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Profiles of initially sinusoidal waves propagating in nonlinear microinhomogeneous materials
Ultrasonics 44 (2006) e1335–e1338 www.elsevier.com/locate/ultras
Profiles of initially sinusoidal waves propagating in nonlinear microinhomogeneous materials Vitalyi Gusev
*
Universite´ du Maine, UMR CNRS 6087, Av. O. Messiaen, 72085 Le Mans Cedex 09, France Available online 2 June 2006
Abstract The asymptotic analytical theory predicting acoustic wave profiles in microinhomogeneous materials with hysteretic quadratic nonlinearity and attenuation proportional to an even power of frequency is developed. The theory predicts that the influence on the nonlinear wave of the Rayleigh scattering of acoustic waves, which is proportional to the forth power of frequency, results in the net diminishing of wave attenuation. This is due to the suppression (diminishing) by scattering of the nonlinear hysteretic losses which is more important than direct increase in linear losses added by scattering. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Microinhomogeneous materials; Hysteretic nonlinearity; Scattering; Nonlinear waves
1. Introduction Theoretical analysis of finite-amplitude waves propagation in microinhomogeneous materials is stimulated by their much higher acoustical nonlinearity in comparison with amorphous materials and perfect crystals, and the fact that the manifestations of the hysteretic quadratic nonlinearity, which is a specific feature of these materials, differ from manifestations of classical elastic nonlinearities [1– 5]. One important direction of theory development is the analysis of different possible nonlinear phenomena in their ‘pure’ form [4–11] neglecting the role of such linear phenomena as wave absorption, dispersion, and scattering inevitably accompanying wave propagation in microinhomogeneous materials. Another important theoretical problem is to understand how these linear phenomena influence the nonlinear phenomena in materials with hysteretic nonlinearity. The influence of the linear absorption (quadratic in wave frequency) and of the linear relaxation processes was analyzed in [4,12], respectively. Below the influence on nonlinear wave of the sound attenuation increasing pro*
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0041-624X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2006.05.012
portionally to the forth power of frequency, characteristic to regime of Rayleigh scattering of acoustic waves in microinhomogeneous materials, is revealed. Evolution equation describing propagation of initially sinusoidal plane acoustic wave in microinhomogeneous material with hysteretic quadratic nonlinearity had been derived in [4] Vn + VAVh + Vj Vhj = 0. Here n is the propagation distance x normalized by the characteristic nonlinear length, h = 2p (t x/c)/T is the accompanying time normalized using the wave period T, V is the particle velocity normalized by its amplitude value at the boundary, VA(n) is the local wave amplitude, and c is the linear sound Rn speed. The shift of the time variable h ¼ h 0 V A ðn0 Þdn0 transforms this equation into a canonical form Vn + VjVhj = 0. The analytical solutions of this evolution equation in [4] as well as numerical analysis of wave interactions due to hysteretic quadratic nonlinearity in [5] both demonstrated that, although the wave profile V(n, h) keeps being continuous in propagation, the time-derivative of this profile oV/oh becomes discontinuous both in maximum and minimum of the wave. In other words the behavior of particle acceleration oV/oh is not analytic in the reversal points of material loading by sound wave. The examples of a half period of such profiles are marked by curve (1) in Figs. 1
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V. Gusev / Ultrasonics 44 (2006) e1335–e1338
and 2. In Ref. [4] it was demonstrated that to get the wave profiles with continuously varying oV/oh it is sufficient to take into consideration a possible linear absorption of sound which increases proportionally to the square of sound frequency. The generalized equation for this particular case is Vn + VjVhj AVhh = 0, where the nondimensional parameter A > 0 characterizes the importance of absorption relative to the importance of the hysteretic quadratic nonlinearity in the total wave process. From the mathematics point of view the inclusion in the equation of the linear term proportional to the derivative onV/ohn of the order n of the wave profile over time should provide 1 0.8
PARTICLE VELOCITY
0.6
oV =on þ V joV =ohj þ P on V =ohn ¼ 0
ð1Þ
in the case of even n, which can be applied to the analysis of the influence of wave attenuation on the nonlinear processes, is proposed. To describe wave attenuation and not the amplification the sign of the parameter P should be fixed by sign P = (1)n/2. In particular jPj = A in the previous case of n = 2. Another case of practical importance is one with n = 4, jPj = S > 0, which describes wave attenuation caused by scattering in microinhomogeneous media [13,14]. The nondimensional parameter S characterizes in the latter case the importance of Rayleigh scattering relative to the importance of the hysteretic quadratic nonlinearity in the wave process.
