Exact solutions to the Telegraph equation in terms of Airy functions

TL;DR

This paper derives two exact solutions to the three-dimensional Telegraph equation using Airy functions, revealing a unique propagation pattern and resulting in Schrödinger-like equations that are solvable in closed form.

Contribution

It introduces novel exact solutions to the Telegraph equation in three dimensions expressed via Airy functions, highlighting a distinctive propagation pattern.

Findings

Solutions exhibit a speed-cone-like propagation pattern

Derivation of Schrödinger-like equations from the solutions

Exact Airy function solutions for the Telegraph equation

Abstract

Two exact different solutions to the Telegraph equation in three-dimensional space are obtained in terms of Airy functions. As a result, these solutions unveil a distinctive propagation pattern along a coordinate that resembles a speed-cone-like coordinate of the system. This unique characteristic leads to effective Schr\"odinger-like equations, amenable to exact solutions through Airy functions.