Vibrations of a segment of a variable-length longitudinally-moving string (Q1090140)

The problem of string vibrations on a section of uniformly changing length was examined by \textit{E. L. Nikolai} [Works on mechanics (1955)]. Transformation belonging to the one-dimensional wave equation group were utilized by \textit{A. I. Vesnitskij} and \textit{A. I. Potapov} [System dynamics. 13, Izd., Gor'k. Univ., Gor'kii (1978)] for the solution in the case of an arbitrarily changing length. An analogous problem is examined for an arbitrarily moving string. By the successive application of the Galileo and Lorentz transformations it is reduced to a boundary value problem for the wave equation on a section with one moving end, which results in a problem for a segment with fixed ends. The general solution of the problem on waves on a section of variable-length moving string is represented in the form of a sum of natural vibrations. A procedure is proposed for determining the coefficients of the eigenfunction expansion of the solution for initial conditions given in the original variables.