0.4
2. Theory
0.2 0 -0.2 (3)
-0.4 -0.6
(2) -0.8 (1)
-1 -3
-2
-1
0 TIME
1
2
3
Fig. 1. Modification of the profile of initially sinusoidal wave in material with hysteretic quadratic nonlinearity when sound absorption proportional to square of frequency increases. The profiles are presented for the values of the nondimensional parameter A (absorption/nonlinearity) equal to 0 (1), 0.1 (2) and 0.3 (3).
1 0.8 0.6 PARTICLE VELOCITY
opportunity to get the nonlinear wave profiles where all the derivatives up to on1V/ohn1 are continuous. In the present communication the general approach to the analysis of the wave equation
0.4 0.2 0
Here below a method of analysis of Eq. (1) in the asymptotic limit jPj 1 is presented. The method is an extension to the general case of even n of the approach proposed in [4] for the case n = 2. It should be clearly stated from the beginning that even for jPj 1 the role of the term with high-order derivative in Eq. (1) is of crucial importance for the elimination of the non analiticity of the solutions derived for jPj = 0. Near the maximum and the minimum of the wave profile (near the turning points in loading/unloading history) the magnitude of the term with high derivative is comparable with the magnitude of the nonlinear term. However the method of the successful approximation can still be developed taking into account that higher order derivatives are important only in the vicinity of the wave extrema in a time interval which is much shorter than the wave period. In other words the term with higher order derivative introduces an additional time scale in the phenomena which is much shorter than the wave period. The latter is equal to 2p in the accepted normalization. Based on these arguments an approximate solution of Eq. (1) is proposed in the form a
b
V ¼ V ð0Þ ðn; hÞ þ jP j V ð1Þ ðn; h=jP j Þ;
-0.2 -0.4
(1)
-0.6
(3)
-0.8
(2)
-1 -3
-2
-1
0 TIME
1
2
3
Fig. 2. Modification of the profile of initially sinusoidal wave in material with hysteretic quadratic nonlinearity when sound scattering proportional to the forth power of frequency increases. The profiles are presented for the values of the nondimensional parameter S (scattering/nonlinearity) equal to 0 (1), 0.001 (2) and 0.008 (3).
ð2Þ
where the first term is the known analytical solution [4] obtained in the absence of wave attenuation (for jPj = 0) which incorporates the shocks in particle acceleration near wave extrema, and the second term is an additional contribution caused by attenuation. Note that for a > 0, b > 0 the second term in Eq. (2) is small due to jPja 1 but its n-th derivative over time (/jPjanb) can provide an important contribution to wave equation. The substitution of Eq. ð0Þ a ð1Þ a (2) into Eq. (1) leads to V n þ j P j V f þ ðV ð0Þ þ jP j V ð1Þ Þ ð0Þ
ab
1þanb
jV h þ jP j oV ð1Þ =oh1 j þ signðP ÞjP j on V ð1Þ =ohn1 ¼ 0; b where h1 h/jPj denotes the fast time. Due to the assumpð0Þ ð0Þ tion jPja 1 the last equation simplifies V n þ V ð0Þ jV h þ
V. Gusev / Ultrasonics 44 (2006) e1335–e1338
jP jab oV ð1Þ =oh1 j þ signðP ÞjP j1þanb on V ð1Þ =ohn1 ¼ 0. Taking into account that V(0)(n, h) is the solution of Vn + VAVh + ð0Þ VjVhj = 0, the equation is transformed into V ð0Þ ½jV h þ ð0Þ ab 1þanb n ð1Þ @ V =ohn1 ¼ 0. jP j oV ð1Þ =oh1 j jV h j þ signðP ÞjP j All the terms here are of the same order of magnitude if a b = 0 and 1 + a nb = 0 simultaneously, leading to a = b = 1/(n 1). Thus the equation for the unknown function V(1)(n, h1) to be solved is ð0Þ
ð0Þ
V ð0Þ ½jV h þ oV ð1Þ =oh1 j jV h j þ signðP Þon V ð1Þ =ohn1 ¼ 0: ð3Þ (1)
