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A stochastic model for acousticattenuationB. Lacaze aa TéSA/Enseeiht , 14-16 Port St-Etienne, Toulouse, 31000, FrancePublished online: 03 Jul 2007.

To cite this article: B. Lacaze (2007) A stochastic model for acoustic attenuation, Waves in Randomand Complex Media, 17:3, 343-356, DOI: 10.1080/17455030701247919

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Waves in Random and Complex MediaVol. 17, No. 3, August 2007, 343–356

A stochastic model for acoustic attenuation

B. LACAZE∗

TeSA/Enseeiht, 14-16 Port St-Etienne, 31000 Toulouse, France

(Received 9 November 2006; in final form 27 January 2007)

Propagation of waves in gases or liquids has been observed for a long time in homogeneous or non-homogeneous media. In acoustics, attenuation is a significant problem which is studied mainly throughthe ‘equations of change’ of fluid mechanics. These equations are based only on the macroscopiccharacteristics of the medium. Microscopic variations, related to other phenomena like Brownianmotion or critical opalescence, are not taken into account. This paper provides a random one-raymodel. This model explains the proportionality between the standard attenuation and the productbetween length and frequency squared in a logarithmic scale. The wave is shown to be necessarilyassociated with noise, even if this noise cannot be observed by devices. Furthermore, the ‘coefficientof variation’ defined in turbulent environments can be explained as a random version of the usualcoefficient of attenuation.

1. Introduction

Sound propagation in a continuous medium is explained through the ‘equations of change’ offluid mechanics [1, 2], which are made up of the equations of continuity, the equation of motionand the equation of energy balance. These equations are linearized, neglecting squares of smallquantities. Then, particular solutions at equilibrium are sought. For example, in the case ofpropagation without absorption of a monochromatic wave of frequency ω0/2π , a solution foreach variable is found in the form φ = φ0 exp i[−αx + ω0t], x being a coordinate, t the time,α = ω0/c the wave number, and c the celerity of the wave. When there is ‘ absorption’, thesame type of solution is obtained, with a complex α = ω0/c − iβ, where β is the absorptioncoefficient.

The propagation has reached an equilibrium state when the absorption of the wave is aconstant at each point of the space. At this time, each large enough volume has to give backto the medium the energy which has been received. This is the meaning of the equation ofenergy balance. Consequently, it is not true that the molecules absorb the energy; rather, theyabsorb the energy and exchange it with other molecules. Otherwise, the temperature of themedium would increase without limit. Furthermore, it is not reasonable to admit that a stablestate exists everywhere for a monochromatic wave without a constant mean time energy permolecule.

Then, each surface of the medium is crossed by an energy which is cut into two parts. Thefirst is organized, like a signal in communications; it is the weakened monochromatic wave.

∗Corresponding author. E-mail: bernard.lacaze@tesa.prd.fr

Waves in Random and Complex MediaISSN: 1745-5030 (print), 1745-5049 (online) c© 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/17455030701247919

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344 B. Lacaze

The second one is like a noise, and is a consequence of the random nature of the medium. Atequilibrium, the sum of the powers crossing a closed surface centred at the transmitter has tobe equal to the emitted power. In the case of a gas, it is this random nature which explains themacroscopic parameters, like pressure, temperature, diffusion coefficients, etc. If the receiversdo not show the second part of the energy, it is because their frequency window covers onlythe acoustic band (up to few MHz), and is too narrow to take into account frequencies farabove. Because the number of shocks of molecules is very large (about 1010 for each moleculeby second in air at normal conditions), the noise power spectrum is likely to be too sparseto be detected (see Section 2.1 and Appendix 1). This noise does not appear in the literature.A possible reason is that the equations of change are linearized and the researched solutionsare monochromatic (because the observed result is monochromatic). Moreover, microscopicfluctuations are not taken into account in these equations; instead macroscopic quantities liketemperature, viscosity coefficient, etc. are used. For orders of magnitude of distances used inacoustics, it is not obvious that it is a sufficient model, as is the case for Brownian motion, theblue sky, critical opalescence, etc. But these latter phenomena are observed, and have to beexplained, while the acoustic noise is not observed, and is then neglected.

A comparison with circuit theory is instructive. The current which crosses an electricalcircuit is held by electrons which circulate in some families of elements. In simple cases, thecircuit is a system composed by resistances, capacities, self-inductions, etc. and is modeledby a set of linear differential equations L. Solutions are found using Laplace or Fouriertransforms and linear algebra, which give the amplitude and the phase of the monochromaticcircuit output as a function of the monochromatic input. Of course, these quantities dependon the wave frequency ω0/2π which belongs to some frequency band, determined by someapplication.

