Combinatorial-analytical method for wave equation with abrupt parameter changes
https://doi.org/10.22405/2226-8383-2025-26-4-344-356
Abstract
The paper presents a complete analytical solution to the problem of free vibrations of a string with an arbitrary number of abrupt changes in the wave propagation velocity. A novel combinatorial-analytical method is proposed that allows representing the solution in the form of a compact explicit formula. It is proved that the solution represents a superposition of 2𝑁−1 waves, each corresponding to one of the possible paths of disturbance propagation through the velocity switching moments. It is established that the coefficients in the obtained formula have a clear physical meaning and represent products of transmission and reflection coefficients at the interfaces. The method is generalized to the case of a finite string with zero Dirichlet boundary conditions. The solution is constructed in closed form and confirmed by two independent methods:
the Fourier method and the method of mathematical induction. The obtained results allow analyzing complex wave processes in media with piecewise constant parameters and can be
used in problems of acoustics, seismology, and control theory.
About the Authors
Alexander Ivanovich Nizhnikov
Moscow State Pedagogical University
Russian Federation
Russian Federation
Oleg Emmanuilovich Yaremko
Moscow State Technical University “Stankin”
Russian Federation
Russian Federation
Natalya Nikolaevna Yaremko
National Research Technological University “MISiS”
Russian Federation
Russian Federation
Sergey Alexandrovigh Mukhanov
Russian Technological University “MIREA”
Russian Federation
Russian Federation
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Review
For citations:
Nizhnikov A.I., Yaremko O.E., Yaremko N.N., Mukhanov S.A. Combinatorial-analytical method for wave equation with abrupt parameter changes. Chebyshevskii Sbornik. 2025;26(4):344-356. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-4-344-356
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