A New Construction for Scalar Wave Equations in Inhomogeneous Media

TL;DR

This paper introduces a unified discrete formulation for scalar wave equations in inhomogeneous media, capable of deriving Klein-Gordon or Schrödinger equations through a single tunable parameter, with potential extensions to other wave types.

Contribution

It presents a novel systematic derivation of a unified discrete wave equation that can be tuned to produce different fundamental equations, extending applicability to various lattice structures.

Findings

Unified discrete wave equation derived

Equation can be tuned to Klein-Gordon or Schrödinger

Method applicable to various lattice types

Abstract

The paper describes a formulation of discrete scalar wave propagation in an inhomogeneous medium by the use of elementary processes obeying a discrete Huygens' principle and satisfying fundamental symmetries such as time-reversal, reciprocity and isotropy. Its novelty is the systematic derivation of a unified equation which, properly tuned by a single parameter, leads to either the Klein-Gordon equation or the Schr\"{o}dinger equation. The generality of this method enables one to consider its extension to other types of discrete wave equations on any kind of discrete lattice.