong Zeng

This article summarizes the electromagnetic time-reversal (EMTR) technique for fault location, and further numerically validates its effectiveness when the fault impedance is negligible. In addition, a specific EMTR model considering the fault impedance is derived, and the correctness of the model derivation is verified by various calculation methods. Based on this, we found that when the fault impedance is large, the existing EMTR methods might fail to accurately locate the fault. We propose an EMTR method that improves the location effect of high-impedance faults by injecting double-ended signals simultaneously. Theoretical calculations show that this method can achieve an accurate location for high-impedance faults. To further illustrate the effectiveness, the proposed method is compared with the existing EMTR methods and the most commonly used traveling wave-based method using wavelet transform. The simulation results show that the proposed double-ended EMTR method can effectively locate high-impedance faults, and it is more robust against synchronization errors compared with the traveling wave method. In addition, the proposed method does not require the knowledge or a priori guess of the unknown fault impedance.

2021

Related concepts (2)

Related lectures (5)

Periodic travelling wave

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