Sorry about that. It was switched over to a pdf format and seems located on Amazon servers.

A few highlights;

"This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration.

Abstract

The present paper deals with the effect of dissipation on the propagation of wave packets governed by a wave equation of Jeffrey type. We show that all packets undergo a shift of the central frequency (the mode with maximal amplitude) towards the lower frequencies (‘‘redshift’’ in theory of light or ‘‘baseshift’’ in acoustics). Packets with Gaussian apodization function do not change their shape and remain Gaussian but undergo redshift and spread. The possible applications of the results are discussed.

1. Introduction

The propagation of waves in linear dissipative systems is well studied but most of the investigations are concerned with the propagation of a single-frequency wave. On the other hand, in any of the practical situ- ations, one is faced actually with a wave packet, albeit with a very narrow spread around the central fre- quency. This means that one should take a special care to separate the effects of dispersion and dissipation on the propagation of the wave packet from the similar effects on a single frequency signal.

The effect of dissipation of the propagation of wave packets seems important because their constitution can change during the evolution and these changes can be used to evaluate the dissipation.

Especially elegant is the theory of propagation of packets with Gaussian apodization function.

Eq. (12) shows that an initial distribution of the energy as function of k will change in time in the sense that the amplitudes of the shorter waves will diminish faster in time than the amplitudes of the longer waves. This will lead to redistribution of the amplitudes and to a change of the apodization function of a wave packet that is subject to evolution according to Jeffrey’s equation. Therefore a general shift of the central wave number towards longer waves (smaller wave numbers k) is to be expected. In the case of light, this is called ‘‘redshift’’. The quantitative values for the redshift for different apodization functions may differ. The most interesting case appears to be the Gaussian distribution of the packet and we focus in this short note on the said case.

The last expression means that the central wave length of the packet is shifted to the longer waves. Borrowing a coinage from the theory of light we can say that the packet is redshifted. Naturally, for acoustic waves it should be called baseshift. Eq. (21) shows that the redshift depends linearly on the time of propagation. As witnessed by the Eq. (17), the group velocity of the packet is not constant. However, for very tightly localized packets, we can assume that the group velocity is well approximated by the phase velocity cg 1⁄4 xð~kÞ=~k 1⁄4 c and to assess the propagation time as T 􏰃 c􏰀1L, where L is the length traveled by the packet. Then Eq. (21) adopts the formz1⁄4Dk1⁄4 d L1⁄4HL; where H 1⁄4 d : ð22Þ ~k cb bc

The relation Eq. (22) can be called ‘‘Hubble law’’ for redshifting of a Gaussian wave packet. More precisely we shall call it ‘‘distance-Hubble law’’, because the redshift is proportional to the distance traveled by the wave packet. Then Eq. (21) can be named ‘‘time-Hubble law’’. Note that due to the intricate dependence of the group velocity on wave number, the time-Hubble law is strictly linear, while the distance-Hubble law can be slightly nonlinear because the Hubble constant can be function of ~k through cg and hence of distance L for a given T. For c, d 􏰂 c one gets that cg 􏰃 c which means that the distance-Hubble law is also virtually linear. Even if we assume that c, d 􏰂 c, Hubble constant is a function of the central wave number ~k through the dependence b 1⁄4 bð~kÞ which specifies the width of a packet as a function of its central wave number.

The magnitude of packet’s maximum decays exponentially with time with an exponent that depends linearly on the attenuation c, but the role of the dispersive dissipation d is more intricate. For very large times T, the role of dissipation amounts to an additional attenuation b/2 which means that a packet that is narrower in terms of wave number (wider in wave length) is attenuated more. Clearly, for non-zero values of T, the magnitude of the maximal amplitude decreases exponentially. It is convenient to scale the wave packet by Em(T) and to concern ourselves with the renormalized spectrum

E^ðk; T Þ 1⁄4 Eðk; T Þ 1⁄4 e􏰀12ðbþdT Þðk􏰀􏰊kÞ2 ; ð24Þ EmðTÞ which is once again a Gaussian distribution with a larger coefficient b, namely bT 1⁄4bþdT 􏰄bð1þzÞ: ð25Þ

The main conclusion from this subsection is that the dissipation acts to shift the central wave length of the packet to the longer waves and to increase its width while preserving the Gaussianity. Preserving the shape of the spectral density is a unique property of the Gaussian apodization functions.

5. Conclusions

In the present work, the effect of attenuation and dissipation on propagation of waves governed by the Jeffrey equation is addressed. When packets of small but finite breadth are considered the presence of dissipation changes the central wave number of the packet. The distribution of the wave length around the central length is assumed to be Gaussian which is the most frequently encountered case in cosmology when hot stars are observed. Dispersion relation for the damped wave equation is derived and the evolution of the packet density is investigated in time (or space). It is shown that the attenuation acts merely to decrease the amplitude of the shifts packed, while the dissipation damps the higher frequencies stronger than the lower frequencies and shifts the maximal frequency of the packet to lower frequencies (longer wave lengths), i.e., the packet appears redshifted upon its arrival. For Gaussian wave packets, this kind of redshift is linearly proportional to the time passed or the distance traveled. The coefficient of proportionality contains the ratio of the dissipation coefficient and the initial width of the distribution which means that the thicker packets are redshifted more than the narrower ones for the same distance or for the same time. We call this liner relationship ‘‘Hub- ble Law’’ for redshifting of wave packets.

The new approach can be used in acoustics for devising methods for estimating the bulk viscosity of air, or other slightly compressible liquids based on the relationship between the ‘‘baseshift’’ and viscosity coefficient. An application to cosmology is also possible because the spectral lines measured in the experiments are wave packets, and never a single isolated wave comprising it. Thus, one has to take special care to distinguish between the redshift of the packet (as outlined in the present work) and the redshift due to the dilation of a single wave."

The implications of which are that we are sampling a wave front, not seeing individual photons, that traveled billions of lightyears. so the quantification of light would be a function of its absorption and measurement, not fundamental to the light itself.