Combinatorial-analytical method for wave equation with abrupt parameter changes

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 $2^{N-1}$ 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.