The solution V (n, h1) of Eq. (3) should satisfy the conditions of the continuity of the wave profile Eq. (2) and of all its time derivatives up to the order n 1. Note that Eq. (3) is not a partial differential equation, but an ordinary differential equation. The dependence of its solution V(1)(n, h1) on the co-ordinate is just parametrical through the known dependence of V(0)(n, h) on the co-ordinate. Thus the method of successive approximations has provided important simplification of the evolution equation in comparison with its initial form in Eq. (1). In the case of wave attenuation (even n) the solution should describe a modification of the acoustic filed in the vicinity of the turning points. If h = 0 is chosen to coincide with a position of local wave extrema then the localization condition for the field V(1)(n, h1) has the form V ð1Þ ðn; h1 ! 1Þ ! 0:
ð4Þ (0)
For the initial sinusoidal excitation the function V (n, h) can be approximated near its extrema h = 0 by linear dependence on time V ð0Þ ðn; hÞ V ð0Þ ðn; 0Þ signðV ð0Þ ð0ÞÞj@V ð0Þ ðn; 0Þ=@hjjhj: In the following, for compactness of formulas, we will not write more the dependence on the co-ordinate explicitly, because it influences the final solution only parametrically. Assuming that attenuation does not change the sign of the first derivative oV(n, h)/oh of the total acoustic field near the extrema, Eqs. (3) and (5) lead to on V ð1Þ =ohn1 ð1Þn=2 signðhÞjV ð0Þ ð0ÞjoV ð1Þ =oh1 ¼ 0, where (1)n/2 is substituted for sign(P). The first integration of this equan=2 tion leads to on1 V ð1Þ =ohn1 ð1Þ signðhÞjV ð0Þ ð0Þj 1 ð1Þ V ¼ 0, where the integration constant has been eliminated by virtue of the condition Eq. (4). The possible solutions of this equation have the form n=2
signðhÞjV ð0Þ ð0Þj
1=ðn1Þ
:
exponent V(1) = C exp (jV(0)j jh1j). Determining the integration constant C from continuity of oV/oh in h = 0 the description of the wave profile near the extrema is obtained by adding V(1) to Eq. (5): V ðn; hÞ V ð0Þ ð0Þ signðV ð0Þ ð0Þ ð0Þ ð0ÞÞjoV h ðn; 0Þjjhj A½jV h ð0Þj=V ð0Þ ð0Þ expðjV ð0Þ ð0Þjjhj= AÞ. This solution of the equation Vn + VjVhj AVhh = 0 has been derived earlier in [4]. For its graphical illustration it is rewritten by introducing normalization to local amplitude value V V =V ð0Þ ðn; 0Þ, A ¼ A=jV ð0Þ ðn; 0Þj: V ¼ 1 ð0Þ jV h ð0Þjjhjð1 þ expðjhj=AÞA=jhjÞ. The profile V of a half period of the initially sinusoidal wave predicted by this solution at distances n 1, where the profile V(0)(n, h) is composed of purely linear parts [4] and joV ð0Þ ð0Þ= ohj ffi 2=p does not vary any more with distance, is presented in Fig. 1 for different values of the local parameter A. In the case n = 4 of sound attenuation proportional to forth power of frequency (6) predicts pffiffiEq. ffi pffiffiffi three possible values c1;2;3 ¼ fð1 i 3Þ=2; ð1 þ i 3Þ=2; 1gsignðhÞ 1=3 jV ð0Þ ð0Þj and a solution in form of a linear superposition 1=3 of V ð1Þ ¼ C 1 exp½jVpð0Þ pffiffiffi three exponents. ð0Þ ffiffiffi ð0Þj ð1=2 1=3 i 3=2Þjh1 j þ C 2 exp½jV ð0Þj ð1=2 þ i 3=2Þjh1 j þ C 3 exp½jV ð0Þ ð0Þj1=3 jh1 j. The integration constant C3 should be zero by virtue of Eq. (4). The integration constant C1 and C2 can be obtained from the continuity condition of oV/oh and o3V/oh3 in h = 0, which for the even function of time are equivalent to oV(1)/oh (0) = 0 and o3V(1)/oh3(0) = 0. By adding the obtained solution for V(1) to Eq. (5) it is derived V ðn; hÞ V ð0Þ ð0Þ signðV ð0Þ ð0ÞÞjoV ð0Þ ðn; 0Þ=ohjjhj pffiffiffi ð0Þ þ ð2= 3ÞS 1=3 fjV h j=½V ð0Þ ð0Þ1=3 g 1=3
ð5Þ
V ð1Þ / expðch1 Þ; c ¼ ½ð1Þ
e1337
ð6Þ
In the following the derived general solution will be specified for the important particular cases of n = 2 and n = 4. 3. Analysis In the case n = 2 of sound absorption proportional to square of frequency Eq. (6) predicts a single value c = sign(h)jV(0)(0)j and a solution in form of a single
expðjV ð0Þ ð0Þj jhj=2=S 1=3 Þ pffiffiffi 1=3
