Illuminating finite-amplitude wave propagation: Hargrove’s use of recurrence relations

Abstract

A classic problem in the theory of finite-amplitude wave propagation is that of the spectral evolution of a plane wave. An initial sinusoidal wave becomes triangular, with a vertical face at the shock distance. The wave in the pre-shock regime was represented by a Fourier series in the seminal work by Fubini, i.e., Fubini Ghiron [Alta Frequenza, 4, 530 (1935)]. Independently, Hargrove performed a similar derivation, also invoking recurrence relations for cylindrical Bessel functions, but presenting this with exceptional heuristic clarity and brevity [J. Acoust. Soc. Am. 32, 511 (1960)]. This was coincident with a third paper, by Keck and Beyer [Phys. Fluids 3, 346 (1960)], which knowingly followed Fubini. All of this work has facilitated understanding, providing cogent reminders of the power of recurrence relations when dealing with special functions.