But, we know that the movement of electrons has a random character, and this results inelectronic noise (Johnson noise, shot noise, etc.). This noise is not explained by the system ofequations L. For example, two resistances with the same value (in ohms) give the same systemL but different noise following the composition and/or the temperature of the resistances. Infact, the propagation of waves in an electrical circuit or in the atmosphere obeys a similarmechanism. In the first case, electrons move electronic charges, though molecules movekinetic energy in the second case. Obviously, a random character has to be associated to bothphenomena. Then, the noise has to come with a sound, like noise accompanies an electricalcurrent. The system of equations L for an electrical network is like the equations of changefor an acoustical medium. They explain only the well-ordered transformations of waves butnot the noise which may go with it.

The main difference between both situations is that electrical noise can be observed and canrepresent a drawback for transmission. Processes are developed to minimize it and improvethe signal/noise ratio (SNR). In the case of sound propagation, the noise is imbedded in otherexternal noise, and its frequency band is likely to be far away from the frequency of the soundor too spread out. Then, devices cannot distinguish it.

In a medium homogeneous in temperature, pressure, etc. the linearized equations havecoefficients independent of the time and of the coordinates, and can then be solved, even whentransport coefficients are taken into account [1]. But the macroscopic conditions can vary asa function of the time and/or of the coordinates. In this case, the monochromatic solutionof the linearized equations of change is written as an integral equation [3–5]. The solution isindependent of time, or is slowly varying, because the factor exp[iω0t] has disappeared (due tothe a priori choice of the class of solutions and because of the linearization). The result ends inthe amplitude A and in the phase � of (for instance) the pressure p at a given point. The solutiondepends on the values of the macroscopic parameters at the points crossed by the wave andwhich appear in the integral equation. When these parameters are random, it is the same for the

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A stochastic model for acoustic attenuation 345

solutions A and �. It is not possible to know the exact value of the parameters in each intervalof time large in front of the period of the wave. But the knowledge of statistical properties ofparameters allows us to calculate the moments of A and �. In particular, the moment of ordertwo depends on the spatial correlation functions. Several models exist for these quantities,based on Gaussian or Lorentzian shapes or on the Kolmogorov equations, depending on thecharacteristic lengths linked to the turbulence. But it is not reasonable to take these functionsindependent of the coordinates, particularly for long distances, where A can depend stronglyand randomly on the coordinates [6]. Consequently, solutions of the linearized equations arefunctions of a large number of unknown quantities, which cannot be measured in most cases [7].

In many situations, the value of the amplitude a at distance l of the emitter satisfies

a = a0e−βlω20 (1)

where β is a function which varies slowly compared with ω0. The geometrical weakening dueto the limited surface of the receivers is nowhere taken into account in this paper. In a quietatmosphere or water, figures in [8–10] and empirical formulas are in agreement with (1). Thegap in β from a constant is attributed to the presence of H2O, CO2, etc. in the atmosphereand MgSO4, B(OH)3, etc. in sea water. The behavior of these molecules β depends on thecharacteristics of the fluid, the concentration of species, and the values of the temperature,viscosity and heat conductivity. In normal conditions, typical values are β = 0.6.10−12 s2 m−1

for air and β = 0.6.10−15 s2 m−1 for water [8]. The weakening constant β is lower in the caseof liquids than in the case of gases, because the energy transmission takes place in a mediumwhere the molecules are more linked. The same reason may be put forward for the celerity inthe medium. Hirschfelder et al. [1] studied the additive influence of viscosity, thermal conduc-tivity and diffusion. Each of these causes leads to a weakening such as (1). When the medium isturbulent, (1) is still available but the amplitude becomes random, and a is no longer the ampli-tude but is defined from the second moment of the amplitude [4]. Formulas are given in this lastpaper, under Gaussian or Kolmogorov hypotheses, but other situations could be taken, leadingto different formulas, and we do not know what is the best fit. Results can be theoreticallyapplied on propagation on paths of a few meters up to many kilometres (where the distanceis no longer Euclidian, and is unknown). Obviously, these models take into account unknownparameters which cannot be measured at all times or places, when the distance is large.

It is quite surprising that, in free space, equation (1) seems to fit well the attenuation of amonochromatic wave versus the distance and the frequency, whatever the state of the medium,and whatever the model used.

In a medium at rest, homogeneous or not, the amplitude of the received wave is constant.This is no longer the case when turbulence is created in the crossed medium. Experimentsin [11–14] were done in the following framework. A tank of water is heated from below byresistances, which leads to eddies of temperature on the (collimated) wave trajectory, dueto the difference of temperature between the top and the bottom of the tank. The frequencyω0/2π goes up to 9 MHz, and the trajectory length l is of the order of a meter. A generatoremits monochromatic waves within bursts containing hundreds of periods and separated byintervals of the order of a second. An amplitude demodulation is made in each pulse. Theresult is a sequence of samples of the (random) amplitude A(t), say a(t1), a(2t1), . . . , a(nt1),where t1 is the time between two pulses. These samples allow us to estimate the first andsecond moments of A(t).

