Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://acoustics.phys.msu.ru/teachers/gusev_files/diff2011.pdf Äàòà èçìåíåíèÿ: Wed Nov 7 11:23:33 2012 Äàòà èíäåêñèðîâàíèÿ: Sat Feb 2 23:20:56 2013 Êîäèðîâêà:

DAYS on DIFFRACTION' 2011

77

Analytical solutions for diffraction problem of nonlinear acoustic wave b eam in the stratified atmosphere
Vladimir A. Gusev, Ruslan A. Zhostkow
Department of Acoustics, Physical Faculty, Lomonosov's Moscow State University, Russia; e-mail: vgusev@bk.ru
The nonlinear wave equation and mo dified Khokhlov­Zab olotskaya typ e equation for high intensive acoustics wave b eams propagating in stratified atmosphere with inhomogeneous of sound sp eed is set up. Some approaches to find analytical solutions of this equation are develop ed. The geometrical acoustics approximation and mo dified Raley integral for this problem is suggested. The asymptotical pro cedure is develop ed for describing of wave profile near the axis of wave b eam. This metho d allows to take into account phase distortion due to diffraction and nonlinear effects and improve the nonlinear geometrical acoustics solution.

where u -- particle velocity, -- medium density, p -- pressure, g -- free fall acceleration, c -- local sound speed. The equation (3) is equation of the adiabatic process s = const or dp/dt = 0, written with taking into account change of equilibrium state with height. The local sound speed depends on temperature and in the case of intensive waves includes the nonlinear terms due to nonlinear of state equation (Poisson adiabat). Assume that wave beam propagates vertically upward along axis z . The equilibrium state is defined by the following equations: p0 = -0 g , p0 = 0 RT . (4) z For the isothermal atmosphere one can obtain the equilibrium density distribution 0 = 00 exp(-z /H ), where 00 -- density near Earth surface and H -- the width of standard atmosphere. The following nonlinear equation for the vertical component of particle velocity can be obtained ( ) 2 2w 1 c2 w 2 - c2 w - + c2 B W w 0 t2 t2 z z (2 )( ) W R = - c2 Q- + c2 . (5) t2 t z Here the left side contains all linear terms and nonlinear term due to physical nonlinearity ( = 0 + , is acoustic perturbation of density), the right side contains only nonlinear terms Q, W , R of the complicated form. It is reasonable to simplify equation (5) to construct some analytical solutions. The most appropriative approach is the method of slowly changing wave profile. This method demands the wave profile changes weak at scale of wave length . This condition is satisfied in the problem under discussion because the characteristic scale H . In accordance with this method we look for the solu tion of this form p = p( = t - z /c, µz , µx, µy ),

1

Introduction

The problem of intensive acoustic wave propagation in the stratified atmosphere is connected with many important applications. Among them the influence of seismic processes under Earth surface on the high layers of atmosphere and interaction and energy exchange between different geospheres can be mentioned. The main feature of this problem is that the presence of gravity leads to equilibrium density decreasing with increasing of height. As a result the amplitude of acoustic velocity increases exponentially with height, so the nonlinear effects become important even for small initial amplitudes. 2 The main equation for the intensive wave beam

The equation for intensive wave beam propagation in the stratified atmosphere can be derived from the hydrodynamic equation system: 1 u + (u)u = - p - g, t + (u) + divu = 0, t ( ) p 2 + (u)p = c + (u) , t t (1)

(2) (3)


78

DAYS on DIFFRACTION' 2011

simplifying one So we obtain the exact equation for acoustic field along the wave beam axis ) c 2u 2c w =- 2 (z u) (13) 2 z a z 4 w2 2 + c2 B W w = 2 . (6) and the simplified model equation for field along 0 2c 4 the axis 2u 2c At usual conditions the Brent frequency B W is = - 2 u. (14) z a small and for high frequency acoustic waves one can Now let return to the nonlinear equation (7) and use the following equation use here expression (12). Then for the isothermal 1 c2 0 w c 2 w2 2w 0 atmosphere we can write the model nonlinear equa+ - w = 2 . z 20 c2 z 2 2c 2 tion (7) ( ) w w w w 2c Equations (5)­(7) can be used for description of in- -2 (z u) (15) =- 2 z 2H c a z tensive acoustic beams in the stratified atmosphere. 3 The model nonlinear equation for the acoustic field at the axis of the gaussian wave beam and simplified model nonlinear equation ( ) w w w w 2c - -2 = - 2 u. z 2H c a

µ 1 -- small parameter. After can obtain (2 2 w 1 c2 0 w 0 + - 2 z 20 c2 z

(16)

Now let us consider the linearized equation (7). For Equation (16) can be written in new variables isothermal atmosphere we obtain equation taking into account the exponential increasing with heght 2w 1 w c - = w . (8) w(z , ) = u(z , ) exp(z /2H ), z 2H 2 z After introducing new variable u = w exp(-z /2H ) x1 = exp(z /2H )dz = 2H (exp(z /2H ) - 1) 0 equation (8) has the form of linearized Khokhlov­ in following form Zabolotskaya equation ( ) 2u c u u u 2c u = w . (9) -2 =- 2 . (17) z 2 x1 c a 1 + x1 /2H Let consider the gaussian wave beam u = The dimensionless variables can be introduced exp(-r2 /a2 )u0 (t) as the initial condition. Then the general solution for arbitrary function u0 is znl = c2 0 /u0 , V = U /u0 , x = x1 /znl , 0 1 u= 2


Equation (16) becomes ]) ([ ( ) r2 /a2 V NV V . (10) â exp i - t - -V =- . 2cz /a2 + i x 1 + x/x0
- -

u0 (t) dt

d 2cz /a2 + i

= /0 ,

x0 = 2H /znl .

