A comparison of the factorization approach to temporal and spatial propagation in the case of some acoustic waves

Abstract

The evolution of acoustic waves can be evaluated in two ways: either as a temporal, or a spatial propagation. Propagating in space provides the considerable advantage of being able to handle dispersion and propagation across interfaces with remarkable efficiency; but propagating in time is more physical and gives correctly behaved reflections and scattering without effort. Which should be chosen in a given situation, and what compromises might have to be made? Here the natural behaviors of each choice of propagation are compared and contrasted for an ordinary second order wave equation, the time-dependent diffusion wave equation, an elastic rod wave equation, and the Stokes'/ van Wijngaarden's equations, each case illuminating a characteristic feature of the technique. Either choice of propagation axis enables a partitioning the wave equation that gives rise to a directional factorization based on a natural 'reference' dispersion relation. The resulting exact coupled bidirectional equations then reduce to a single unidirectional first-order wave equation using a simple 'slow evolution' assumption that minimizes effect of subsequent approximations, while allowing a direct term-to-term comparison between exact and approximate theories.

The surprising persistence of time-dependent quantum entanglement

“…These Fokker-Planck equation terms can be converted into temporal SDEs using standard techniques [37][38][39], and by then transforming them into a co-moving frame [34], we can arrive at the set of spatially propagated SDE's ( 2)-( 7) used in the simulation model. Although reasonable in the case of weak dispersion, as is the case here, in general the conversion of a material's dispersive response between the temporally propagated and spatially propagated domains is not straightforward [48,49].…”

An introduction to spatial dispersion: revisiting the basic concepts

“…Any physical meaning(s) that is attributable to such negative spectral quantities is comprehensively discussed elsewhere [12], so for simplicity we will only consider positive values here. Typically the sign choices correspond to the different direction in which propagating waveforms will evolve [13][14][15]. Dr.Paul.Kinsler@physics.org http://www.kinsler.org/physics/…”

“…As a passing note, the spatial dispersion due to the structural configuration of an optical fibre but instead temporal in origin. This is because a majority of optical pulse propagation techniques are propagated along a spatial axis [12,13,[20][21][22][23][24], which means that the dispersion computations are more convenient in ω than they would be in the k available in temporally propagated techniques [14,15].…”

“…One might consider quite a wide variety of wave models when treating dynamic spatial dispersion; for example, in acoustic or elastic media, we might think that any of the somewhat eclectic selection in [15] could be suitable. Indeed, if used as a priori mechanisms, they might be, but it is worth noting that such wave models result from applying approximations to an underlying, more complicated microscopic description.…”

Temporal boundaries in electromagnetic materials

“…Research into temporal boundaries dates back to the 1950s. Morgenthaler [1] demonstrated a temporal change of the permittivity produces both forward and backward propagating waves; a result echoed in modern directional formulations for wave propagation [2][3][4][5][6]. Temporal boundaries can act as a time-reversing mirror in acoustics [7], and in electromagnetism an instantaneous time mirror with a sign-change in permittivity has been predicted [8] to cause field amplification.…”