Exact expressions for the sums of the non-Fourier trigonometric series arising from time-dependent eigenvalue problems

Abstract

Exact analytic expressions are found for the sums of certain non-Fourier trigonometric series which are encountered in the mathematical analysis of a problem of classical dynamics. The physical situation considered here is as follows: When a bullet collides with an elastic rod with a finite length in the direction of its length, a density wave in the elastic rod is produced by this collision. Since the wave obeys the spatially one-dimensional wave equation, the solution is constructed by expanding it into a series of trigonometric functions. The boundary conditions imposed are as follows: At the left end the bullet sticks to the elastic rod after the collision, while at the right end the elastic rod is free or fixed, or “intermediate”. The eigenvalue problem, which becomes time-dependent because of the motion of the bullet, yields a complicated non-Fourier trigonometric series. The main problem is to find the sum of this series explicitly. Since concrete non-Fourier trigonometric series are hardly discussed in mathematics, no standard method is available for solving this problem. By using various techniques of analysis, an analytic expression containing Laguerre’s polynomials is found in closed form for the sum of the series, in spite of the fact that it has an infinite number of discontinuity points. This expression reproduces the solution directly constructed by the general solution of the wave equation in the time before the reflection wave arrives. Numerical analysis is also undertaken; it confirms the validity of the analytic solution.