sinð 3jV ð0Þ ð0Þj jhj=2=S 1=3 Þ For graphical illustration of the wave profile this description is rewritten by introducing normalization to local amplitude value V Vp =Vffiffiffi ð0Þ ðn; 0Þ, S ¼ S=jV ð0Þ ðn; 0Þj V ¼ ð0Þ 1 p jVffiffiffih ð0Þjjhjf1 ð2= 3ÞðS 1=3 =jhjÞ exp½ð1=2Þðjhj=S 1=3 Þ sin½ð 3=2Þðjhj=S 1=3 Þg: The profile of a half period of the initially sinusoidal wave predicted by this solution at distances n 1 is presented in Fig. 2 for different values of the local parameter S 1=3 . Please note that at distances n 6 1, where the initially sinusoidal wave is not yet fully transformed in a triangular form, the obtained solutions are still locally valid near the wave extrema although joV ð0Þ ð0Þ=ohj is, in general, a function of n at these distances. 4. Discussion The obtained analytical solution for the profile of the initially sinusoidal wave in materials with hysteretic quadratic nonlinearity and Rayleigh type scattering demonstrates (Fig. 2) that in a weak scattering regime (characterized by inequality S 1=3 1Þ the scattering does
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V. Gusev / Ultrasonics 44 (2006) e1335–e1338
not modify the nonlinear wave amplitude. Moreover, the scattering increases the particle velocity V(n, h) in the vicinity (h 0, h 5 0) of the extrema. It can be verified that due to this the amplitude of the fundamental frequency component increases with increasing magnitude of the parameter S. It can be also seen from Fig. 2 and verified by integration of the obtained analytical solution that the area of a half period of the profile increases when the scattering of the acoustic waves by micro-inhomogeneities is taken into account. The acoustic intensity proportional to the integral of V2(n,h) over the wave period also increases. Both these findings (the increase in the fundamental frequency amplitude and in the acoustic wave intensity) indicate that the Rayleigh scattering significantly suppresses the nonlinear hysteretic processes which are causing wave spectrum transformation and nonlinear energy losses. This suppression of nonlinear attenuation processes by wave scattering overcompensates additional linear attenuation of acoustic waves introduced by scattering directly. The effect of suppression of the nonlinear losses by introducing additional linear losses for higher harmonics (at frequencies higher than a fundamental one) is well understood in the case of quadratic elastic nonlinearity [15–18]. It is due to suppression of cascade processes of harmonics excitation by additional linear absorption of higher harmonics. The analytical solutions illustrated in Figs. 1 and 2 demonstrate that suppression of nonlinear hysteretic losses by including additional linear losses overcompensates these additional linear losses if they are of forth power in frequency, and does not compensate these
additional losses if they are quadratic in frequency. Thus linear attenuation should increase sufficiently fast with wave frequency in order to be capable of diminishing the total (nonlinear plus linear) losses in the micro-inhomogeneous material. References [1] V.E. Nazarov, L.A. Ostrovsky, I.A. Soustova, A.M. Sutin, Phys. Earth Planet. Inter. 50 (1988) 65–73. [2] R.A. Guyer, P.A. Johnson, Phys. Today 52 (1999) 30–35. [3] L.A. Ostrovsky, P.A. Johnson, Rivista del Nuovo Cimento 24 (2001) 1–46. [4] V. Gusev, C. Glorieux, W. Lauriks, J. Thoen, Phys. Lett. A 232 (1997) 77–86. [5] K.E.-A. van den Abeele, P.A. Johnson, R.A. Guyer, K.R. McCall, J. Acoust. Soc. Am. 101 (1997) 1885–1898. [6] V. Gusev, J. Acoust. Soc. Am. 107 (2000) 2047–2058. [7] V. Gusev, V.Yu. Zaitsev, Phys. Lett. A 314 (2003) 117–125. [8] V. Aleshin, V. Gusev, V.Yu. Zaitsev, J. Comput. Acoust. 12 (2004) 319–354. [9] V.Yu. Zaitsev, V.E. Gusev, Yu. Zaitsev, Ultrasonics 43 (2005) 183– 195. [10] V. Gusev, J. Acoust. Soc. Am. 117 (2005) 1850–1857. [11] V. Gusev, Wave Motion 42 (2005) 97–108. [12] V. Gusev, W. Lauriks, J. Thoen, J. Acoust. Soc. Am. 103 (1998) 3216–3226. [13] H. Sato, M.C. Fehler, Seismic Wave Propagation and Scattering in the Heterogeneous Earth, AIP Press, Springer, NY, 1998. [14] O.V. Rudenko, V.A. Robsman, Doklady Phys. 47 (2002) 443–446. [15] M.B. Moffett, R.H. Mellen, J. Sound Vibr. 76 (1981) 295–297. [16] H.C. Woodsum, J. Sound Vib. 76 (1981) 297–298. [17] O.V. Rudenko, Sov. Phys. Acoust. 29 (1983) 234–236. [18] V.E. Gusev, Moscow Univ. Phys. Bull. 39 (1984) 29–34.
