Singularities and spectral asymptotics of a random nonlinear wave in a nondispersive system

Journal Pre-proof Singularities and spectral asymptotics of a random nonlinear wave in a nondispersive system S.N. Gurbatov, O.V. Rudenko, A.V. Tyurina

PII: DOI: Reference:

S0165-2125(19)30272-0 https://doi.org/10.1016/j.wavemoti.2020.102519 WAMOT 102519

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Wave Motion

Received date : 29 July 2019 Revised date : 28 November 2019 Accepted date : 17 January 2020 Please cite this article as: S.N. Gurbatov, O.V. Rudenko and A.V. Tyurina, Singularities and spectral asymptotics of a random nonlinear wave in a nondispersive system, Wave Motion (2020), doi: https://doi.org/10.1016/j.wavemoti.2020.102519. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Singularities and spectral asymptotics of a random nonlinear wave in a nondispersive system S.N. Gurbatov1,*, O.V. Rudenko2,3,4, A.V. Tyurina1 Lobachevsky State University of Nizhni Novgorod, Gagarin Ave. 23, 603950 Nizhni Novgorod, Russia 2 Lomonosov Moscow State University, Vorob’ev Hills, Moscow 119991, Russia 3 Prokhorov General Physics Institute, Russian Academy of Sciences, 119991 Moscow, Russia 4 Institute of Physics of the Earth, Russian Academy of Sciences, 123242 Moscow, Russia

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1

Abstract

In this paper, we study the propagation of high-intensity acoustic noise in free

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space and in waveguide systems. A mathematical model generalizing the Burgers equation is used. It describes the nonlinear wave evolution inside tubes of variable cross-section, as well as in ray tubes, if the geometric approximation for heterogeneous media is used. The generalized equation transforms to the common Burgers equation with a dissipative parameter, known as the “Reynolds-Goldberg number”. In our model, this number depends on the distance travelled by the wave. With a zero “viscous” dissipative term, the model reduces to the Riemann (or

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Hopf) equation. Its solution presents the field by an implicit function. The spectral form of this solution makes it possible to derive explicit expressions for both dynamic and statistical characteristics of intense waves. The use of a spectral

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approach allowed us to describe the high-intensity noise in media with zero and finite viscosity. Applicability conditions of these solutions are defined. Since the phase matching is fulfilled for any triplet of interacting spectral components, there is an avalanche-like increase in the number of harmonics and the formation of shocks. The relationship between these discontinuities and other singularities and

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the high-frequency asymptotic of intense noise is studied. The possibility is shown to enhance nonlinear effects in waveguide systems during the evolution of noise. *

Corresponding author. E-mail address: [email protected]

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Keywords: nonlinearity, noise, spectrum, Burgers equation.

1.

Introduction

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The problem of nonlinear transformations of spectra is important for both theoretical physics and for applications. Among the latter, one can specify effects of signal attenuation in a noise field, of low-frequency noise suppression when

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exposed to an intense harmonic signal, and some other phenomena [1]. Obviously, of the greatest interest are two groups of such phenomena, in which: (i) nonlinear effects are strong; (ii) nonlinearity is weak, but can accumulate in time or space, leading to strongly pronounced nonlinear effects [2]. It is important to describe the dynamics of spectral transformations and to invent ways to control them, as well as

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to find stationary (steady-state) spectral distributions, including universal highfrequency “tails”. As shown below, these tails are associated with the appearance of various singularities in the spatio-temporal structure of the wave field. This paper studies the spectra of random (noise) waves of high intensity, spaced in the form of a directional beam. The main attention is paid to the strong manifestations of nonlinear phenomena, which are observed at high acoustic Reynolds numbers (or Goldberg numbers) [3, 4]. It is known that the dispersion or

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frequency dependence of the sound speed is insignificant in a wide frequency range. Therefore, the conditions of synchronism (or of wave resonance) are

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fulfilled for any triplet of interacting harmonics. This leads to an avalanche-like growth in the number of spectral components and, as a result, to the formation of discontinuities in the initially smooth wave profile. For noise waves, a strong broadening of the spectrum takes place and the associated process of formation of high-frequency asymptotic occurs. From the mathematical point of view, the

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description of noise requires finding the statistical solutions of nonlinear models like the Riemann (Hopf), Burgers or Khokhlov-Zabolotskaya equations. The statistics at the input of the medium is assumed to be known [3-5]. In the theory of random hydrodynamic and acoustic fields it is of particular

interest to find high-frequency asymptotic of the spectrum. This asymptotic is

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commonly associated with the singularities of the spatio-temporal structure of the field. In the theory of turbulence, for example, the Kolmogorov spectrum plays an important role. In the theory of non-dispersive waves, an exponentially decreasing dissipative “tail” of the spectrum transforms at high intensities in an “omega-to-

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minus-second” power spectrum, which is associated with the formation of shock fronts. A strongly distorted nonlinear wave may contain not only discontinuities,

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but also singularities of different types, which form a new asymptotic of the spectrum. Examples include the following features: an inflection in the wave profile immediately preceding the appearance of a discontinuity; profile ambiguity with strong distortion in the absence of high-frequency dissipation; the sharpening of the maximum (jump of the derivative or “cusp”), observed in the focal region

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when diffraction is taken into account. In mathematics, such features are known as “catastrophes” (see [6]). For better understanding the physics of nonlinear wave processes, we will not deal with the general mathematical theory, but limit ourselves to analysing the real and artificial features of the noise spectra resulting from the use of approximate methods.

Consider for example the one-dimensional wave described by the generalized Burgers equation [3, 4, 7]:

𝑝 𝑑 𝑙𝑛𝑆 𝑠 2 𝑑𝑠

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𝜕𝑝 𝜕𝑠

𝜀 𝜕𝑝 𝑝 𝜌𝑐 𝜕𝜏

𝑏 𝜕 𝑝 2𝜌𝑐 𝜕𝜏

(1)

In Eq. (1), p is the acoustic pressure, s is the distance travelled by the wave, =t–s/c

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is the time in accompanying coordinate system moving at the sound speed c. The parameters of the medium, namely: the density , the nonlinearity , the effective viscosity b, and the speed c, can depend on the coordinate s. Eq. (1) also can be used as the transport equation in the nonlinear geometric acoustics of an

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inhomogeneous medium. One can derive it, for example, by going to the highfrequency limit in the Khokhlov-Zabolotskaya equation [8]. In this case, S(s) is the cross section of the ray tube, and the distance s is measured along the curvilinear

central beam [9].