Specialists in the subject have a particular interest in the ‘coefficient of variation’ µ, definedas

µ =√

VarA(t)

E[A(t)]. (2)

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E[. . .] denotes the mathematical expectation (the ensemble average) and Var[. . .] denotes thevariance. The cited authors are in agreement with the fact that µ is proportional to

√l (for low

µ) [12]. In this last paper, the range was 0.9–2.2 meters. At the same time, in accordance withthe theoretical paper [5], they found that µ is proportional to ω0. This point is clearly detailedin [12] where the range of variation of the frequency is 100–600 kHz. Mintzer gave µ as afunction of parameters describing the strength of the turbulence (see Section 2.2). Equivalentexperiments and similar results were done for air in grid generated turbulence [15]. Note thatthe coefficient µ has its equivalent in other domains of physics, like the ‘scintillation index’in optics [16].

As explained above, a model of propagation for attenuation would exhibit a noisy partadded to the informative part, to explain the disorganized energy. The noise is hidden becausethe equations of change are built on macroscopic considerations, without studying molecularfluctuations. Moreover, equations are linearized and only elementary solutions are researched(but only this kind of solutions is detected by devices). Finally, solutions contain too manyparameters of the medium which are not known most of the time.

In the following section, we establish a simple stochastic model which explains the normalattenuation for a fluid at rest. The model takes into account microscopic and random fluctua-tions of the transit time of the wave. Secondly, we show how to explain the random increaseof the attenuation which leads to the coefficient of variation, when inhomogeneities or eddiesappear in the fluid. This kind of method was successfully applied to other phenomena [17],like backscattering on trees [18], backscattering on sea of radar waves [19], or HF propagation[20].

2. A stochastic model

Let us assume that the wave a0eiω0t is emitted in a quiet medium. The wave is received ata point at a distance l. We look for a stochastic model Z = {Z (t), t ∈ R}, which exhibits amonochromatic part added to a random part, and which satisfies properties of weakening like(1), of energy preserving and of noise invisibility. The simplest model which fulfils theseconditions is in the form

Z (t) = a0eiω0(t−L(t)). (3)

L(t) can be considered as an equivalent random time for the wave to reach the point at thedistance l.

We assume that L = {L(t), t ∈ R} is a random process with stationary laws defined by thecharacteristic functions (in the probability sense)

ψ(ω) = E[e−iωL(t)] φ(ω, τ ) = E[e−iω(L(t)−L(t−τ ))]; (4)

ψ(ω) and φ(ω, τ ) are the Fourier transforms of the probability densities of the random variables(r.v.) L(t) and (L(t) − L(t − τ )). It appears that Z (t) can be decomposed into the shape

Z (t) = a0ψ(ω0)eiω0t + V (t) (5)

where V = {V (t), t ∈ R} is a zero mean random process with autocorrelation function

E[V (t)V ∗(t − τ )] = a20[φ(ω0, τ ) − |ψ(ω0)|2]eiω0τ . (6)

If the r.v. L(t) and L(t − τ ) tend to become independent when τ is large (this is obviously thecase for a phenomenon like Brownian motion), then the process V will have a band spectrum[21]. Simulations show that the faster the correlation between the L(t) decreases, the faster the

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A stochastic model for acoustic attenuation 347

broadening of the V spectrum. If the received wave is filtered around the emitted frequency(also for eliminating other noise), the process V is not taken into account (see Appendix 1).In these conditions, (5) proves that the measured amplitude a is equal to a0|ψ(ω0)|, which isnot random.

Now, we know that for a medium at rest, it is the formula (1) which gives the wave amplitudeat distance l. Then, from (5) the model which is studied here has to satisfy

|ψ(ω0)| = e−βlω20 . (7)

The only characteristic function ψ(ω) which satisfies (7) for all ω0 corresponds to the Gaussianlaw [30], where

ψ(ω) = e−imω−(σω)2/2.

Obviously, m = E[L] (the mean) and σ = √VarL (the standard deviation) have to satisfy

m = l/c (c is the celerity in the medium) and σ 2 = 2βl. Then

ψ(ω) = e−iωl/c−βlω2, σ 2 = 2βl, m = l/c. (8)

To sum up, the attenuation of a wave in a gas or a liquid at rest can be modeled consideringthat the (equivalent) transit time L(t) is a random Gaussian process, with a mean m = E[L]and a variance σ 2 = VarL proportional to l. The lost energy (the process V in (5)) has a bandspectrum, and is eliminated because of the limited frequency range of receivers. In all cases,the additional noise V cannot be seen by the receiver (see Appendix 1).

In water in normal conditions, we have β = 6.10−16 s2 m−1. The model leads to 35.10−9

s, for 1 metre as an order of magnitude for the standard deviation σ, compared with 6.10−4 swhich is the mean time of the path. Then a relative variation of 10−4 in this duration suffices toexplain the experimental data (3.10−4 for the atmosphere). Obviously, this is a strong argumentin favour of the proposed model.