(18)

Solution at the beam axis

Give for comparison the exact equation for stratified atmosphere ) ( 2cz 2cz ( ) u = u0 ( ) - 2 u0 (t) exp (t - ) dt. V N V V a a2 -V =- (19) - (11) x 1 + x/x0 Now one can write the expression for the Laplacian and the standard Khokhlov­Zabolotskaya equation at the axis ( ) c 2u 2c V V u = (z u ) . (12) =- 2 -V = -N V . (20) 2 z r=0 a z r =0 x r =0


DAYS on DIFFRACTION' 2011

79

~ It is obviously that for case N 1 diffraction effects. Function S satisfies to equation ( effects prevail and for case N 1 nonlinear effects ~ ~ N S0 S prevail. For case x0 strafication effects can be S +V = ·V dV x 1 + x/x0 V V neglected. The most interesting is that stratifica) tion leads to relatively decreasing diffraction effects ~ ~ ~ S S0 S S in comparison with nonlinear effects. At heights + ·V dV + ·V dV . (23) V V V V x/x0 1 diffraction effects can be neglected. It means at large heights the nonlinear geometrical It is interesting that function f (x) from S0 doesn't acoustics will give good description. influence on nonlinear phase. Only distortion of initial function (V , x) can influence on this phase. It is obviously that this part of full nonlinear phase S 4 Equation for nonlinear diffraction is connected with change of wave amplitude. And phase other part is connected with the "true phase". Of course there are also terms responsible for interacThe main goal is to construct analytical solutions tion. for diffracting nonlinear wave. This case corre~ ~ First, let expand S as a series S = S1 + N S2 , so sponds to small but finite N . If N 1 the pertur S1 bation method will give good results where the non= -V , S1 = -xV (24) x linear terms are consider as small value. If N 0 and wave will be close to Riemann type wave. ( Some additional information gives the following S2 1 S0 S1 = ·V dV approach. Let consider variable as dependent on x 1 + x/x0 V V all other variables V = V (, x) = S (V , x). ) S1 S0 S1 S1 The equation for phase S is + ·V dV + ·V dV . (25) V V V V ( ) S/ x + V NV S For initial condition V (x = 0) = sin solution = (21) V S/ V 1 + x/x0 V S0 = arcsin V - N x. Therefore (22 ) S2 1 xV 3V 2 - 2 or, after integrating = -x . (26) x 1 + x/x0 2 1-V2 S N S S +V = ·V dV . (22) The main contribution is given by diffractionx 1 + x/x0 V V nonlinear interaction and is quadratic at small x. It is important that with diffraction taking into acFor case N = 0 one can obtain the solution count phase of nonlinear wave contains not only the S = -xV that is the phase of plane nondiffrac- first power of V but all other powers. tive Riemann wave. On the other hand the linear Another approach to obtain analytical solution is problem correspond to neglecting term V in the left the nonlinear geometrical acoustics. This method side of (22). This solution S0 can be obtain from leads to equation which can be solved exactly. But linearized Khokhlov­Zabolotskaya equation. nonlinear part of phase of wave in this approach coincide with phase of plane Riemann wave. Using the improved expression (24)­(26) for phase in the 5 Solution for nonlinear diffraction nonlinear geometrical acoustics solution can give phase more accurate solution for diffractive intensive wave beam at its axis. The idea of constructing analytical solution is as follows. Let expand function S as a series S = Acknowledgements ~~ S0 + S , S = S1 + N S2 on small parameter N , where S0 is the solution of linear equation. Solution S0 The work was financially supported by grants of the has the form S0 = (V , x) + f (x), f (0) = 0. Func- RF Presidential Program in Support of Leading Scition (V , x) is connected with initial condition. entific Schools (NSh-4590.2010.2) and the Russian ~ Function S describes phase shift due to nonlinear Foundation for Basic Research (pro ject no 09-02effects and interactions of nonlinear and diffractive 00925-a).


80

DAYS on DIFFRACTION' 2011

References [1] Golicyn G.S., Romanova N.N., 1968, Vertical propagation of acoustical waves in atmosphere with variable on heights viscosity, Izv. AN USSR, Physics of atmosphere and ocean, V. 4, No 2, pp. 210­214. [2] Romanova N.N., 1970, On vertical propagation of short acoustical waves in the atmosphere, Izv. AN USSR, Physics of atmosphere and ocean, V. 6, No 2, pp. 134­145. [3] Gusev V.A., 2010, The atmosphere heating due to wideband acoustic and shock waves propagating, Geophysical Journal, V. 32, No 4, pp. 56­57.

[4] Gusev V.A., Sobissevitch A.L., 2010, Propagation of wideband and shock waves induced by seismic activity in the stratified atmosphere, Proceedings of 28th International Conference on Mathematical Geophysics, Pisa, Italy, pp. 65. [5] Rudenko O.V., Soluyan S.I., 1977, Theoretical Foundations of Nonlinear Acoustics, New York: Plenum, Consultants Bureau. [6] 34. Gusev V.A., Sobissevitch A.L., 2010, Lowfrequency wave processes in geospheres foregoing intensive seismic events, Extreme nature phenomena and catastrophes, V. 1, Moscow: IEP RAS, pp. 65­80.