One-dimensional numericalsimulations of random sound impulses

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING Waves Random Media 13 (003) WAES IN RANDOM MEDIA PII: S (03)616-0 One-dimensional numericalsimulations of random sound impulses KMurawski 1 and M Mȩdrek 1 Institute of Physics, UMCS, ulica Radziszewskiego 10, Lublin, Poland WIBIS, Technical University of Lublin, ulica Nadbystrzycka 40, Lublin, Poland Received 4 March 003, in final form 1 July 003 Published 8 August 003 Online at stacks.iop.org/wrm/13/311 Abstract We perform one-dimensional numerical simulations of small-amplitude acoustic pulses in space- and time-dependent random mass density and timedependent velocity fields. Numerical results reveal that: (a) random fields affect thespeeds, amplitudes and, consequently, shapes of sound pulses; (b) for weak random fields and short propagation times the numerical data converge with the analytical results of the mean field theory which says that a space-dependent (time-dependent) random field leads to wave attenuation (amplification) and all random fields speed up sound pulses; (c) for sufficiently strong random fields and long propagation times numerical simulations reveal pulse splitting into smaller components, parts of which propagate much slower than a wave pulse in a non-random medium. These slow waves build an initial stage of a wave localizationphenomenon. However, thiseffectcanbeveryweak inareal three-dimensional medium. 1. Introduction Being the most common and simplest waves we can envisage, sound waves have been the subject of intensive studies for a long time. Although sound wave propagation is well understood in homogeneous media,the compleity of the phenomena in inhomogeneous media causes such waves to be largely unknown. In particular, many of the most interesting problems in sound waves are those that can be viewed as wave propagation through a random medium which is a family of media together with a probability distribution for this family. Sound wave propagation through a random medium then refers to the wave propagation through each member of this family. Some work has been done on random sound waves. For instance, the analytical study of sound waves in a space-dependent random mass density field reveals that these waves are accelerated and attenuated, with shorter waves affected more [1]. On the other hand, the effect of a time-dependentrandom mass density field is to accelerate and amplify the sound waves [] /03/ $ IOP Publishing Ltd Printed in the UK 311

2 31 KMurawskiandMMȩdrek Accordingly, a space-dependent (time-dependent) random velocity field slows down (speeds up) and attenuates (amplifies) sound waves [3, 4]. The above analytical results have been derived for periodic sound waves. It is illustrative to consider impulsively generated waves, as an impulse consists of a packet of periodicwaves and because sound waves can be ecited impulsively. These studies stem from our wish to understand the influence of random fields on impulsively generated sound waves. Such an attempt has already been undertaken in the past (e.g. [5 10]). However, the authors considered the case of sound pulses in a random mass density with space-dependent step-wise profiles although it seems natural to discuss smooth profiles too []. Belzer et al [6] adopted the causality principle epressed by the Kramers Kröning relation to investigate the pulse propagation in random media. Wenzel [5] described the propagation of pulses by the use of perturbation theory. Shapiro and Hubral [7] discussed in great mathematical and eperimental detail random pulse propagation. Müller et al [8] discussed the pulse evolution in a single realization of two- and three-dimensional random media but they paid attention to a self-averaging phenomenon. This paper is organized as follows. In section we present the numerical model for random sound waves. Sections 3 and 4 contain results of the numerical simulations for random mass density and velocity fields, respectively. This paper concludes with the presentation and discussion of the main results in section 5.. Numerical model for sound waves in random fields We consider sound waves which are described by the one-dimensional Euler equations: ϱ t + (ϱ ) = S ϱ, (1) (ϱ ) + t (ϱ + p) = S + ϱs, () E + [ (E + p)] = t S ϱ + ϱs, (3) where E p/(γ 1) + ϱ /isthetotal energy density, ϱ is the mass density, is the component of the flow velocity, p is the pressure, γ is the adiabatic inde and S ϱ (t) and S (t) are the source terms which are used here to seed appropriate random fields (e.g. [, 11]) at equilibrium which consists of the non-random uniform quantities ϱ 0 = constant, p 0 = constant, and random mass density ϱ r (, t) and velocity r (t) fields: ϱ 0 + ϱ r (, t), r (t), p 0. (4) The random fields are chosen to have a Gaussian correlation function: f r ( 1, t 1 ) f r (, t ) =σ ep ( 1 ) ep ( t 1 t ), (5) where σ corresponds to the strength of the random field, l and l t are, respectively, the correlation length and the correlation time and f r ( i, t i ), i = 1,, is a statistically homogeneous random quantity (in this case ϱ r or r )with a vanishing ensemble average, f r =0. Henceforth we work with normalized quantities. For instance, a spatial distance and time t are epressed in units of l and l /c 0,respectively. Here c 0 = γ p 0 /ϱ 0 is the sound speed in the uniform medium. This sound speed is used to normalize the velocity while ϱ is epressed in units of ϱ 0. 4l 4l t

