On the Wave Equation

Abstract

Uchida Keitaroh

This paper presents a structural reinterpretation of the wave equation based on the interplay among second-order hyper- exponential functions, Hamilton’s quaternions, and a geometric model of revolution around a line segment. Owing to the natural compatibility between these hyper-exponential functions and quaternions, the proposed framework yields a decomposition of the solution into two distinct terms. The main result is that, under this hyper-exponential representation, the d’Alembertian operator admits a quaternionic decomposition into two mutually cancelling components. This reformulation provides a new perspective on the underlying structure of partial differential operators.

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