Profiles of initially sinusoidal waves propagating in nonlinear microinhomogeneous materials Vitalyi Gusev
*
Universite´ du Maine, UMR CNRS 6087, Av. O. Messiaen, 72085 Le Mans Cedex 09, France Available online 2 June 2006
Abstract The asymptotic analytical theory predicting acoustic wave profiles in microinhomogeneous materials with hysteretic quadratic nonlinearity and attenuation proportional to an even power of frequency is developed. The theory predicts that the influence on the nonlinear wave of the Rayleigh scattering of acoustic waves, which is proportional to the forth power of frequency, results in the net diminishing of wave attenuation. This is due to the suppression (diminishing) by scattering of the nonlinear hysteretic losses which is more important than direct increase in linear losses added by scattering. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Microinhomogeneous materials; Hysteretic nonlinearity; Scattering; Nonlinear waves
1. Introduction Theoretical analysis of finite-amplitude waves propagation in microinhomogeneous materials is stimulated by their much higher acoustical nonlinearity in comparison with amorphous materials and perfect crystals, and the fact that the manifestations of the hysteretic quadratic nonlinearity, which is a specific feature of these materials, differ from manifestations of classical elastic nonlinearities [1– 5]. One important direction of theory development is the analysis of different possible nonlinear phenomena in their ‘pure’ form [4–11] neglecting the role of such linear phenomena as wave absorption, dispersion, and scattering inevitably accompanying wave propagation in microinhomogeneous materials. Another important theoretical problem is to understand how these linear phenomena influence the nonlinear phenomena in materials with hysteretic nonlinearity. The influence of the linear absorption (quadratic in wave frequency) and of the linear relaxation processes was analyzed in [4,12], respectively. Below the influence on nonlinear wave of the sound attenuation increasing pro*
Tel.: +33 243833976; fax: +33 243833518. E-mail address: [email protected]
0041-624X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2006.05.012
portionally to the forth power of frequency, characteristic to regime of Rayleigh scattering of acoustic waves in microinhomogeneous materials, is revealed. Evolution equation describing propagation of initially sinusoidal plane acoustic wave in microinhomogeneous material with hysteretic quadratic nonlinearity had been derived in [4] Vn + VAVh + Vj Vhj = 0. Here n is the propagation distance x normalized by the characteristic nonlinear length, h = 2p (t x/c)/T is the accompanying time normalized using the wave period T, V is the particle velocity normalized by its amplitude value at the boundary, VA(n) is the local wave amplitude, and c is the linear sound Rn speed. The shift of the time variable h ¼ h 0 V A ðn0 Þdn0 transforms this equation into a canonical form Vn + VjVhj = 0. The analytical solutions of this evolution equation in [4] as well as numerical analysis of wave interactions due to hysteretic quadratic nonlinearity in [5] both demonstrated that, although the wave profile V(n, h) keeps being continuous in propagation, the time-derivative of this profile oV/oh becomes discontinuous both in maximum and minimum of the wave. In other words the behavior of particle acceleration oV/oh is not analytic in the reversal points of material loading by sound wave. The examples of a half period of such profiles are marked by curve (1) in Figs. 1
e1336
V. Gusev / Ultrasonics 44 (2006) e1335–e1338
and 2. In Ref. [4] it was demonstrated that to get the wave profiles with continuously varying oV/oh it is sufficient to take into consideration a possible linear absorption of sound which increases proportionally to the square of sound frequency. The generalized equation for this particular case is Vn + VjVhj AVhh = 0, where the nondimensional parameter A > 0 characterizes the importance of absorption relative to the importance of the hysteretic quadratic nonlinearity in the total wave process. From the mathematics point of view the inclusion in the equation of the linear term proportional to the derivative onV/ohn of the order n of the wave profile over time should provide 1 0.8
PARTICLE VELOCITY
0.6
oV =on þ V joV =ohj þ P on V =ohn ¼ 0
ð1Þ
in the case of even n, which can be applied to the analysis of the influence of wave attenuation on the nonlinear processes, is proposed. To describe wave attenuation and not the amplification the sign of the parameter P should be fixed by sign P = (1)n/2. In particular jPj = A in the previous case of n = 2. Another case of practical importance is one with n = 4, jPj = S > 0, which describes wave attenuation caused by scattering in microinhomogeneous media [13,14]. The nondimensional parameter S characterizes in the latter case the importance of Rayleigh scattering relative to the importance of the hysteretic quadratic nonlinearity in the wave process.
