On the solutions of a factorized wave equation

TL;DR

This paper investigates conditions under which a factorized wave equation in one dimension admits solutions that are pure traveling waves, enabling reflectionless energy transmission in inhomogeneous media.

Contribution

It identifies specific conditions for both solutions of a second-order factorized wave equation to be traveling waves, advancing understanding of reflectionless wave propagation.

Findings

Conditions for reflectionless wave solutions are derived.

Different wave structures are classified and analyzed.

Results extend previous studies on wave propagation in inhomogeneous media.

Abstract

Long-distance transmission of energy by waves is a key mechanism for many natural processes. It becomes possible when the inhomogeneous medium is arranged in such a manner that it enables a specific type of waves to propagate with virtually no reflection or scattering. If the corresponding wave equation admits factorization, at least one of the waves it describes propagates without reflection. The paper is devoted to searching for conditions under which both solutions of a one-dimensional factorized wave equation of the second order describe traveling waves, that is, waves propagating without reflection. Possible variants of wave structure are found and the results are compared with those obtained in previous studies.