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The analysis of solutions to Eq. (1) will be discussed in the following sections. Here we start with the simplest case of a homogeneous medium in the absence of canalizing walls, that is, at S(s)=const, s≡x. In this case, Eq. (1) turns into the common Burgers equation 𝜀 𝜕𝑝 𝑝 𝜌𝑐 𝜕𝜏

𝑏 𝜕 𝑝 2𝜌𝑐 𝜕𝜏

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𝜕𝑝 𝜕𝑥

(2)

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Further, it is important to discuss one exact particular solution of Eq. (2), describing the shock compression front of finite width (see Problem 3.5, Eq. (3.16) in [10]): 𝑝

𝑝

𝑝

𝑝

𝑝

2

2

tanh

𝑝 𝜀 𝑝 ∙ 2 𝑏

𝜏

𝑝 𝜀 𝑝 ∙ 𝑥 2 𝜌𝑐

(3)

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From formula (3) it follows that at →-∞ p→p1, and at →+∞ p→p2>p1. Thus, the magnitude of the “jump” at the shock front equals (p2−p1). When this value tends to infinity, the hyperbolic tangent turns into a function sgn() describing the discontinuity − a shock front of zero width. The front of a finite width turns into an ideal discontinuity even at a different limiting transition, when the dissipation is negligible, that is b→0. The presence of singularities of the “function discontinuity” type leads to the appearance of a power asymptotic of the wave

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spectrum. To verify this, we expand solution (3) into Fourier spectrum. We use the technique described in the textbook [11] “Theory of Waves” (see Section 4, Chapter 9).

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The frequency spectrum of solution (3) is calculated by the general formula 𝑝 𝜔

Here, the constants a, a1,

𝑎

𝑎 tanh 𝛽𝜏 exp

𝑖𝜔𝜏 𝑑𝜏

(4)

are introduced for brevity. One can easily establish

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connections with old constants by comparing formulas (3) and (4). Further,

differentiating by parameter 𝑑𝑝 𝜔 𝑑𝛽

2𝑖𝑎

, we reduce (4) to the table integral: 𝜏sin 𝜔𝜏 𝑑𝜏 cosh 𝛽𝜏

𝑖𝜋𝑎

𝜋 𝑑 𝛽 sinh 𝜔 2𝛽 𝑑𝛽

(5)

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This formula shows that the high-frequency section of the wave spectrum and the spectrum of its intensity are equal:

𝐺 𝜔

𝑖𝜋 sinh

𝑝𝑝

𝜋𝑎 𝛽

∗

𝜔 ,

sinh

Returning to the old notation, we find: 𝜋𝑏 𝜀

𝜋𝑏 𝜔 𝜀 𝑝 𝑝

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𝐺 𝜔

(6)

𝜋 𝜔 2𝛽

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𝑝 𝜔

sinh

(7)

With weak and strong dissipation, this formula shows the universal laws of decreasing the spectrum at high frequencies: 𝑝

𝑝

𝜔 ,𝐺 𝜔

2𝜋𝑏 𝜀

exp

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𝐺 𝜔

2𝜋𝑏 𝜔 𝑝 𝜀 𝑝

(8)

In the case of a noise signal, the shocks are randomly located in the wave profile and have different “amplitude” (p2−p1). Therefore, it is necessary to perform averaging in formulas (8) with the corresponding distribution function for the jump amplitude.

Another singularity of the “derivative discontinuity” type may appear in the focal area for the focused wave [12, 13]. It was shown [12], that the wave near the

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focus is described by the projection of the Khokhlov-Zabolotskaya equation onto the axis of an acoustic beam:

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𝜕 𝜕𝑝 𝜕𝜏 𝜕𝑥

𝜀 𝜕𝑝 𝑝 𝜌𝑐 𝜕𝜏

2𝑐 𝑝 𝑎∗

(9)

Here a* is the width of the focal waist. The solution of Eq. (9) for a periodic wave is expressed in terms of incomplete elliptic integrals. For large amplitudes, it turns

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into a periodic sequence of parabolas of the form: 𝑝

𝐴

3 𝜔𝜏 𝜋

2𝜋𝑛

1 , 𝜋

𝜋𝑛

𝜔𝜏

𝜋

𝜋𝑛

(10)

At points =+2n where the parabolas intersect in pairs, discontinuities of the derivative appear. Diffraction eliminates this singularity, and there forms a smooth

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“sharpening” of the local maxima of the noise implementation. Obviously, the high frequency asymptotic of this feature has the form: 𝑝 𝜔 ~𝜔

(11)

, 𝐺 𝜔 ~𝜔

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Interestingly, the same asymptotic behaviour was observed when intense acoustic noise propagates in pipes [14]. Here, due to the wall viscosity and weak dispersion, features like discontinuity of derivative or “cusp” appear.

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However, the most important and frequently encountered singularity is the one that arises when the Riemann equation or its modifications are used to describe the noise waves at the stage where discontinuities appear. The study of the applicability of approximate methods under strongly pronounced nonlinearity is of theoretical importance.

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For further presentation, it is useful to draw an analogy with the usual hydrodynamic turbulence. Start with the well-known and simplest mathematical model:

𝜕𝑣 𝜕𝑡

𝑣

𝜕𝑣 𝜕𝑥

𝜇

𝜕 𝑣 𝜕𝑥

(12)

The nonlinear diffusion equation in its classical form (12) proposed by Burgers [15] takes into account the competition between inertial nonlinearity and viscosity

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when strong hydrodynamic turbulence is formed. Random fields described by this equation are known as “Burgers turbulence” or are even denoted by the special

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term “Burgulence” [16].

One of the most important applications of statistical nonlinear acoustics is the high-intensity jet aircraft noise or rocket exhaust noise [17–19]. However, applications are much more diverse and include problems of fundamental physics. An example is the nonlinear attenuation of sound in solids due to the interaction of

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coherent phonons with the thermal noise of a crystal lattice (Landau-Rumer mechanism). When considering these diverse applications of the Burgers equation to the intense noise, the random solutions of the Burgers equation are usually referred to as “acoustic turbulence”.

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A large number of studies have been focused on the dynamic and statistical properties of one-dimensional, and, more recently, 3D Burgers equations. These works are listed in monographs and reviews (see, for example, [3-5, 20-22]). Note that 3D Burgers equation is used for the model description of the cosmological

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turbulence of stellar matter and the formation of the large-scale structure of the Universe. For the 1D turbulence and for a certain class of initial conditions, it is

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possible to give an exhaustive statistical description [3-5, 20-24]. In particular, single-point and two-point probability distributions of the turbulence are found. Even N-point probabilities and corresponding multipoint moment functions are calculated.

Most theoretical papers consider limiting cases where a small or large

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parameter exists. At the initial stage, such a small parameter is the number of shocks per unit duration of the temporal realization of a random process. In this case, the effect of discontinuities is weak and it is possible to use the nondissipative Riemann wave model. At the late stage, the time scale becomes much larger than the initial correlation time, as a result of multiple merging of discontinuities. These fusions make it possible to apply the limit theorem of the probability theory to the asymptotic solutions of the Burgers equation and to show

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that the field passes into a self-similar mode of evolution. An effective method for the numerical simulation of non-viscous asymptotic

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solutions of the Burgers equation is the Legendre transform (FLT) [25]. The use of FLT made it possible to study self-similar regimes of the spectral evolution for different types of initial perturbations [26]. However, intermediate cases of small but finite number of shocks are also of interest. Approximate dynamic solutions, which allow one to obtain analytical

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results for the spectra and correlation functions, play an important role in the statistical description of noise. One of such solutions is the spectral representation of the solution to the Riemann equation [27-29]. It is accurate until shock formation. However, the conditions for its use for statistical problems require detailed consideration. These conditions are studied and discussed below.