0.4
2. Theory
0.2 0 -0.2 (3)
-0.4 -0.6
(2) -0.8 (1)
-1 -3
-2
-1
0 TIME
1
2
3
Fig. 1. Modification of the profile of initially sinusoidal wave in material with hysteretic quadratic nonlinearity when sound absorption proportional to square of frequency increases. The profiles are presented for the values of the nondimensional parameter A (absorption/nonlinearity) equal to 0 (1), 0.1 (2) and 0.3 (3).
1 0.8 0.6 PARTICLE VELOCITY
opportunity to get the nonlinear wave profiles where all the derivatives up to on1V/ohn1 are continuous. In the present communication the general approach to the analysis of the wave equation
0.4 0.2 0
Here below a method of analysis of Eq. (1) in the asymptotic limit jPj 1 is presented. The method is an extension to the general case of even n of the approach proposed in [4] for the case n = 2. It should be clearly stated from the beginning that even for jPj 1 the role of the term with high-order derivative in Eq. (1) is of crucial importance for the elimination of the non analiticity of the solutions derived for jPj = 0. Near the maximum and the minimum of the wave profile (near the turning points in loading/unloading history) the magnitude of the term with high derivative is comparable with the magnitude of the nonlinear term. However the method of the successful approximation can still be developed taking into account that higher order derivatives are important only in the vicinity of the wave extrema in a time interval which is much shorter than the wave period. In other words the term with higher order derivative introduces an additional time scale in the phenomena which is much shorter than the wave period. The latter is equal to 2p in the accepted normalization. Based on these arguments an approximate solution of Eq. (1) is proposed in the form a
b
V ¼ V ð0Þ ðn; hÞ þ jP j V ð1Þ ðn; h=jP j Þ;
-0.2 -0.4
(1)
-0.6
(3)
-0.8
(2)
-1 -3
-2
-1
0 TIME
1
2
3
Fig. 2. Modification of the profile of initially sinusoidal wave in material with hysteretic quadratic nonlinearity when sound scattering proportional to the forth power of frequency increases. The profiles are presented for the values of the nondimensional parameter S (scattering/nonlinearity) equal to 0 (1), 0.001 (2) and 0.008 (3).
ð2Þ
where the first term is the known analytical solution [4] obtained in the absence of wave attenuation (for jPj = 0) which incorporates the shocks in particle acceleration near wave extrema, and the second term is an additional contribution caused by attenuation. Note that for a > 0, b > 0 the second term in Eq. (2) is small due to jPja 1 but its n-th derivative over time (/jPjanb) can provide an important contribution to wave equation. The substitution of Eq. ð0Þ a ð1Þ a (2) into Eq. (1) leads to V n þ j P j V f þ ðV ð0Þ þ jP j V ð1Þ Þ ð0Þ
ab
1þanb
jV h þ jP j oV ð1Þ =oh1 j þ signðP ÞjP j on V ð1Þ =ohn1 ¼ 0; b where h1 h/jPj denotes the fast time. Due to the assumpð0Þ ð0Þ tion jPja 1 the last equation simplifies V n þ V ð0Þ jV h þ
V. Gusev / Ultrasonics 44 (2006) e1335–e1338
jP jab oV ð1Þ =oh1 j þ signðP ÞjP j1þanb on V ð1Þ =ohn1 ¼ 0. Taking into account that V(0)(n, h) is the solution of Vn + VAVh + ð0Þ VjVhj = 0, the equation is transformed into V ð0Þ ½jV h þ ð0Þ ab 1þanb n ð1Þ @ V =ohn1 ¼ 0. jP j oV ð1Þ =oh1 j jV h j þ signðP ÞjP j All the terms here are of the same order of magnitude if a b = 0 and 1 + a nb = 0 simultaneously, leading to a = b = 1/(n 1). Thus the equation for the unknown function V(1)(n, h1) to be solved is ð0Þ
ð0Þ
V ð0Þ ½jV h þ oV ð1Þ =oh1 j jV h j þ signðP Þon V ð1Þ =ohn1 ¼ 0: ð3Þ (1)
