Dimensionless solutions of the wave equation

TL;DR

This paper introduces a dimensionless approach to the wave equation that treats space and time equally, leading to new geometric projections and verifiable acoustic phenomena, with potential implications for broader physical theories.

Contribution

The paper presents an alternative, dimensionless formulation of the wave equation that reveals new geometric projections and experimentally verifies their consequences in acoustics.

Findings

Verified phase flow predictions using acoustic waves

Identified alternative geometric projections of standing waves

Demonstrated equal treatment of space and time in wave solutions

Abstract

Plane waves are regarded as the general solution of the wave equation. However the plane wave expansion of standing waves by means of complex phasors leads to a theory in which the time coordinate does not receive the same treatment as the three space coordinates. An equal treatment is possible using our alternative approach built upon the dimensionless version of the wave equation. As a result, the usual standing wave solution written as sum of plane waves is just one of the available geometrical projections and therefore removes a part of the available information. The existence of these alternative projections and the constraints that they introduce, produce verifiable consequences. We present an experimental verification of one of this consequences by means of acoustic waves. In particular a resonant cavity is radiated from an external source through a squared aperture. The predicted flows of phase based on P\'olya potentials allow us to find the direction of arrival without using temporal coordinates. Although this work is limited to the wave equation, the background concept is the relationship between space and time and therefore could have far reaching consequences in other physical models.