Actually, the parameter β depends on the frequency ω0/2π, as a slowly varying function.This is explained notably by time differences in energy exchanges according to the speciesof molecules, and on chemical processes. For example, in sea water, the following formula isused ( f = ω0/2π in kHz and β by km)

β = 0.003 + 0.001

f 2+ 0.1094

f 2 + 1+ 43.75

f 2 + 4100

but β depends on the temperature, depth, salinity, etc. which are linked quantities. Each seaor ocean has its particular expressions, and all the parameters can vary along the path of thewave.

To take into account the dependence of β on ω0, it suffices to consider that the way timeL(t) obeys probability laws (and characteristic functions) depending on ω0. For example, wecan replace L(t) by L(t, ω0), and ψ(ω) by ψ(ω, ω0), so that (c is also a function of ω0)

ψ(ω, ω0) = e−iωl/c(ω0)−β(ω0)lω2, σ 2 = 2β(ω0)l, m = l/c(ω0).

The spectral density sV (ω) of the noise V is the Fourier transform of E[V (t)V ∗(t − τ )]given in (6). When L is a Gaussian process, we can write [28, 29]

E[V (t)V ∗(t − τ )] = a2

0eiω0τ−(ω0σ )2[e(ω0σ )2ρ(τ ) − 1]

sV (ω + ω0) = a20

π

∫ ∞0 e−(ω0σ )2

[e(ω0σ )2ρ(τ ) − 1] cos ωτdτ

σ 2ρ(τ ) = Cov[L(t), L(t − τ )].

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Then, ρ(τ ) is the correlation coefficient of the process L. A very simple model for ρ(τ ) is

ρ(τ ) ={

1 − ττ0

for 0 < τ < τ0

0 for τ > τ0.

τ0 can be interpreted as a correlation radius, i.e L(t) is linked with L(t + τ ) on a time intervalof length τ0. Other ρ(τ ) will give similar qualitative results. Fast variations of L(t) are forsmall τ0 with ‘white noise’ as the extreme limit, when L(t) and L(t ′) become independent fort different of t ′. On the contrary, relatively slow variations correspond to relatively large τ0. Ofcourse, accurate information about τ0 is not available. However, τ0 is probably a decreasingfunction of the temperature, because of the increasing number of shocks. Moreover, τ0 shouldbe smaller for a gas than for a liquid.

Straightforward computations lead to

sV (ω) = a20τ0

πbgb

[τ0

b(ω − ω0)

]b = 2βlω2

0

gb(x) = b

x2 + b2

[1 − e−b cos x − be−b sin x

x

].

These expressions lead to the relative noise power �(δ) in the frequency band (ω0, ω0 + δ)

�(δ) = SV (ω0 + δ) − SV (ω0)

a20(1 − e−b)

= 1

π (1 − e−b)

∫ τ0δ

0gb(x)dx

where SV (ω) is the power in (−∞, ω). The structural function gb(x) is even and independentof τ0, but depends on b. Recall that a0e−b/2 is the amplitude of the received monochromaticwave. The following table gives the dimensionless quantity τ0δ for values of � in (0, 0.4) ande−b going from 0.01 to 0.99.

e−b\� 0.01 0.03 0.06 0.1 0.2 0.3 0.4

0.01 0.16 0.46 0.92 1.56 3.42 6.42 14.30.05 0.11 0.34 0.67 1.14 2.45 4.39 9.740.1 0.10 0.29 0.59 0.99 2.11 3.66 8.010.5 0.07 0.21 0.43 0.72 1.50 2.46 4.160.9 0.06 0.19 0.38 0.65 1.34 2.17 3.460.95 0.07 0.19 0.38 0.64 1.33 2.15 3.410.99 0.06 0.19 0.38 0.64 1.32 2.13 3.37

For example, when e−b = 0.1, 10% of the noise power is in (ω0, ω0 + δ), where τ0δ � 1.

The table and other computations show that the power spectrum widens when the noise partincreases. Unfortunately, τ0 is an unknown physical parameter. Consequently, the bandwidthrequired to highlight the noise is undetermined. τ0 is probably dependent on the number ofmolecular shocks, which is very dependent of the pressure for a gas, and could be increasedin a gas at low temperature or in low pressure (down to the Knudsen regime). To find enoughnoise power in the frequency band (ω0, ω0 + δ) reachable by low-noise devices, it would benecessary to have large enough τ0, which depends on an unknown parameter.