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The second section of this article deals with a generalized Burgers equation governing the propagation of nonlinear waves in pipes, horns, concentrators, and other guiding systems. Its asymptotic solution for a vanishingly low viscosity and its connection with the solution of the Riemann equation are given. In the third

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section, a spectral representation of the solution to the Riemann equation and an approximate solution of the Burgers equation are given. The behaviour of these

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solutions outside the formal domain of their applicability is discussed. It is shown that despite the absence of viscosity, the spectral solution of the Riemann equation leads to some nonlinear attenuation. The fourth section discusses the evolution of intense acoustic noise based on the spectral representation of the Riemann solution and a comparison with discontinuous solutions of the Burgers equation. The fifth

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section compares the approximate solution with the results of numerical simulation of the Burgers equation. The sixth section provides some results for the evolution of noise in waveguide systems. 2.

Initial equations. Lagrange and Euler description of the acoustic field The propagation of nonlinear waves in pipes, horns, concentrators and other

guide systems with a variable cross section S(s) is described by the generalized

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Burgers equation (1). Equation (1) can also describe cylindrically- and spherically-

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symmetric waves in a homogeneous medium. Putting in it 𝑆 𝑠

𝑆 0

1

𝑠 𝑠

(13)

we come to the equations of Naugolnykh-Soluyan-Khokhlov (see, for example, [6]), which generalize the Burgers equation. Here n=2 corresponds to spherical, and n=1 to cylindrical waves. For converging waves (s0>s>0) in formula (13), the

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plus sign should be taken, and for divergent waves (s0
We assume that the wave is set at the input of nonlinear medium or at the

boundary s=0:

𝑝 𝜏, 𝑠

0

𝑝∗ 𝛷 𝜔∗ 𝜏

(14)

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Here, p*, * are the characteristic amplitude and frequency of the signal, the function  describes the initial temporal waveform. In dimensionless variables 𝑝 𝑆 𝑠 ,𝜃 𝑝∗ 𝑆 0

𝜔𝜏, 𝑧

𝑑𝑠′

𝜔𝑝∗

Eq. (1) reduces to the common Burgers equation: 𝑉

𝜕𝑉 𝜕𝜃

𝛤 𝑧

𝜕 𝑉 ,𝑉 𝜃 𝜕𝜃

𝑆 0 𝑆 𝑠′

𝛷 𝜃

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𝜕𝑉 𝜕𝑧

𝜀 𝜌𝑐

of

𝑉

but with a single variable coefficient

𝑏𝜔𝜔∗ 𝑆 𝑠 𝑧 𝑆 0 2𝜀𝑝∗

𝛤 𝑧

(15)

(16)

(17)

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In the common Burgers equation, is a constant called the inverse acoustic Reynolds number (or the Goldberg number) [3, 7]. By analogy, it makes sense to call hereafter the function z a “local inverse acoustic Reynolds-Goldberg number” or a “Reynolds-Goldberg function”.

Eq. (16) describes the acoustic field V(, z) at a distance z from the source as a function of retarded time θ. In hydrodynamics, such an approach is known as the Euler description. At zero viscosity (b=0), Eq. (16) transforms into the Riemann

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equation (also known as Hopf equation or simple wave equation):

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𝜕𝑉 𝜕𝑧

𝑉

𝜕𝑉 𝜕𝜃

0,

𝑉 𝜃

𝛷 𝜃

(18)

Using the method of characteristics: 𝑑𝑉 𝑑𝑧

0,

𝑑𝑇 𝑑𝑧

(19)

𝑉

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one can get a solution to this equation in a parametric form: 𝑉 𝜏, 𝑧

𝑉 𝜏 , 𝑇 𝜏, 𝑧

𝜏

𝑉 𝜏 𝑧

(20)

Equations (19) and (20) describe the motion of individual points on the wave profile. In hydrodynamics, such an approach is known as the Lagrange description. From (19), (20) it can be seen that the behaviour of the wave in the Lagrange

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description is trivial: the velocity of a single point of the profile is constant, and its temporal displacement is proportional to the travelled distance. To find the velocity field V(, z) as a function of θ at a distance z from the

𝜃

𝜏∗

𝑉 𝜏∗ 𝑧, 𝜏∗

𝜏∗ 𝜃, 𝑧

Then the velocity field can be represented as follows: 𝜃

𝜏∗ 𝜃, 𝑧 𝑧

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𝑉 𝜏∗ 𝜃, 𝑧

𝑉 𝜃, 𝑧

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input, it is necessary to solve the nonlinear equation: (21)

(22)

If Eq. (21) has a single root, the solution of the Riemann equation is unique. If there are several roots, then the solution becomes multi-valued Vi(,z)=V0(i(,z)); here i is the number of root.

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We note that the asymptotic solution to Burgers Eq. (16) is written in the same form at limiting transition to a vanishingly low viscosity →0. In this case,

**(, z) is the absolute maximum coordinate of the following functional [3, 5, 10]: 𝐺 𝜏, 𝜃, 𝑧

𝑆 𝜏

𝜏

𝜃 2𝑧

,𝑆 𝜏

𝑉 𝑡 𝑑𝑡

(23)

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Based on this solution, we can analyse the statistical characteristics of noise at large distances from the source, when multiple merging of shocks leads to the selfsimilar evolution of the noise spectrum [3, 5, 16, 22].

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In the Euler description, the solution for velocity field has the implicit form: 𝑉 𝜃, 𝑧

𝑉 𝑡

𝑧𝑉 𝜃, 𝑧

(24)

The problem of the statistical description of random Riemann waves in the Euler representation is associated with the implicit form of the solution (24), or with the

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need to solve the nonlinear equation (21), where V0() is a random function. At the same time, in the Lagrangian representation (19), (20) the statistical description is trivial. Exact solutions for the spectra, correlation functions, and probability distributions of a random field [3, 5] were found by means of formulas connecting Lagrange and Euler solutions. However, these solutions are correct as long as the

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evolution can be described by the Riemann equation (18), that is, until discontinuities form in the wave. With the Gaussian statistics of the input field, discontinuities form at arbitrarily small distances from the input, and therefore the statistical description of a random field based on the Riemann equation cannot be

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exact. Nevertheless, the number of these discontinuities at the initial stage is obviously small. Therefore, the quantitative estimates of applicability of equation

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(18) are of special interest in the description of intense noise at the first stage of evolution. 3.

Spectral form of the solution of the Riemann equation and an

approximate solution of the Burgers equation

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It is well known that, using the direct and inverse Fourier transforms, we can write the solution of the Riemann equation (18) explicitly [27-29]. However, this solution in the case when the formal solution (22), (24) of the Riemann equation becomes multi-valued, requires a more detailed consideration. Using the Fourier transform of the field V(, z) in the form 1 2𝜋

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𝐶 𝜔, 𝑧

𝑉 𝜃, 𝑧 𝑒

𝑑𝜃

(25)

and passing to integrate from Lagrangian to Eulerian coordinates, we obtain:

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𝐶 𝜔, 𝑧

1 2𝜋

𝑉 𝜏 𝑒

,

𝜕𝑇 𝜏, 𝑧 𝑑𝜏 𝜕𝜏

(26)

Here T(, z)=V0()z. Integrating (26) two times in parts, we obtain the following expression for the Fourier transform:

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𝐶 𝜔, 𝑧

1 2𝜋𝑖𝑧

𝑒

1 𝑒

𝑑𝜏

(27)

Inverse Fourier transform of (27) 𝑉

𝜃, 𝑧

𝐶 𝜔, 𝑧 𝑒

𝑑𝜔

(28)

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which we call the spectral solution of the Riemann equation, is always unique and coincides with the solution of the Riemann equation Vsp(, z) =V(, z) while this solution is unique. After the appearance of ambiguity, it represents the alternating

𝑉

𝜃, 𝑧

𝑉 𝜃, 𝑧

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sum of the branches Vi(, z) of this solution [5, 30]: 1

𝑉 𝜏 𝜃, 𝑧

1

(29)

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Here n is the number of branches of a many-valued solution of the Riemann equation.

Indeed, formula (26) represents the Fourier transform of the Euler velocity field through the Fourier transform of the initial wave V0(), taking into account

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the shift of the initial profile points and their stretching or compression – a factor in parentheses:

𝜕𝑇 𝜏, 𝑧 𝜕𝜏

1

𝑧

𝜕𝑉 𝜏 𝜕𝜏

(30)

The areas of the initial profile, where this factor is negative, give a negative contribution to the solution (28), (29). In this case, the energy of the field (29) decreases.

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In the course of random wave evolution, discontinuities form at arbitrarily small distances from the input of the medium, and therefore, in order to evaluate the correctness of the spectral solution (27), (28) in statistical problems, we

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consider the behaviour of this solution outside the formal domain of its applicability.

For the harmonic input signal V0()=aꞏsin() we get from (27), (28) the well-

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known Bessel-Fubini solution [3, 4, 10]: 𝑉

𝜃, 𝑧

2 𝐽 𝑎𝑛𝑧 sin 𝑛𝜃 𝑛𝑧

(31)

Where Jn(x) are Bessel functions. Fig. 1 shows the multi-valued solution of the Riemann equation V(, z) (20), (24), and the solution Vsp(, z) (31) after the

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formation of a discontinuity at z=2. Thus, the graph shows that artificial nonlinear

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damping occurs, although formally the dissipation is zero, =0.

Fig. 1. Comparison of the multi-valued solution of the Riemann equation V(, z) (24) (dashed curve) and Bessel-Fubini solution Vsp(, z) (31) (solid line) for z=2

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(a=1).

For a quantitative assessment, we compare the attenuation of the average intensity

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within the spectral solution. 𝐸

𝑧, 𝑎

𝑉

𝜃, 𝑧

2 𝑛𝑧

𝐽 𝑎𝑧𝑛

(32)

and the discontinuous solution of the Burgers equation. Transforming at the halfperiod from the Euler variables to the Lagrange variables, for the average wave

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intensity we have

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1 𝜋

𝐸 𝑧, 𝑎

𝑉 𝜃, 𝑧 𝑑𝜃

1 𝜋

𝑉 𝜏

𝑑𝑇 𝑑𝜏 𝑑𝜏

∗

𝑎 𝜋

𝑎𝑧 ∙ cos 𝑎𝜏 𝑑𝜏

of

sin 𝜏 1

(33)

∗

where* is the Lagrangian coordinate of the discontinuity (the point of the initial nonlinear equation: 0

𝜏∗ 𝑎, 𝑧

and correspondingly, 𝑎 𝜋 𝜏∗ 𝜋 2

𝑎𝑧 ∙ sin 𝜏∗ 𝑎, 𝑧 sin 2𝜏∗ 4

𝑎𝑧 ∙ sin 𝜏∗ 3

(34)

(35)

Pr e-

𝐸 𝑧, 𝑎

p ro

profile that falls into the discontinuity) determined from the solution of the

It is seen that for az≤1, the solution (34) will be *=0 and Esp(z)=E(z)=a2/2. When

Jo

urn

al

az≥1, this equation is solved by numerical methods.

Fig. 2. The energy Esp(z, a) of the spectral solution of the Riemann equation (32)

(solid line) and the energy E(z, a) of a discontinuous solution of the Riemann

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equation (35) (dashed line) for a=1.

Thus, the spectral solution of the Riemann equation imitates nonlinear damping,

of

but this damping is stronger than in a discontinuous solution. So, for z=2 we have Esp(2)/E(2)≈0.56.

We now consider an approximate solution of the Burgers equation obtained

p ro

through the spectral solution of the Riemann equation. In [4] it was proposed to use the solution of the linear diffusion equation Vlin(, z) in the implicit solution of the Riemann equation (24) to model dissipative effects in the Burgers equation. In the Lagrange representation (19), this leads to the following solution 𝑉

𝜏, 𝑧 , 𝑇 𝜏, 𝑧

𝜏

𝑉

Pr e-

𝑉 𝜏, 𝑧

𝜏, 𝑧 𝑧

(36)

With this in mind, the spectral solution (27) takes the form 𝐶 𝜔, 𝑧

1 2𝜋𝑖𝑧

𝑒

,

1 𝑒

𝑑𝜏

(37)

From (37), for the velocity field of the harmonic signal at the input we get

and

2 𝐽 𝑛𝑧𝑒 𝑛𝑧

𝜃, 𝑧

,

𝑧

𝑉

,

urn

𝐸

,

al

𝑉

𝜃, 𝑧

2 𝑛𝑧

sin 𝑛𝜃

(38)

𝐽 𝑛𝑧𝑒

(39)

We present below the obtained approximate solution in parallel with the exact solution of the Burgers equation. In Fig. 3, the exact solution Vexact(, z) (dashed curve) and the approximate solution Vsp,dis(, z) (solid curve) are compared; the

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distance z=2 after the formation of discontinuity is considered, and =0.1. Thus, the spectral solution describes the field evolution fairly well, with the exception of the fine structure of the shock front.

Pr e-

p ro

of

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Fig. 3. The dashed curve denotes the exact solution Vexact(, z), the solid one shows the Rudenko spectral solution Vsp,dis(, z) (both for z=2 and=0.1).

The general case of the evolution of energy is shown in Fig. 4. The dotted line

al

denotes the energy E(z) decrease at =0, the dashed line indicates the exact solution Eexact(z), and the solid line represents the approximate one Esp,dis(z) (the last two are shown at=0.1 in Fig. 4.1 and at =0.2 in Fig. 4.2). As the coefficient

urn

 increases, the boundary of the nonlinear damping regime shifts to the right, and

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the larger the  the better the matching of the solutions.

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urn

al

Pr e-

p ro

of

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Fig. 4. The dashed curve shows the exact solution Eexact(z), the solid one – the

solution in the spectral approximation Esp,dis(z), the dotted one is the energy E(z)

with=0. Curves 4.1 are constructed for =0.1, and curves 4.2 – for =0.2.

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4.

Evolution of high-intensity acoustic noise We first discuss the evolution of intense acoustic noise using the spectral

representation of the Riemann solution. Start with the evolution of a narrowband small, we can define the noise energy as 〈𝐸

𝑧

2 𝑛𝑧

𝑧, 𝑎 〉

〈𝐽 𝑎𝑛𝑧 〉

p ro

𝜎

of

noise signal having Gaussian input statistics. If the frequency fluctuations are

(40)

Here, the triangular brackets denote statistical averaging. In this formula, the averaging must be performed over random amplitude, which for Gaussian statistics has a Rayleigh distribution. From (40), for the energy of the noise we have 𝑛 𝑧 2 𝑛𝑧

Pr e-

𝜎

𝑧

𝑒𝑥𝑝

𝐼

𝑛 𝑧 2

(41)

Here, In(z) is the modified Bessel functions. We assumed that the dimensionless energy of noise equals 1/2, or is equal to the energy of the harmonic signal. The Rayleigh distribution Wa(a)=2exp(a2) was used. The attenuation law for quasimonochromatic noise, taking into account the formation of discontinuities, was

al

constructed numerically based on the solution (33), (35). Averaging the exact solution (35) over a random amplitude using the Rayleigh distribution, we obtain expressions for the energy of a quasi-monochromatic noise with regard to the

urn

formation of discontinuities 𝜎

𝑧

𝐸 𝑧, 𝑎 𝑊 𝑎 𝑑𝑎

𝐸 𝑧, 𝑎 2𝑎𝑒𝑥𝑝

𝑎 𝑑𝑎

(42)

Fig. 5 shows the attenuation of energy of the monochromatic signal E(z, a) (35),

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and the noise in the framework of the spectral solution 𝜎

solution accounting for the formation of discontinuities 𝜎

𝑧 (41), and the

𝑧 (42).

Pr e-

p ro

of

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Fig. 5. Attenuation of the energy of a monochromatic signal E(z, a) (35) (dasheddotted curve) and noise in the framework of the spectral solution 𝜎

𝑧 (41) (solid

curve) and the solution taking into account the formation of discontinuities 𝜎

al

(42) (dashed curve).

𝑧

Due to the appearance of discontinuities in realizations of a random signal, where the local signal amplitude exceeds the average amplitude, the noise attenuates

urn

faster than the harmonic signal. In particular, at z=1 we have 𝜎

𝑧 /𝐸 𝑧

0.885. It follows from Fig. 5 that the spectral Riemann solution gives a qualitatively correct description of noise attenuation, but this solution gives a faster attenuation of noise than the discontinuous solution. For example, for z=1 we have 𝑧 /𝜎

𝑧

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𝜎

0.877. Note that at sufficiently large distances, the noise energy

and the energy of the harmonic input signal no longer depend on the initial amplitude and 𝜎

𝑧

𝐸 𝑧 .

When calculating an important physical characteristic the intensity

spectrum G(, z), it is convenient to use formula (27) and the well-known formula

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for the coupling of Fourier transforms and the spectrum. For stationary noise =G(, z)(') where the triangular brackets denote statistical averaging, the asterisk is complex conjugation, and (x) is the delta function. In the case of Gaussian input statistics, we have the following expression

1 2𝜋 𝜔𝑧

𝑒

1 𝑒

𝑒

𝑑𝑠

p ro

𝐺 𝜔, 𝑧

of

for the intensity spectrum [3, 5]

(43)

Here, B0(s) is the correlation function of the input signal 〈𝑉 𝜃

𝐵 𝑠

𝑠 𝑉 𝜃 〉, 𝜎

𝐵 0

(44)

For the correlation function from (43), we have 1 2

𝑦 𝜕 𝐸𝑟𝑓 𝜕𝑦 2𝑧 𝜎 1 2

where

𝜃

Pr e-

𝐵 𝜃, 𝑧

𝑒𝑟𝑓

𝑦

2𝑧 𝜎

2

al

𝐸𝑟𝑓 𝑧

𝐵 𝑦

√𝜋

𝑒

𝐵 𝑦 𝑑𝑦

𝜃

𝑑𝐵 𝑦 𝑑𝑦 𝑑𝑦

𝐵 𝑦

𝑑𝑥

(45)

(46)

Formula (45) gives a nonlinear functional relation 𝐵 𝜃, 𝑧

𝐹 𝐵 𝜃 . The

urn

transitions B(, z)→B0() at z→0 (to the boundary of the medium) or 0→0 (to a weak wave, to a linear problem) are obvious. As noted above, the spectral Riemann solution leads to wave damping despite the absence of a viscous term in the equation. For noise, singularities in the solution arise at an arbitrarily small distance from the input. The attenuation curves for narrowband noise are shown

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above (Fig. 5). To quantify the attenuation of broadband noise at the initial stage, we use formula (45). By decomposing the correlation function of the initial field in a power series

𝐵 𝑠

𝜎

1

𝜔 𝑠 2!

𝜔 𝑠 4!

⋯

(47)

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from formula (45) for the integral noise intensity we have 𝜎

𝑧

𝐵 0, 𝑧

𝜎

1

𝑧

𝛥

,

(48) 𝑧 𝑧 𝜔 𝜔 𝑒𝑥𝑝 𝛥 𝑧 𝑧 2𝑧 √2𝜋 where znl=1/01 is the characteristic scale of non-linearity. As it was expected, at

of

24

the initial stage the attenuation is associated with the appearance of singularities in random realization, the average of which per unit length can be estimated by their 𝑧𝑣 𝜏

0, and proportionally exp

𝑧 /2𝑧 . The noise integral

p ro

equations 1

intensity and the coefficients of the series (47) are also conveniently expressed in terms of the initial spectrum 1 2𝜋

𝐵 𝜃 𝑒

𝑑𝜔,

Pr e-

𝐺 𝜔

𝜎

𝐺 𝜔 𝑑𝜔,

𝜎 𝜔

(49)

𝜔 𝐺 𝜔 𝑑𝜔, 𝑛

1,2

The attenuation described by the spectral Riemann solution is associated with the formation of singularities in this solution, which correspond to discontinuities in the acoustic wave. Below we discuss the comparison of this expression with the

al

formula for acoustic wave attenuation with regard to the formation of discontinuities. The calculation of integral (45) can be performed analytically for the initial correlation function 𝐵 𝜃

𝜎 1

𝜃

[28]. In this case, the

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intensity is almost constant in the region 0
/𝜎

0.85 [28].

We pass now to the propagation of noise, taking into account the dissipative properties of the medium. As is shown in [4], in this case an approximate approach is effective, based on replacing of the original correlation function B0() (44) by the correlation function obtained from solving a linear diffusion equation 𝜃, 𝑧

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𝐵

𝐺

𝜔, 𝑧 𝑒

𝐺

𝜔, 𝑧

𝑑𝜔, 𝐺 𝜔 𝑒

𝜎

𝑧

𝐵

0, 𝑧 ,

(50)

This formula is valid for =const. In the general case, one should replace 𝛤𝑧 → 𝛤 𝑥 𝑑𝑥 . This approach gives good accuracy for (z)>1 values and any

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distances z, or for strongly pronounced nonlinearity (z)<<1 in the region of small values of z, where the number of shock fronts in the noise realization is small. Some examples of this approach and preliminary analysis of its applicability are given in the review [4]. In [30], this approach was applied to nonlinear integro-

of

differential equations describing the propagation of intense waves in homogeneous media with a weak dispersion of the relaxation type (in particular, in soft biological

p ro

tissues).

Using general expressions (43), (50), the following asymptotic formulas for the low and high frequency spectrum are derived. For 0z<<1, one can expand the exponential term in (43) in Taylor series and restrict oneself to the first two terms:

2𝐺

𝐺

𝜔, 𝑧 𝜎

1 𝜔𝑧 2

Pr e-

𝐺 𝜔, 𝑧

𝜔, 𝑧

𝑧

𝐺

𝜔, 𝑧 ⨂𝐺

𝜔, 𝑧



(51)

…

The symbol ⊗ signifies a convolution operation. Taking into account only the first term of the right-hand-side corresponds to neglecting nonlinear effects. If two terms are taken into account, the nonlinear one-time interaction of pairs of harmonics of the initial field takes place, leading to the appearance of spectral

al

components with difference and sum wave numbers. For z<<1, we can neglect the linear damping and assume in the expression (51) Glin(, z)=G0(). It follows

urn

from (51) that the shape of the spectrum is preserved in the low frequency region if 𝐺 𝜔, 𝑧

𝐺 𝜔

𝛼 𝜔 ,𝑛

2. The universal low-frequency asymptotic G(,

z)=2(z) forms in the initial spectrum at n≥2. At the initial stage, from (51) we have 2(z)~z

In the high-frequency region, using the saddle point method for calculating

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the integral (43) and taking into account (47), we have [3,5] 𝐺 𝜔, 𝑧

𝑧 𝑧 √2𝜋𝜔

𝑧 2𝑧

𝑒𝑥𝑝

(52)

The formation of the universal asymptoticis connected with the

appearance of singularities of the form

𝑡

𝑡

in noise realizations [5]. The

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exponential factor is proportional to the number of singularities. Indeed, singularities arise when the derivative ∂, z)/∂ (30) tends to zero. Using the filtering properties of the delta function for the average number of singularities

〈 𝛿

〈

〈𝑁〉

1 𝜕 𝑉 𝜏 𝜕 𝑉 𝜏 𝑑𝜏〉 𝜕𝜏 𝜕𝜏 𝑧

𝜕𝑉 𝜏 𝜕𝜏

𝜕 𝑉 𝜏 𝜕𝜏

𝜔 𝑒𝑥𝑝 𝜋𝜔

〉

〈

1〉,

(53)

1 2𝑧 𝜎 𝜔

,

𝑧𝑣 𝜏

0, averaging is performed over the

𝑧

1 𝜎𝜔

Pr e-

Here n are roots of the equation 1

𝜕 𝑉 𝜏 𝜕𝜏

p ro

〈𝑁〉

of

per unit length, we have [5]

joint two-point probability distribution of the first and second derivatives of the initial field, and 𝜎 , n are determined by (49).

From (52) it follows that at a fixed frequency ω, the increase in the amplitude of the spectrum due to the generation of high-frequency components at the initial stage will change to decreasing at z=znl/√3≈znl 0.58, which is associated with the total attenuation of the random field. Approximate consideration of attenuation

al

leads to the fact that the characteristic attenuation length in (48), (52) begins to depend on the distance and is determined from relations (49), (50) 𝜎 𝑧 𝜔 𝑧

urn

1

𝑧

𝑧

𝜔 𝐺 𝜔 𝑒

𝑑𝜔

(54)

From (54), it can be seen that taking damping into account leads to an increase in znl(z) and, accordingly, to a weakening of nonlinear effects. As can be seen from the behaviour of the harmonic signal, the spectral

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representation leads to a faster damping of the wave than the true discontinuous solution (Fig. 2). The estimation of the noise attenuation at the initial stage can be carried out by introducing shock fronts in the implicit Riemann solution [31]. This solution in the implicit form is V(, z)=V0(zV(, z)), or (V, z)= 0(V)Vz. Here,

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 0(V)is the inverse function to V0(. At short distances after the formation of the singularity, the function 0(V) can be approximated by a cubic parabola. Its parameters are determined by the first derivatives of the initial field. For the energy

of

of the noise field, we have [31] 𝑧 54 𝑧 𝜔 𝜔 exp (55) 𝑧 𝜋 2𝑧 Comparing this formula with expression (48), we see that the expression for 𝜎 1

∆ 𝑧 ,∆ 𝑧

p ro

𝜎 𝑧

attenuation in the spectral representation has the same functional dependence as the discontinuous representation, but gives a somewhat overestimated attenuation of

urn

al

Pr e-

the energy (see Fig. 6).

Fig. 6. Attenuation of the noise energy described by the spectral solution (41) ∆𝜎

𝑧

𝜎

0

𝜎

𝑧

(solid curve), solutions taking into account the

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formation of discontinuities ∆𝜎

𝑧

𝜎

0

𝜎

𝑧

(42) (dashed curve).

Asymptotic formulas of energy E at small distances: attenuation in the spectral representation (48) (dotted curve), in the discontinuous representation (55) (dashdotted curve).

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The appearance of discontinuities also leads to a power-law decrease of the spectrum ~ but slower than the decrease which appears because of the power 𝑡 , for which the spectrum is ~. As the distance

𝑡

of

feature of the form

increases, the power spectrum associated with the formation of discontinuities

5.

p ro

shifts to lower frequencies.

Numerical modelling of the evolution of noise waves

For numerical simulation of the evolution of noise, the equation for the Fourier component of the field was used 1 2𝜋

𝑉 𝜃, 𝑧 𝑒

𝑑𝜃

Pr e-

𝐶 𝜔, 𝑧

𝐹𝑉

(56)

The nonlinear term was taken into account using fast forward F and inverse F–1 transformation: 𝜕𝐶 𝜔, 𝑧 𝜕𝑧

1 𝑖𝜔𝐹 𝐹 2

𝐶 𝜔, 𝑧

Γ𝜔 𝐶 𝜔, 𝑧

(57)

Testing of the numerical scheme was carried out for the case of a regular harmonic initial profile V0(sin(). The complete agreement was obtained between the

al

exact numerical solution and the exact Hopf-Cole solution in the range of inverse

Jo

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acoustic Reynolds numbers  =0.0050.1 and distances z/znl<10.

Pr e-

p ro

of

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Fig. 7. Comparison of the exact Hopf-Cole solution (dots) and the numerical solution (solid curve) for  =0.01 and the distances z equal to 0, 0.5, 1, 5.

C0()= 𝐺

al

As the initial condition C(, z=0)=C0(), we considered the random process 𝜔 , where  is a random Gaussian process with zero mean < >=0

and unit variance <2>=1; G0() is the initial spectrum. The averaging was carried

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out over 500 realizations, each of them containing 214 numerical counts. The results of numerical simulation were compared with the analytical expression (43) for the spectral solution of the Riemann equation. At the same time, noise dispersions at different distances from the input were compared in a numerical and spectral analytical solution and the spectra behaviour was compared in different

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frequency ranges.

Below the graphs are shown comparing the analytical solution (43) and the

results of a numerical experiment for the initial spectrum of the form

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𝜔 𝜔∗ √2𝜋 𝜔∗ 𝜎

𝐺 𝜔 𝜎

1, 𝜔∗

1, 𝑛

𝑒

∗

,

(58)

1

of

The spectrum maximum is at the frequency , and the characteristic nonlinear scale is znl=1/01/0√3 The effective inverse Reynolds

spectrum: 𝜎

p ro

number  was 0.1, 0.01 and 0.005 in the experiments. The parameters of the initial 1, =1.

Fig. 8 shows the dependence of energy on the distance z for three values of

 The solid line indicates the analytical solution (43), the dots indicate the numerical solution at  =0.005, =0.01 and  =0.1. It can be seen from Fig. 8 that

Pr e-

for small  the energy calculated numerically decays more slowly than the analytical one. It can be seen also that at low Reynolds numbers the main mechanism of noise attenuation is nonlinear absorption in the vicinities of shock

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fronts.

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Fig. 8. Comparison of the energy of a numerical solution and an analytical one (43). The solid line denotes the analytical solution (43) (=0), the dots show the numerical one at =0.005, =0.01 and =0.1. The dotted line indicates the

of

analytical solution in the linear approximation for =0.01. Below we consider the evolution of the spectrum (58), which was compared with

p ro

the analytical solution (43). The corresponding curves are shown in Figs. 911 for

 =0.01. The solid line “z=0” indicates the numerical solution for z=0. The solid line “numerical” is the numerical solution and the solid line “analytical” is the analytical one for distances z equal to 0.21=0.37znl, 0.42=0.73znl, 1.07=1.84znl, respectively. The dots denote the asymptotic behavior of the spectrum, calculated

Pr e-

by formula (52). It also shows the energy values for numerical simulation – Em(z) and the spectral analytical model – Esp(z). For the analytical and numerical solutions in the high-frequency region, various asymptotic approximations of the

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urn

al

spectrum are given.

Fig. 9. The noise spectrum at a distance z=0.21=0.37znl from the input. The solid

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line “z=0” indicates the input spectrum at z=0 (E0=0.5). The solid line “numerical” is the numerical solution (58) (Em=0.4898) and the solid line “analytical” is the analytical one (43) (Esp=0.4983). The dots correspond to the asymptotic

al

Pr e-

p ro

of

expressions:  in the low-frequency and  in the high-frequency region.

Fig. 10. The noise spectrum at a distance z=0.42=0.73znl from the input. The solid line “z=0” indicates the input spectrum at z=0 (E0=0.5). The solid line “numerical”

urn

is the numerical solution (58) (Em=0.4736) and the solid line “analytical” is the analytical one (43) (Esp=0.4165). The dots correspond to the asymptotic expressions for the analytical solution:  in the low-frequency and  in the high-frequency region; dashed lines are the asymptotic expressions for the

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numerical solution: and exp(in the high-frequency region.

Pr e-

p ro

of

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Fig. 11. The noise spectrum at a distance z=1.07=1.84znl from the input. The black solid line “z=0” indicates the input spectrum at z=0 (E0=0.5). The solid line “numerical” is the numerical solution (58) (Em=0.3391) and the solid line “analytical” is the analytical one (43) (Esp=0.1520). The dots correspond to the asymptotic expressions for the analytical solution:  in the low-frequency and 

al

in the high-frequency region; dashed lines are the asymptotic expressions for the

urn

numerical solution: and exp(in the high-frequency region. From Fig. 9, it is clear that at the initial stage the analytical spectral solution (43) gives an overestimated value of the spectrum in the high-frequency region. At the initial stage, the number of singularities of the form 𝑣~ 𝑡

𝑡 is small, and

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they are just beginning to form. High-frequency attenuation smoothest the features, which leads to a slower decrease of the spectrum. Figure 10 corresponds to the distances where the features 𝑣~ 𝑡

𝑡 form the high-frequency asymptotic of the

spectrum G(, z)~. This asymptotic is observed both in the analytical spectral solution (43) and in the numerical result. The slower decay of the spectrum in a

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numerical experiment is due to the fact that high-frequency attenuation leads to the formation of discontinuities. Nevertheless, due to the relatively small amplitude of discontinuities, the power law G(, z)~ does not yet have time to form. At high frequencies, due to the finite width of the shock front, an exponential law replaces

of

the power law of the spectrum decay. Moreover, due to fluctuations of the discontinuity amplitude, the fall-off at high frequencies is slower than for a

p ro

harmonic initial perturbation, where G(, z)~exp(z).

Fig. 11 corresponds to distances greater than the characteristic distance for the formation of discontinuities, and therefore the high-frequency asymptotic of the spectrum G(, z)~ is determined precisely by the discontinuities. At these distances, the spectral solution gives a faster damping of the field than the solution

Pr e-

of the Burgers equation (see Section 3). Accordingly, the spectrum of the

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urn

al

approximate analytical solution decreases significantly slower.

Fig. 12. The noise spectrum at a distance z=0.42=0.73znl from the input for three

values of the inverse acoustic Reynolds number =0.005, 0.01, 0.1. The dashed curve is the analytical solution (43). The three values of Reynolds number

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correspond to the energy Em equal to 0.4834, 0.4736, 0.3802 while the energy for the initial spectrum initial E0=0.5. Fig. 12 shows the noise spectra for different values of the attenuation

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coefficient. As the dissipation factor  decreases from 0.01 to 0.005, only the behavior of the spectrum at high frequencies changes. Namely, the power portion to the law G(, z)~,

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of the decay of the spectrum G(, z)~ has a more distinct tendency to go over associated with the formation of discontinuities.

Accordingly, the transition from the power law to the exponential decay occurs at higher frequencies. With an increase in attenuation by an order of magnitude (from

=0.01 to =0.1), the high-frequency components are still effectively generated,

6.

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but the power part of the spectrum does not have time to form.

Evolution of intense acoustic noise in waveguide systems In waveguide systems, the evolution of the correlation function is described

by expression (45), where the original correlation function B0() (44) is replaced by the correlation function obtained from the solution of the linear diffusion

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equation (50), and the spectrum is described by the expression 𝐺

𝜔, 𝑧

(59)

𝐺 𝜔 𝑒

Consider an important case of an exponential concentrator used in medicine

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and ultrasound technology [32, 33]. In this case, from (15), (17) we get: 𝑠 𝑠 𝑆 𝑠 𝑆 0 𝑒𝑥𝑝 2 , 𝑧 𝑧 𝑒𝑥𝑝 1, 𝑠 𝑠 𝑧

𝜀𝜔∗ 𝑝∗ 𝑠 𝑐 𝜌

,Γ 𝑧

Γ

1

𝑧 𝑧

(60)

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Here, *, p* are the characteristic amplitude and frequency of the signal. We first consider how the acoustic Reynolds number behaves for weak perturbations, when the linear approximation of equation (16) can be used. For the harmonic input signal V0()=a0sin(0) for the current acoustic Reynolds number, we have

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𝑅𝑒 𝑧

𝑧 𝑙𝑛 1 𝑙 𝑧 1 𝑧

𝑒𝑥𝑝

𝑅𝑒

𝑎 , 𝜔 𝛤

𝑅𝑒

𝑧 𝑧

𝑅𝑒

1 𝜔 𝛤

𝑙

𝑧 𝑧

1

(61)

of

𝑎 𝑧 𝜔 𝛤 𝑧

Thus, if a sufficiently strong signal amplification occurs at the concentrator (z0
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the input. Consequently, at large distances, the wave enters a nonlinear mode of evolution.

In the linear approximation, the evolution of noise with the initial spectrum (58) is described by the same expression, but due to linear attenuation, both the

𝜔

𝑧

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characteristic frequency and the noise dispersion decrease 𝜔∗

1

𝑙

𝑧 ,𝜎

𝑧

1

𝜎 𝑧 𝑙

/

(62)

1 4Γ𝜔∗

𝑙

For the current acoustic Reynolds number for the concentrator in the linear mode

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from (62), we have 𝜎

𝑅𝑒 𝑧

𝑧 𝛤 1

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𝜔

𝑧

𝑙

𝑧 𝑧

1

𝑅𝑒

1 , 𝑅𝑒 4𝛤 𝜔∗

1

𝑧 𝑧

𝑧 𝑙𝑛 1 𝑙

𝑧 𝑧

/

, (63)

𝜎 𝜔∗ 𝛤

That is, in the linear approximation, an increase in the noise Reynolds number is observed, and a nonlinear stage should replace the linear one. Estimates of the

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behavior of the noise field in the concentrator can be made on the basis of the spectral solution based on replacing the initial correlation function B0() (44) in

formula (43) with the correlation function (50) obtained from the solution of the

linear diffusion equation. One of the important criteria for the manifestation of

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nonlinear effects is the establishment of the high-frequency asymptotic of the spectrum (43). From (43), (63) we have

𝑧

𝛼 𝑧 2

exp

√2𝜋𝜔

,

1/𝜎

𝑧 𝜔 𝑧

𝑧

𝑧

1/𝜎 𝜔

1/𝜎 𝜔∗ 3, 𝑙

1

𝑧 /𝑙

𝑧

𝛼 𝑧

𝑧 𝑧

,

ln 1

𝑧 𝑧

/

(64)

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𝑧

𝑧 𝑧

of

𝑧

𝐺 𝜔, 𝑧

∗

In the absence of a concentrator or with z<
𝑧 𝑧

1

/

𝑧/𝑙

. The maximum exponent is reached at zm=4ldis,

exp

Pr e-

and the exponential factor is 5 / 2 𝑅𝑒

exp

1.75 , 𝑅𝑒 𝑅𝑒

𝑙 𝑧

(65)

Thus, at low Reynolds numbers, the amplitude of the high-frequency components is exponentially small. However, in the concentrator with z0<
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the spectrum (64) is formed. Fig. 13 shows the dependence of the amplitude of the high-frequency components 𝐴 𝑧

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for different ratios z0, ldis.

√2𝜋𝜔 𝐺 𝜔, 𝑧 as a function of the distance z

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p ro

of

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Fig. 13. Dependence of the amplitude 𝐴 𝑧

√2𝜋𝜔 𝐺 𝜔, 𝑧

of the high-

frequency part of the spectrum (64) on the distance z in free space (dashed curve) and in the hub (continuous curve) ldis/znl=1 and the concentrator z0=0.01.

parameter

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Figs. 14, 15 depict the noise spectrum in the concentrator at different ratios z0, ldis . It can be seen from the figure that a greater number of high-frequency components

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are generated more efficiently in the concentrator than in free space.

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p ro

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Fig. 14. The noise spectrum G(, z) in free space (line “free space”) and in the concentrator (line “concentrator”) for ldis/znl=1, the concentrator parameter z0=0.01, the initial Reynolds-Goldberg number 𝛤

√3/4 at a distance of z=0.35=0.61znl.

The solid line “z=0” shows the spectrum at the input G0(), the dashed line shows

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al

the high-frequency asymptotics for the concentrator 3.

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p ro

of

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Fig. 15. The noise spectrum G(, z) in free space (line “free space”) and in the concentrator (line “concentrator”) for ldis/znl=1, the concentrator parameter z0=0.01 for the initial Reynolds-Goldberg number 𝛤

√3/4 at a distance of

z=1.07=1.84znl. The solid line “z=0” shows the spectrum at the input G0(), the

Conclusion

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7.

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dashed line shows the high-frequency asymptotics for the concentrator 2.

The paper studies the evolution of high intensity noise waves in waveguide systems, described by the generalized Burgers equation. This equation is a universal one-dimensional model. It also governs the field in inhomogeneous media in the approximation of non-linear geometric acoustics and can be derived

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by going to the high-frequency limit in the Khokhlov-Zabolotskaya equation. It is reduced to the usual Burgers equation, but with a coordinate-dependent dissipative term that describes the relative role of nonlinear and dissipative effects due to inhomogeneity of the medium or change in the cross section of the ray tube. Its

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asymptotic solution for a vanishingly low viscosity is given; its relation to the solution of the Riemann (Hopf) equation is noted. It is shown that the spectral representation of the Riemann solution plays a special role in statistical problems. With this representation (by using a direct and

of

inverse Fourier transform), it is possible to derive an implicit solution and obtain a simple description for the field. It provides convenient analytical expressions for

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the correlation functions and spectra of the noise field intensity. In dynamic problems, one can strictly determine the limits of applicability of the spectral representation. However, with noise disturbances due to large outliers of the initial amplitude, singularities can arise at arbitrarily small distances from the input. In particular, such a solution leads to nonlinear damping — a decrease in the energy,

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despite the absence of viscosity in the original equation. This paper compares the spectral solution with the solution of the Burgers equation in the case of both vanishingly low and finite viscosity. It is shown that the spectral solution gives a qualitatively correct description of the evolution of the field outside the formal range of its applicability.

For high-intensity waves in media without dispersion, an avalanche-like increase in the number of interacting harmonics takes place. As a result, there

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occurs a strong spectral broadening and singularities arise in the field realizations. At vanishingly low viscosity, both the discontinuity and the inflection in the wave

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profile appear. For the Fourier transform, these features lead to the appearance of the high-frequency asymptotics “omega to the minus first power” and “omega to the minus three-halves power”, where omega is the signal frequency. In a medium with a moderate viscosity, the shock front has a finite width, which leads to an exponential decay of the spectrum at high frequencies. For noise waves, due to

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fluctuations and “bursts” of the signal amplitude, different types of features can occur at the same time, which results in a complex structure of the energy spectrum.

The paper presents a series of numerical experiments on the simulation of

intense noise at different acoustic Reynolds numbers. It is shown how the

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generation of high-frequency components and the change of the power law of the energy spectrum “omega to the minus third” to “omega to the minus second power” with increasing distance occur. Moreover, there is a good agreement between the numerical calculation and the results obtained from the spectral

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representation of the Riemann solution. It was found that due to fluctuations in the width of the shock front, the exponential damping occurs slower than for a

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harmonic input signal.

By using the example of an exponential concentrator, it is shown that in waveguide systems it is possible to significantly enhance the manifestation of nonlinear effects in the evolution of noise. In particular, if a noticeable amplification of the signal occurs, then the current acoustic Reynolds number

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increases with distance from the input. In this case, a nonlinear mode of evolution can replace the linear one at large distances. Acknowledgements

The work is supported by the grant from the Russian Science Foundation: No. 19-

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New York, 1964.

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DOI: 10.1126/science.145.3639.1424-a

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Dear editors, We agree with all remarks given by reviewers. Sincerely, S.N. Gurbatov, O.V. Rudenko, A.V. Tyurina

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• Statistical characteristics of acoustic noise are studied based on Burgers equation • Spectral representation of the Riemann solution let obtain a simple field description • Spectral approach is used to study high-intensity noise at zero and finite viscosity • Generation of high-frequency components in spectrum occurs with distance increasing • In guiding systems it is possible to enhance the manifestation of nonlinear effects