3 Random sound impulses 313 We solve the Euler equations (1) (3) numerically byadopting the appropriate codefromthe CLAWPACK package [1]. This package uses modern shock capturing schemes for solving hyperbolic equations such as the Euler equations [1, 13]. These schemes are designed to satisfy conservation and low numerical error constraints. In particular, the total energy is conserved in the system, in accordance with equation (3). Numerical simulations are performed in the region 0 64π. (6) At = 0and64π free boundary conditions are set. So, a sound wave can freely leave the simulation region. These simulations are carried over time 0 t 190. (7) Initially, at t = 0, we launch a small pulse: (, t = 0) = 10 3 ep [ ( 5) 4 ]. (8) This pulse splits into two counter-propagating pulses. In the non-random medium, for which σ = 0, these pulses are of identical amplitudes,i.e. equal to Theleftward propagating wave reaches the left boundary of the simulation region at t = 5andislost from sight as this boundary is transparent to any outgoing signal. The above pulse is synthesized from Fourier components whose spectrum is Gaussian. As theshortest waves are affected most by a random field [] we epect that pulses will be altered by the presence of a random field. For instance, in a random medium a pulse will generally be shifted to a different spatial position and its amplitude will be altered. 3. Numerical results for random mass density fields In this part of the paper we present and discuss the numerical results for the case of a random mass density. First we consider the case of ϱ r () and subsequently ϱ r (t) in the following part A space-dependent random mass density field Figure 1 displays numerically the obtained pulses n (, t = 171) for eight different realizations of the random medium ϱ r (). Heren denotes a particular realization of the random medium. It is seen that a scenario of pulse propagation varies over a particular realization of the random field as the pulse amplitude and position are different for each different realization. However, for most of these realizations particular pulses are shifted towards higher values of and their amplitudes are reduced. This effect is enhanced for a stronger random field (right panel of figure 1). Behind the pulses dispersive tails are present. In seismology and rock physics such atail is called a coda [17]. For a given realization of the random field ϱ r () atailhas been generated from the original pulse as a result of back-scattering from ϱ r (). Thisback-scattering is a part of the interaction process, which gives way to diffusive transport akin to Brownian motion. During this interaction, pulse energy is transferred into a dispersive tail. The energy transfer is higher as the pulse progresses and the propagation distance is long. Interference of the multiple backward scattered wave field (which is always partially constructive) inevitably leads to an accumulation of the wave field energy within local potential wells. This effect is enhanced in the case of a stronger random field (right panel of figure 1).

4 314 KMurawskiandMMȩdrek 6e-04 5e-04 4e-04 3e-04 e-04 1e-04 0e n e-04 5e-04 4e-04 3e-04 e-04 1e-04 0e n Figure 1. Spatial profiles of the pulses (, t = 171) for eight different realizations of ϱ r () with σ = (leftpanel) and σ = (right panel). The broken line that is perpendicular to the ais corresponds to the position of the pulse in the deterministic medium for which σ = Figure. The spatial profiles of the ensemble averaged pulses (, t = 190) in ϱ r () with σ = (long broken curve), (short broken curve) and (dotted curve). The full curve represents the non-random pulse (σ = 0). Particular realizations of the pulses n,suchasthose, for instance, from figure 1, are used to obtain the ensemble averaged pulse, namely (, t) = 1 n ma n (, t). (9) n ma n=1 Such ensemble averaged pulses which are obtained from this formula with n ma = 100 are presented in figure for several strengths of the random field, σ =, and. A pulse is synthesized from Fourier components from which the shortest ones propagate fastest [1]. Positions of the pulses do not line up over different realizations of the random medium. Consequently, averaging over realizations results in a diffusion of the pulse. A pulse width increases with the strength of ϱ r (), σ. The random pulses are shifted forward and their amplitudes are reduced (broken curves of figure ). This effect is higher for a stronger random mass density field ϱ r () and is presented quantitatively in figure 3, which displays the relative phase shift [10]: X = ma(σ = 0, t = 190) ma (σ 0, t = 190) 100% (10) ma (σ = 0, t = 190)

5 Random sound impulses 315 Χ [%] A r - σ σ Figure 3. The relative phase shift X (left panel) and the relative amplitude A r (right panel) as functions of the strength of the random field ϱ r (), σ. The crosses correspond to particular realizations of the random field and ensemble averaged values are displayed by the squares which are joined by the broken lines. A f (K) 1e-05 9e-06 8e-06 7e-06 6e-06 5e-06 4e-06 3e-06 e-06 1e e-06 3e-06.5e-06 e e-06 1e-06 5e K A f (K) 1e-05 9e-06 8e-06 7e-06 6e-06 5e-06 4e-06 3e-06 e-06 1e K Figure 4. Fourier spectra of (, t = 171) for the random field ϱ r () with σ = (leftpanel) and (right panel). The full curves correspond to the non-random medium with σ = 0. Note that, as a result of wave attenuation, shorter random sound waves contribute less to the spectrum (chain curves) than their non-random counterparts (full curves). and the relative amplitude A r = ma(σ 0, t = 190) (11) ma (σ = 0, t = 190) versus σ.here ma corresponds to the spatial position of a pulse maimum ma.thecrosses correspond to particular realizations of the random field and the squares are associated with ensemble averaged signals. We see that the effect of the random field is to speed up and attenuate waves and this effect increases with the strength of the random field σ.att = 190, in the case of σ =, the phase shift isclose to % and the pulse amplitude is reduced by about 15%. As a consequenceofthat, we deduce that ϱ r () eerts a stronger effect on a wave amplitude than on its phase. Awaveattenuation is discernible in figure 4 which compares Fourier spectra for the nonrandom medium (full curves) and random media (chain curves). As a consequence of wave attenuation Fourier components with high K contribute less to the power spectra than their non-random counterparts and this effect increases with the strength of the random field, σ. It is noteworthy that the effect which is displayed in figure 4 is partially due to numerical diffusion, as a consequence of which short waves are removed from the system. However,

6 316 KMurawskiandMMȩdrek 6e-04 5e-04 4e-04 3e-04 e-04 1e-04 0e+00-1e n Figure 5. Spatial profiles of the pulses at t = 171 in eight realizations of ϱ r (t) with σ =. numerical tests have shown that, for the chosen grid, the numerical diffusion is negligibly small in comparison to the wave attenuation [14]. 3.. A time-dependent random mass density field In this part of the paper we present numerical results for impulsively generated sound waves that propagate in a time-dependent random mass density field. Figure 5 shows spatial profiles of the sound pulses for eight particular realizations of the random field. Comparing figures 5 and 1 we conclude that, for the weak random fields with σ = andforatime-dependent random mass density field, pulses are spread out more spatially than in the case of their spacedependent counterparts, ϱ r (). Hence we conclude that a time-dependent random field eerts astronger effect on wave pulses than a space-dependent field. Figure 6 displays the ensemble averaged pulses which have been obtained with the use of n ma = 150 realizations in equation (9). These pulses are drawn for the weak random field with σ = (toppanels) and a stronger random field of σ = (bottom panels). A stronger random field leads to faster splitting of the original pulse. For instance, in the case of σ = (top right panel) even at t = 190 the random pulse (full curve) resembles the non-random pulse (broken curve) despite its lower amplitude and larger width. On the other hand, the case of σ = reveals the main pulse with its maimum located close to = 40 and two smaller pulses at = 50 and 60 (bottom middle panel). These small pulses have resulted from high Fourier components which are speeded up and amplified by a time-dependent random mass density. Such small pulses have not been observed in the case of ϱ r () for which high Fourier components are attenuated. At t = 15 in the case of σ = (toppanels) the small pulse is generated from short Fourier harmonics which are speeded up and amplified by a time-dependent random mass density field ϱ r (t). In the case of σ = (bottom panels) a strong back-scattering is present. As a result of that much of the wave signal lags behind as, at t = 38, the maimum of this signal is located at = 40, while the non-random pulse has its maimum at = 43. At t = 190 this effect is even stronger (right bottom panel of figure 6). A part of the pulse which lags much behind the positions of the non-random pulse is a result of the presence in the system of a significant amount of Fourier components which propagate slower than their coherent counterparts. Such slow waves have been observed for periodic sound waves in a time-dependent random mass density field [14]. In the present case they build an initial stage of the phenomenon which is reminiscent of wave localization [15].

7 Random sound impulses (t=95) (t=19) (t=15) (t=38) (t=171) znorm. amplituda amplituda (t=190) (t=190) Figure 6. Temporal evolutions of the ensemble averaged pulses in a time-dependent random mass density field ϱ r (t) with its strength σ = (toppanels)and (bottom panels). The broken curves correspond to the pulse in the non-random medium for which σ = 0. In 1958 Anderson [15] was the first who showed that a particle in a random potential can be localized in space, turning a conductor into aninsulator. As a consequence of this phenomenon, which is known as Anderson localization, thewavesdonot propagate at all but instead they are trapped in a spatial region as a result of back-scattering that produces a standing wave. In this case the wave back-scattering is so strong that the net effect is to form standing waves that oscillate back and forth in a bounded region of space. It is astonishing that in a one-dimensional medium this localization always takes place, even if the scattering is weak and the inhomogeneities are sparsely distributed [16]. It is onlyfor space dimensions higher than two that it is thought that localization occurs only when the scattering is sufficiently strong. The pulse spreading may suggest that, as in the case of ϱ r (), waveattenuation takes place. However, this conclusion is misleading as, in fact, sound waves are amplified by ϱ r (t). Figure 7 confirms this statement as it shows that Fourier components with high K contribute more to the power spectrum than their non-random counterparts. It is noteworthy here that, in the case of ϱ r (), Fourier components with high K contribute less to the power spectrum than for the non-random wave pulse. Compare figure 7 with figure 4. This behaviour is a consequence of wave amplification which works in the case of ϱ r (t). This effect increases with K and σ (not shown). 4. Numerical results for sound waves in a time-dependent random velocity field It is noteworthy that the amplitude of the initial pulse is equal to 10 3 (see equation 8) and the strengths of the random fields we have worked with are at least σ =. For a weaker field,random effects are obviously lower and they are more difficult to trace numerically. For sufficiently high values of σ,wave pulses are overlaid by the random field and we eperienced problems with tracing pulse development in time. As a consequence of that we have limited ourselves to the case of a time-dependent random velocity field r (t) for which there have not been such problems. The case of r () is devoted to future studies with the use of the linearized version ofequations (1) (3).

8 318 KMurawskiandMMȩdrek 9e-06 8e-06 7e e-07 A f (K) 6e-06 5e-06 4e-06 3e-06 8e-08 e e-06 1e K Figure 7. Fourier spectra of (, t = 171) for the random field ϱ r (t) with σ =. The full curves correspond to the non-random medium with σ = 0. Note that, as a result of wave amplification, shorter random sound waves contribute more to the spectrum (broken curves) than their non-random counterparts (full curves). 6e-04 5e-04 4e-04 3e-04 e-04 1e-04 0e n e-04 5e-04 4e-04 3e-04 e-04 1e-04 0e n Figure 8. Spatial profiles of the pulses at t = 171 for eight different realizations of r (t) with σ = (leftpanel) and (right panel). The dotted line that is close to the pulse maima corresponds to the position and amplitude of the pulse in the deterministic medium for which σ = 0. Figure 8 displays a spatial distribution of wave pulses (, t = 171) for eight different realizations of r (t). Notethat positions of the particular pulses varied greatly over particular realizations of the random field and most pulses have not reached the position of the nonrandom pulse, = 176, that is shown by the broken line perpendicular to the ais. Compare with figures 1 and 5. This pulse lagging is stronger for a stronger random field (compare the left and right panels of figure 8) and is a result of strong back-scattering which leads to waves propagating leftwards. Theback-scatteringeffectcan be observedin figure9 which shows time evolution of the pulses for the random field strength σ = (toppanels) and σ = (bottom panels). In the former case, the initially launched pulse is diffused in time and its amplitude is reduced. In the latter case, the field is strong enough that theinitial pulse splits into several low amplitude pulses. Two left pulses of the rightmost bottom panel propagate slower than the coherent pulse that is denoted by the broken curve. This suggests that a process of wave trapping has already begun, similar to what occurred in the case of ϱ r (t). Compare with figure 6.

9 Random sound impulses (t=38) (t=95) (t=171) (t=38) (t=95) (t=171) Figure 9. Temporal evolution of the ensemble averaged pulse in the random velocity field r (t) with σ = (top panels) and (bottom panels). The broken curve corresponds to the impulse which propagates in the deterministic medium for which σ = Summary In this paper we have studied the effect of random mass density and velocity fields on speeds and amplitude alteration of sound waves. The one-dimensional numerical simulations have been performed for impulsively generated waves. The general conclusion is that, for weak random fields and short propagation distances, all random fields considered speed up sound pulses. Space-dependent (time-dependent)random fields lead to wave attenuation (amplification). For astronger random field and long propagation distances a wave pulse eperiences a stronger velocity shift and amplitude alteration as well as the pulse splitting into parts with a greater contribution from those components lagging behind. This effect is clearly observed in the cases of time-dependent random fields and is reminiscent of the initial stage of Anderson localization [15]. However, the one-dimensional nature of our model restricts the range of applications of this conclusion as, in fact, all media are three-dimensional. It has been shown that Anderson localization always takes place in one-dimensional media, weakens significantly in the two-dimensional case, while it becomes etremely weak when etended to three dimensions [15]. Acknowledgment The authors epress their cordial thanks to the unknown referees for their constructive comments. References [1] Nocera L, Mȩdrek M and Murawski K 001 Astron. Astrophys [] Murawski K, Nocera L and Mȩdrek M 001 Astron. Astrophys [3] Murawski K 00 Waves Random Media [4] Murawski K, Mȩdrek M and Ostrowski M 001 Proc. INTAS Workshop on MHD Waves in Astrophysical Plasmas ed J L Ballester and B Roberts p 179 [5] Wenzel A R 1975 J. Math. Phys

10 30 KMurawskiandMMȩdrek [6] Beltzer A I, Wegner J, Tittman B R and Haddow J B 1989 Bull. Seismol. Soc. Am [7] Shapiro S A and Hubral P 1999 Elastic Waves in Random Media (Heidelberg: Springer) [8] Müller T M, Shapiro S A and Sick C M A 00 Waves Random Media 1 3 [9] Mȩdrek M, Michalczyk J, Murawski K and Nocera L 00 Waves Random Media 1 11 [10] Murawski K, Nakariakov N and Pelinovsky E N 000 Astron. Astrophys [11] Juvé D,Blanc-Benon Ph and Wert K 1999 Theoretical and Computational Acoustics 97 ed Y-C Teng et al (Paris: World Scientific) p 653 [1] Leeque R 00 Finite olume Methods for Hyperbolic Problems (Cambridge: Cambridge University Press) [13] Murawski K 00 Analytical and Numerical Methods for Wave Propagation in Fluids (Singapore: World Scientific) [14] Murawski K and Mȩdrek M 003 Waves Random Media submitted [15] Anderson P W 1958 Phys. Rev [16] van der Baan M 001 Geophys. J. Int [17] O Doherty R F and Anstey N A 1971 Geophys. Prospect