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A stochastic model for acoustic attenuation 349

3. The perturbed medium

3.1 Introduction

The model above explains the normal attenuation which appears in equation (1). It is availablewhen the macroscopic parameters of the medium, such as the temperature T , the pressure p,the viscosity η, etc. are well defined and constant on sufficiently large intervals of time. Whenchanges are created in the medium, due for instance to eddies, the amplitude of the wave ata given point is no longer constant. But, if changes have a stationary character, it will be thesame for the amplitude A(t). When the variations are not too fast and the turbulence too rough,it can be admitted that each measured value of attenuation corresponds to a particular value ofβ (and the same property can be assumed for c). At each set of values of the parameters, therewill correspond a value of (c, β). Equivalently, if the characteristics of the medium are random,the parameter β which rules the weakening of the wave, is also random, and is representedby a random process B = {B(t), t ∈ R}. The same is true for the celerity c but its influence ishidden.

Therefore, when the parameters of the medium vary, we are in the following situation.Inside each pulse of experiments p1, p2, . . . , pm around the times t1, t2, . . . , tm (for examplewith a time interval of the order of few milliseconds), we have particular values of B(t), sayb1, b2, . . . , bm , which give particular values of the amplitude a1, a2, . . . , am

a1 = a0e−b1lω20 , . . . , am = a0e−bmlω2

0

which are observed values of a random process slowly varying with time

A(t) = a0 exp[ − B(t)lω2

0

]. (9)

Obviously, this model requires no hypothesis and is perfectly general, because the relationis bijective: each value of A(t) defines one and only one value of B(t). A physical situationis characterized by particular properties of the process B, which represents an equivalentrandom attenuation parameter. The coefficient of variation µ defined in (2) will depend on theprobability law of B(t), which will be assumed independent of t (hypothesis of stationarity).So, we can forget the variable t. When a homogeneous medium is at rest, B is reduced to theconstant β which does not depend on l and ω0 (in a first approximation). It is no longer the casewhen the medium composition changes with the distance, and/or when there is turbulence.In both cases, the parameter depends on l and/or is random. Moreover, l can be a parameterwhich is not the Euclidian distance, for example when the wave trajectory is not a straightline (this is the case for propagation in the sea, for large l). It is a supplementary difficulty forapplying formulas, because the real value of l will be unknown, like the real path of the wave.Being given a medium, and whatever the meaning of l, the main problem is to decide if the Bprobability law depends on l, and also on ω0.

The amplitude A(t) of the wave is characterized by the ‘coefficient of variation’ µ definedby (2):

µ =√

VarA(t)

E[A(t)].

Many experiments have been done on propagation in perturbed media (see Section 1), andtheories have been built for explaining practical results. In the 1950s, Mintzer predicted thatthe coefficient µ had to vary like ω0

√l, for short distances [5]. Measurements are more or

less in agreement with this assertion. Other theories give complicated formulas with too manyparameters to be seriously satisfied [4]. In the next section, we analyze the simplest modelwhere the probability law of B(t) does not depend on l and ω0.

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350 B. Lacaze

3.2 A probability law for B(t)

Let us assume that the probability law of the r.v B is independent of l, in a range (l1, l2) ofvalues of l. If B has a finite mean E[B], then ξ (s) = E[e−s B], s ≥ 0, possesses a derivativesuch that ξ ′(0) is finite [22], and we can write

ξ ′(0) = −E[B] = lims→0

ξ (s)−1s

ξ (2s) − ξ 2(s)

s= 2

ξ (2s) − 1

2s− ξ (s) − 1

s(ξ (s) − 1) − 2

ξ (s) − 1

s→s→0 0.

(10)

Moreover, from (9), we have E[A(t)] = a0ξ (lω20), E[A2(t)] = a2

0ξ (2lω20) and then

µ =√

ξ(2lω2

0

) − ξ 2(lω2

0

)ξ(lω2

0

) ξ (s) = E[e−s B]. (11)

Consequently, if E[B] is finite,√

ξ (2s)−ξ 2(s)s →s→0 0 from (10), ξ (0) = 1, and then

√ξ (2s)−ξ 2(s)

ξ (s)

cannot be equivalent to√

s for small s. Equivalently, from (11), the coefficient of variationµ cannot be of the order of

√α for an amplitude in the shape A = a0e−αB, when the mean

of B is finite. In particular, we have α = lω20 and then it is not possible to have µ ∼ ω0

√l as

predicted by Mintzer [5] and satisfied by experiments in [11–14].Actually, a generating function ξ (s) which behaves, in a neighbourhood of the origin point, as

ξ (s) = 1 + a1s ln s + a2s + o(s)

allows us to verify the Mintzer’s prediction. Such a ξ (s) has no moment. In this case, we have

ξ (2s) − ξ 2(s) = 2a1s ln 2 + o(s).

To my knowledge, the simplest one-sided probability function f (b) with this property isdefined by (see Appendix 2)

f (b) = 2θ

π (θ2 + (b − b0)2), b ≥ b0 > 0. (12)

It is the absolute value of a Cauchy law. Although this law has no moment, it is a modelfor a simple physical situation [22]. f (b) is called the ‘half-Cauchy law’, and recent worksuse it for modelling random parameters, particularly variances (recall that β is closely linkedto VarL(t), see Section 2 and [23]). A larger family of admissible probability laws can bededuced from (12) (see Appendix 2).

With the probability defined in (12), we have (see Appendix 2)

ξ (s) =∫ ∞

b0

f (b)e−sbdb = 1 + 2θ

πs ln s + as + o(s). (13)

where a = 2θπ

(ln θ + γ − 1) − b0. With such a model, we find (using the approximation

2√

ln 2π

∼= 0.94)

µ ∼= 0.94ω0

√θl. (14)

In order to obtain µ = 0.02 with ω0/2π = 208 kHz, l = 2 m like in figure 2 of [12], we haveto take θ = 0.13.10−15 s2 m−1. This value represents the distance between b0 (the smallestvalue possible for B) and the median b0 + θ of B. θ is of the same order of magnitude as thevalue of B at rest, i.e β =0.6.10−15 s2 m−1 [8]. Note that, with these values, the significantterm in (13) is about 10−4, which justifies the use of a limited development. Also, it seems

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reasonable to put b0 = β, if we consider that turbulence increases the attenuation, but this hasno influence on µ.

In the theory of Mintzer, the coefficient of variation is given by the approximation (formula10 of [12])

µ ∼= α√

x0

2πc0ω0

√l (15)

where 2πc0/ω0 is the wavelength. α and x0 are characteristics of the (random and normalized)index of refraction N (x) = 1 + αN0(x) in the fluid, with

E[N0(x)] = 0, E[N 20 (x)] = 1, E[N0(x)N0(x + x0)] = 1

e.

The parameter x0 can be viewed as a correlation length, and the last equality is right when theautocorrelation function E[N0(x)N0(x +u)] is the Gaussian exp[−(u/x0)2] or the exponentialexp[−|u|/x0]. From (14) and (15) we deduce the relation

θ ∼= 0.03α2x0

c20

.

The main drawback of this construction is that the log-amplitude ln A(t) = −B(t)lω20 has

no moment, which is in contradiction with classical works [4]. In Appendix 3, it is proved thatthe equivalence

µ ∼=√

E

[ln2 A(t)

m A

](16)

is true when the variations of A(t) around its mean m A can be assumed small and when themoments of B(t) are finite. If B(t) has no moment, then the term on the right of (16) is notdefinite, which is not the case for µ.

As a consequence, if we want a model which satisfies (16), it is necessary to give to B(t) aprobability law dependent on l and/or ω0. If the variance σ 2

B of B(t) is finite, we obtain, froma limited development of ξ (s),

µ = σBlω20.

To satisfy the Mintzer law, it is necessary that (at least in given ranges for ω0 and l)

VarB(t) ∼ 1/ω0

√l,

a relation which has no physical sense.From a statistical point of view, linking the sections above suppresses the Gaussian character

of the random transit time L . Now, it is the conditional r.v (L(t)|B(t) = b) which follows aGaussian law. We know that L(t) will be a Gaussian if and only if it is the same for B(t). But,this has no importance when the variations of B(t) are slow. Moreover, L(t) represents thefast variations of the transit time due to the molecular agitation, and not the measured transittime, equal to l/c for the medium at rest. When there are changes in the medium, giving arandom character to the attenuation parameter β, they also give a random character to c, whichdefines a new (assumed stationary) random process C = {C(t), t ∈ R}. Obviously, C(t) is notthe celerity in the fluid, which depends generally on the coordinates, but a random averageof the celerity along the trajectory of the wave. The measured transit time is T (t) = l/C(t),assuming also that the distance l is well defined. Then{

E[T ] = lE[1/C]

VarT = l2Var[1/C].

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352 B. Lacaze

The last result is not in accordance with the theory of Kolmogorov and Obukhov which predictsa variation in

√l for the standard deviation

√VarT , but the experiments are not convincing

[25]. Recent work seems to prefer a variation in l [26].

3.3 Remarks

µ is defined from the received amplitude A(t). We know that the notion of amplitude is subtle,except for monochromatic waves [28]. For a complex process Z, the amplitude is identifiedwith the modulus. For a real process X , it is the amplitude of the complex process Z builtthrough the Hilbert transform Y and the ‘analytic signal’ Z = X+ iY. Then, going back to (3),the amplitude of the real process cos ω0[t − Ll(t)] is not the amplitude of the complex processexp[iω0(t − L(t))], because sin ω0[t − L(t)] is not the Hilbert transform of cos ω0[t − L(t)]except for very particular L(t). Fortunately, we consider only the monochromatic part of thisprocess, neglecting the broadband part denoted V (t) in (5), which suppresses ambiguity.

The attenuation formula (1) was proved by experiments on gases or liquids at a giventemperature [10]. In the developments above, it is also used in the turbulent case, for shorttime intervals, where the temperature can vary over the travel time of the wave. In a smallinterval of time (like a pulse in the wave emission), it is currently assumed that the spatialdistribution of the macroscopic parameters is fixed, which ends to a constant value of theattenuation parameter beta in this interval. However, this hypothesis seems questionable forlong travels, and durations longer than a second. In the situation of turbulence, the attenuationis also of the form (1), but is larger, as proved in [4] (for example, see the formulas (49) and (50)therein), even if β depends itself on ω0. Intuition agrees with that, because of the increase ofthe disorganization of the wave. This last remark justifies a one-sided probability law for B(t).

In all situations, we can write the amplitude in the form A = a0e−αB, where α is a functionof ω0 and l, perhaps different from lω2

0, and it is possible that α depends on other parameters.Developments in Appendix 2 suggest that a coefficient of variation µ proportional to

√α

implies no moment for the log-amplitude (see Appendix 3).A Gaussian random transit time is a fair model for explaining the spectra of electromagnetic

waves [18–21]. The product ω0σ is large enough to cancel the monochromatic part, becauseelectromagnetic waves are in frequency bands far above the acoustic band. This is not the casefor the HF band (between 3 MHz and 30 MHz), but the distances of propagation are large(up to a few thousand kilometres) and the ionospheric layers are turbulent enough to yield alarge variance in the transit time. The process Z is reduced to the process V. The width of theV-spectrum is small in front of the emitted frequency. Then, the situation is very different foracoustic waves with respect to electromagnetic waves.

In media other than water or air, the attenuation formula (1) can be wrong. For example, theparameter in ω (and not in ω2) is available for propagation through pig spleen and human liver[27]. Such shapes and others (in the more general form ωη) can be explained by non-Gaussiantransit times, provided that exp[−|ω|η] is the characteristic function of an infinitely divisiblerandom variable (this is the case if and only if 0 < η ≤ 2).

4. Conclusion

The model defined by (3) can be used for acoustic weakening, because the noise which ishighlighted in the model cannot be viewed by devices. In the case of monochromatic wavesa0eiω0t propagating in a quiet medium, the amplitude a is a constant, and the attenuation varieslike a = a0 exp[−βlω2

0]. When the medium becomes turbulent, for example due to temperature

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gradients which create eddies, the amplitude a shows fluctuations which are measured by the‘coefficient of variation’ µ. The attenuation is no longer a constant, but a value can be attributedfor intervals containing hundreds of periods of the emitted wave, i.e., for each burst in theexperiments cited above. It seems equivalent to consider that the parameter of attenuation is ar.v. B and in the same way that the amplitude is a r.v. A = a0 exp[−Blω2

0]. Mintzer has provedthat µ has to vary like ω0

√l, and experiments are in accordance with this law when the medium

is not too turbulent. We have shown that a particular probability distribution independent of land ω0 can be fitted to Mintzer’s law. It was shown in sections 2 and 3 that, both schemes (formotionless and turbulent medium) can be studied separately. The formula (1) can be proved,giving to the transit time a Gaussian character; the coefficient of variation µ can be explainedby giving a random character to β, i.e. to the standard deviation of the transit time. Finally,turbulent and motionless propagation can be approached by a conditional Gaussian process.

Appendices

Appendix 1: A model for the noise

To understand the disappearance of the process V in (5), we construct the process L from ahomogeneous Poisson process {tn, n ∈ Z} with parameter λ [28], and a sequence {Xn, n ∈ Z}of independent Gaussian r.v. with E[Xn] = m > 0,VarXn = σ 2 (and σ/m very small). Now,define L by

L(t) = Xn, tn < t ≤ tn+1.

This means that the travel time L(t) varies at each Poisson time tn , and is unchanged betweentwo successive Poisson times. A quick calculation leads to

ψ(ω) = e−imω−(ωσ )2/2 φ(ω, τ ) = e−ω2σ 2 + e−λ|τ |(1 − e−ω2σ 2).

Whatever λ (the mean number of Poisson events and then the mean number of trajectories pertime unit), the amplitude of the monochromatic received wave is equal to a0e−ω2

0σ2/2 when the

wave a0eiω0t is emitted. The spectral density sV (ω) of the noise V is calculated from (6), bythe Fourier transform [29]

sV (ω) = λa20

π

(1 − e−ω2

0σ2) 1

(ω − ω0)2 + λ2.

The maximum of sV (ω) is reached at the point ω = ω0, and

limλ→∞

sV (ω0) = limλ→∞

a20

1 − e−ω20σ

2

πλ= 0

In other words, the larger λ is, the smaller will be the V power seen by a receiver with alimited frequency band. In this model, the attenuation of the monochromatic wave dependsonly on the variance σ 2 of the transit time, and the noise spectrum depends on the frequencyof changes in the trajectories (led by the Poisson process of parameter λ). This property isobviously true for any probability distribution with finite variance for the Xn .

In air at normal conditions, the number of shocks of molecules on a surface is about 3.1015

by mm2 µs. This implies a value at least of the same order of magnitude for λ, to comparewith 107, corresponding to a few MHz.

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354 B. Lacaze

Appendix 2: The law of B(t)

The problem is to find a generating function ξ (s) = E[e−s B] and a function f (s) which satisfy

ξ (s) = 1 − f (s) + o(s), f (2s) − 2 f (s) = as + o(s) (17)

in a neighbourhood of the origin, with a �= 0, f (0) = 0, because

ξ (2s) − ξ 2(s) ∼ − f (2s) + 2 f (s)

and because we want √ξ (2s) − ξ 2(s) ∼ √

s.

We see that the only function f (s) in the (relatively simple) shape sα lnδ s ln lnγ s . . . whichhas this property is s ln s. Now, mathematical tables give the following formula

α(s) = 2

π

∫ ∞

0

e−sx

1 + x2dx = (1 − 2

πSi(s)) cos s + 2

πCi(s) sin s

with (γ = 0.57 . . . is the Euler constant)

Si(s) =∫ s

0

sin t

tdt, Ci(s) = γ + ln s +

∫ s

0

cos t − 1

tdt = −

∫ ∞

s

cos t

tdt.

Close to the origin, we have

α(s) = 1 + 2

πs ln s + 2(γ − 1)

πs + o(s)

which satisfies (17) and which then explains (13). In dictionaries of Laplace transforms (ofone-sided probability densities), I have found only one law with this property.

Characteristic functions φ(t) = E[eit B] of stable distributions with exponent 1 are of theform [30]

φ(t) = exp

[iat − b|t |

{1 + i

2c

π

t

|t | ln |t |}]

, b > 0, |c| ≤ 1.

These functions have the right behavior near the origin (for c �= 0), but they correspond toprobability densities which are entire functions (see theorem 5.8.5 in [30]). Consequently,they are not one-sided.

The half-Cauchy law is introduced because of its analytical properties, and it is a modelused today in statistics. Moreover, let us consider that C is a random process independent ofB, having a generating function ζ (s) = E[e−sC(t)] well defined for s > 0. We assume that, forsome d ≥ 1, d ′ �= 0

ζ (s) = 1 + d ′sd + o(sd ), s ≥ 0. (18)

This is the case for a � law (d = 1) and for a Gaussian law (d = 1 or 2). If ξ satisfies (13),then the product [ξζ ] has the same property

[ξζ ](s) = 1 + 2θ

πs ln s + a′s + o(s).

Equivalently, the processes B and B + C lead to the same µ. Then, we can take for the B(t)probability density any convolution of (12) with a probability density having a finite moment.The behavior at the origin of this product is dominated by ξ (s), which tends to 0 slowly ats = ∞. Then, it is easy to take ζ (s) with a dominant behavior for large s. Also, this propertyis linked to the self-decomposability of the half-Cauchy law [24, 30].

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A stochastic model for acoustic attenuation 355

This large possibility of choices cannot hide a true drawback. Indeed, r.v. with ξ (s) like[ξζ ](s) as generating functions do not have moments. This implies the same drawback for thelog-amplitude ln A(t), which is not in accordance with the literature [4].

Appendix 3: Coefficient of variation and log-amplitude

Let us assume that the mean m A = E[A] of the r.v A is finite (A ≥ 0, m A > 0). If the variationsof A around m A are ‘small’, then

lnA

m A= ln

[1 + A − m A

m A

]= A − m A

m A− 1

2

(A − m A

m A

)2

. . . (19)

Neglecting terms of order three, we obtain (µ is the coefficient of variation of A)√E

[ln2 A

m A

]∼=

√VarA

E[A]= µ. (20)

This formula relates the coefficient of variation to the log-amplitude second-order moment. Inreal cases, since A is bounded, all the moments of A exist. However, this result is not obviousfor ln A. For example, let consider propagation towards a point which is either shadowedor reachable according to the physical conditions. In this case, the moments of ln A do notexist, but the coefficient of variation µ is perfectly defined. The same situation happens whenA = a0 exp[−αB], but E[B] = ∞. In this case, E[ln2 A

m A] = ∞, but µ is finite, and can follow

Mintzer’s law (see Appendix 2).Finally, if A′ = A+N (N > 0), A′ is a good estimation of A when m A m N and σA σN .

We have also µA′ ∼= µA. N could model the noise due to the measurement devices which isadded to the signal of interest. However, the existence of E[ln2 N ] implies the existence ofE[ln2 A′], even if E[ln2 A] = ∞. In this case, the log-amplitude of A′ is more related to themeasurement noise N than to the wave amplitude A.

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