The solution V (n, h1) of Eq. (3) should satisfy the conditions of the continuity of the wave profile Eq. (2) and of all its time derivatives up to the order n 1. Note that Eq. (3) is not a partial differential equation, but an ordinary differential equation. The dependence of its solution V(1)(n, h1) on the co-ordinate is just parametrical through the known dependence of V(0)(n, h) on the co-ordinate. Thus the method of successive approximations has provided important simplification of the evolution equation in comparison with its initial form in Eq. (1). In the case of wave attenuation (even n) the solution should describe a modification of the acoustic filed in the vicinity of the turning points. If h = 0 is chosen to coincide with a position of local wave extrema then the localization condition for the field V(1)(n, h1) has the form V ð1Þ ðn; h1 ! 1Þ ! 0:
ð4Þ (0)
For the initial sinusoidal excitation the function V (n, h) can be approximated near its extrema h = 0 by linear dependence on time V ð0Þ ðn; hÞ V ð0Þ ðn; 0Þ signðV ð0Þ ð0ÞÞj@V ð0Þ ðn; 0Þ=@hjjhj: In the following, for compactness of formulas, we will not write more the dependence on the co-ordinate explicitly, because it influences the final solution only parametrically. Assuming that attenuation does not change the sign of the first derivative oV(n, h)/oh of the total acoustic field near the extrema, Eqs. (3) and (5) lead to on V ð1Þ =ohn1 ð1Þn=2 signðhÞjV ð0Þ ð0ÞjoV ð1Þ =oh1 ¼ 0, where (1)n/2 is substituted for sign(P). The first integration of this equan=2 tion leads to on1 V ð1Þ =ohn1 ð1Þ signðhÞjV ð0Þ ð0Þj 1 ð1Þ V ¼ 0, where the integration constant has been eliminated by virtue of the condition Eq. (4). The possible solutions of this equation have the form n=2
signðhÞjV ð0Þ ð0Þj
1=ðn1Þ
:
exponent V(1) = C exp (jV(0)j jh1j). Determining the integration constant C from continuity of oV/oh in h = 0 the description of the wave profile near the extrema is obtained by adding V(1) to Eq. (5): V ðn; hÞ V ð0Þ ð0Þ signðV ð0Þ ð0Þ ð0Þ ð0ÞÞjoV h ðn; 0Þjjhj A½jV h ð0Þj=V ð0Þ ð0Þ expðjV ð0Þ ð0Þjjhj= AÞ. This solution of the equation Vn + VjVhj AVhh = 0 has been derived earlier in [4]. For its graphical illustration it is rewritten by introducing normalization to local amplitude value V V =V ð0Þ ðn; 0Þ, A ¼ A=jV ð0Þ ðn; 0Þj: V ¼ 1 ð0Þ jV h ð0Þjjhjð1 þ expðjhj=AÞA=jhjÞ. The profile V of a half period of the initially sinusoidal wave predicted by this solution at distances n 1, where the profile V(0)(n, h) is composed of purely linear parts [4] and joV ð0Þ ð0Þ= ohj ffi 2=p does not vary any more with distance, is presented in Fig. 1 for different values of the local parameter A. In the case n = 4 of sound attenuation proportional to forth power of frequency (6) predicts pffiffiEq. ffi pffiffiffi three possible values c1;2;3 ¼ fð1 i 3Þ=2; ð1 þ i 3Þ=2; 1gsignðhÞ 1=3 jV ð0Þ ð0Þj and a solution in form of a linear superposition 1=3 of V ð1Þ ¼ C 1 exp½jVpð0Þ pffiffiffi three exponents. ð0Þ ffiffiffi ð0Þj ð1=2 1=3 i 3=2Þjh1 j þ C 2 exp½jV ð0Þj ð1=2 þ i 3=2Þjh1 j þ C 3 exp½jV ð0Þ ð0Þj1=3 jh1 j. The integration constant C3 should be zero by virtue of Eq. (4). The integration constant C1 and C2 can be obtained from the continuity condition of oV/oh and o3V/oh3 in h = 0, which for the even function of time are equivalent to oV(1)/oh (0) = 0 and o3V(1)/oh3(0) = 0. By adding the obtained solution for V(1) to Eq. (5) it is derived V ðn; hÞ V ð0Þ ð0Þ signðV ð0Þ ð0ÞÞjoV ð0Þ ðn; 0Þ=ohjjhj pffiffiffi ð0Þ þ ð2= 3ÞS 1=3 fjV h j=½V ð0Þ ð0Þ1=3 g 1=3
ð5Þ
V ð1Þ / expðch1 Þ; c ¼ ½ð1Þ
e1337
ð6Þ
In the following the derived general solution will be specified for the important particular cases of n = 2 and n = 4. 3. Analysis In the case n = 2 of sound absorption proportional to square of frequency Eq. (6) predicts a single value c = sign(h)jV(0)(0)j and a solution in form of a single
expðjV ð0Þ ð0Þj jhj=2=S 1=3 Þ pffiffiffi 1=3
sinð 3jV ð0Þ ð0Þj jhj=2=S 1=3 Þ For graphical illustration of the wave profile this description is rewritten by introducing normalization to local amplitude value V Vp =Vffiffiffi ð0Þ ðn; 0Þ, S ¼ S=jV ð0Þ ðn; 0Þj V ¼ ð0Þ 1 p jVffiffiffih ð0Þjjhjf1 ð2= 3ÞðS 1=3 =jhjÞ exp½ð1=2Þðjhj=S 1=3 Þ sin½ð 3=2Þðjhj=S 1=3 Þg: The profile of a half period of the initially sinusoidal wave predicted by this solution at distances n 1 is presented in Fig. 2 for different values of the local parameter S 1=3 . Please note that at distances n 6 1, where the initially sinusoidal wave is not yet fully transformed in a triangular form, the obtained solutions are still locally valid near the wave extrema although joV ð0Þ ð0Þ=ohj is, in general, a function of n at these distances. 4. Discussion The obtained analytical solution for the profile of the initially sinusoidal wave in materials with hysteretic quadratic nonlinearity and Rayleigh type scattering demonstrates (Fig. 2) that in a weak scattering regime (characterized by inequality S 1=3 1Þ the scattering does
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V. Gusev / Ultrasonics 44 (2006) e1335–e1338
not modify the nonlinear wave amplitude. Moreover, the scattering increases the particle velocity V(n, h) in the vicinity (h 0, h 5 0) of the extrema. It can be verified that due to this the amplitude of the fundamental frequency component increases with increasing magnitude of the parameter S. It can be also seen from Fig. 2 and verified by integration of the obtained analytical solution that the area of a half period of the profile increases when the scattering of the acoustic waves by micro-inhomogeneities is taken into account. The acoustic intensity proportional to the integral of V2(n,h) over the wave period also increases. Both these findings (the increase in the fundamental frequency amplitude and in the acoustic wave intensity) indicate that the Rayleigh scattering significantly suppresses the nonlinear hysteretic processes which are causing wave spectrum transformation and nonlinear energy losses. This suppression of nonlinear attenuation processes by wave scattering overcompensates additional linear attenuation of acoustic waves introduced by scattering directly. The effect of suppression of the nonlinear losses by introducing additional linear losses for higher harmonics (at frequencies higher than a fundamental one) is well understood in the case of quadratic elastic nonlinearity [15–18]. It is due to suppression of cascade processes of harmonics excitation by additional linear absorption of higher harmonics. The analytical solutions illustrated in Figs. 1 and 2 demonstrate that suppression of nonlinear hysteretic losses by including additional linear losses overcompensates these additional linear losses if they are of forth power in frequency, and does not compensate these
additional losses if they are quadratic in frequency. Thus linear attenuation should increase sufficiently fast with wave frequency in order to be capable of diminishing the total (nonlinear plus linear) losses in the micro-inhomogeneous material. References [1] V.E. Nazarov, L.A. Ostrovsky, I.A. Soustova, A.M. Sutin, Phys. Earth Planet. Inter. 50 (1988) 65–73. [2] R.A. Guyer, P.A. Johnson, Phys. Today 52 (1999) 30–35. [3] L.A. Ostrovsky, P.A. Johnson, Rivista del Nuovo Cimento 24 (2001) 1–46. [4] V. Gusev, C. Glorieux, W. Lauriks, J. Thoen, Phys. Lett. A 232 (1997) 77–86. [5] K.E.-A. van den Abeele, P.A. Johnson, R.A. Guyer, K.R. McCall, J. Acoust. Soc. Am. 101 (1997) 1885–1898. [6] V. Gusev, J. Acoust. Soc. Am. 107 (2000) 2047–2058. [7] V. Gusev, V.Yu. Zaitsev, Phys. Lett. A 314 (2003) 117–125. [8] V. Aleshin, V. Gusev, V.Yu. Zaitsev, J. Comput. Acoust. 12 (2004) 319–354. [9] V.Yu. Zaitsev, V.E. Gusev, Yu. Zaitsev, Ultrasonics 43 (2005) 183– 195. [10] V. