Proceedings of Meetings on Acoustics

Volume 19, 2013 http://acousticalsociety.org/

ICA 2013 Montreal

Montreal, Canada

2 - 7 June 2013

Physical AcousticsSession 4aPA: Nonlinear Acoustics I

4aPA7. The numerical simulation of propagation of intensive acoustic noiseIgor Demin*, Sergey Gurbatov, Nikolay Pronchatov-Rubtsov, Oleg Rudenko and Artem Krainov

*Corresponding author's address: Radiophysics, University of Nizhny Novgorod, 23, Gagarin Ave., Nizhny Novgorod, 603950,Nizhny Novgorod, Russia, phdem56@gmail.com The propagation of intensive acoustic noise is of fundamental interest in nonlinear acoustics. Some of the simplest models describing suchphenomena are generalized Burgers' equations for finite amplitude sound waves. An important problem in this field is to find the wave'sbehavior far from the emitting source for stochastic initial waveforms. The method of numerical solution of generalized Burgers equationproposed is step-by-step calculation supported on using Wavelet Analysis. For simulating the wave evolution we used 4096 (212) pointrealizations and took averaging over 1000 realizations.

Published by the Acoustical Society of America through the American Institute of Physics

Demin et al.

© 2013 Acoustical Society of America [DOI: 10.1121/1.4800940]Received 4 Feb 2013; published 2 Jun 2013Proceedings of Meetings on Acoustics, Vol. 19, 045075 (2013) Page 1

INTRODUCTION

The propagation of finite amplitude sound waves is of fundamental interest in nonlinear acoustics. Some of the simplest models describing such phenomena are Burgers’ equation for plane waves and generalized Burgers’ equations for cylindrical and spherical waves [1-3]. An important problem in this field is to find the wave’s behavior far from the emitting source for both regular and stochastic initial waveforms. The propagating wave is damped by absorption and shock wave dissipation and therefore the asymptotic wave may be described by a linear diffusion equation. Solutions of such a kind are called old-age solutions in the following. Burgers’ equation has an exact solution, which is found by reducing it to the linear diffusion equation by mean of the Hopf-Cole transformation. Because of the existence of an exact solution it is relatively easy to find the old age waveform developing from simple initial perturbations like periodic signals and N-waves in the plane wave case. The solution of Burgers’ equation with random initial conditions is often called Burgers turbulence. In the case of vanishing viscosity the continuous random initial field is transformed into a sequence of straight lines with some slope and with random locations of the shocks separating them. Due to the merging of the shocks the internal scale of the turbulence increases and the random wave decreases more slowly than the periodic signal.

NONLINEAR WAVES IN SOFT TISSUES

The study of nonlinear waves in soft biological tissues is necessary both for diagnostic purposes and tissue therapy[4-5]. From the acoustic viewpoint, tissue is a hereditary medium with frequency dependent dissipative properties. However, statistical problems for waves propagating in biological tissues have not been considered. In addition, for medical applications, both noise waves and waves with fluctuating parameters can be used. As well, in a number of cases, statistical solutions prove simpler than regular ones and can be convenient for understanding the physics of the processes.

� � 0,2 0 0

2

2

3 �����

�

���

���

���

�

�������

� dxpt

Kc

mppcx

p . (1)

where p is acoustic pressure, x is the coordinate along which the wave travels, c is the velocity of sound, �= t – x/c is time in a coordinate system moving at the velocity of sound, m is a constant characterizing the “force” of the medium’s hereditary properties, and K(s) is the kernel describing the decrease in the medium’s “memory” with time. We establish the relation between kernel K(s) with the dispersion law, for which we seek a solution in the form

� � kikkxiktip �������� ,exp � , (2)

where k is the wavenumber and k', k'' are its real and imaginary parts. The specific form of the kernel should be established from some physical model or from experimental measurement data. An important model is a relaxation medium (Mandelshtam–Leontovich model) (see [5]). For this model, the kernel has the form of an exponent: K = exp(–s). In this case,

Demin et al.

Proceedings of Meetings on Acoustics, Vol. 19, 045075 (2013) Page 2

� �� �

� �� �2

0

02

0

20

12,

12 tt

cmk

tt

cmk

���

���

����

���� . (3)

Frequency dependences (4) of dispersion k' and attenuation k'' have been multiply confirmed by experiments. We point out that intense noise waves described by Eq. (1) has not yet been studied. Note that for hereditary media, biological tissues are an important class; it is necessary to solve various statistical problems in nonlinear formulation.

WAVELET ANALYSIS OF BURGERS TURBULENCE

In recent years the development of numerical schemes for the solution of partial differential equations was developed new mathematical - wavelet analysis, to compete with the Fourier analysis. The basis of this method is based on special classes of functions - wavelets, which are obtained from each other by the scaling transformation (tension / compression) and shear. In this paper, an example of using wavelet analysis for the numerical solution of one-dimensional Burgers equation (BE) [1-2], which is the simplest model that describes two mechanisms inherent in real turbulence: nonlinear energy transfer over the spectrum and viscous dissipation in a small scale. In a dimensionless form of the model is given by BE

2

2

v v vv �z t t

� � �� �

� � �, (4)

where the parameter � assesses the relative contribution of non-linear and dissipative parameters. So when � ›› 1 (say in the case of vanishingly small viscosity), there is exponential growth of harmonics and the strong distortion of the wave front. Use the scheme of solutions of partial differential equations of the wavelet Galerkin method, we seek a solution BE (4) in the form

1,( , ) ( ) ( )

nj j

k kj k

V z D z� � ���

� � ,

where �jk(�) - wavelet basis functions and unknown functions Dj

k(z) are found by numerical integration of ordinary differential equations [2]. Difference schemes BE will be recorded in the form of

, ,

( ) ( ) ( ) ( ) ( )l l lsj s j lj jm m mrk r k mk k

r k r kD z s D z s D z D z D z

� �� � � � �� �� �

� � � , (5)

here s - coordinate the integration step and �lm - coupling coefficients determined from the integral-

differential equations in terms of the basic wavelet function. To implement the numerical scheme for solving the BE program was written in the algorithmic programming language C++. The algorithm of the program was laid down that the initial random field is specified as a Gaussian random process with zero mean and unit variance. Then, for given values of the propagation z and the nonlinear parameter � wavelet decomposition is carried out and the coupling coefficients are calculated for each level. Then at each time step and for each level explicit scheme is applied, after which a reverse signal recovery.

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Proceedings of Meetings on Acoustics, Vol. 19, 045075 (2013) Page 3

FIGURE 1. Evolution of Burgers turbulence (numerical solution using wavelet analysis).

Figure 1 shows the results of numerical simulation for the length of the implementation of BE 1024, � = 0.04 (which corresponds to the Reynolds number equal to 25) and z = 10. It is clearly seen that the solution is sufficiently stable. In this well traced regularization initial random perturbations, which is also a manifestation of BE at large distances.

ACKNOWLEDGMENTS

This work was supported by the contract 11.G34.31.0066 of Russian Federation Government, by RFBR (11-02-00774) and by Leading Scientific Schools grant N 333.2012.2.

REFERENCES

1. S.N. Gurbatov, O.V. Rudenko, A.I. Saichev, Waves and Structures in Nonlinear Nondispersive Media. General Theory and Applications to Nonlinear Acoustics (Springer, 2011), 500p.

2. S. N. Gurbatov, I. Yu. Demin, Val. V. Cherepennikov, B. O. Enflo, Behavior of intense acoustic noise at large distances, Acoustical Physics, 53, 48-63 (2007)

3. S. Gurbatov, I. Demin, N. Pronchatov-Rubtsov, Computational aspects of propagation of high-intensity modulated noise, Acta Acoustica united with Acoustica, 97(Suppl.1), S69 (2011).

4. K. Kameyama, T. Inoue , I.Yu. Demin, K. Kobayashi, T. Sato, Acoustical tissue nonlinearity characterization using bispectral analysis, Signal Processing, 52, 117-131 (1996).

5. O. V. Rudenko, S. N. Gurbatov, Noise signal propagation in soft biological tissues, Acoustical Physics, 58, 243-245 (2012).

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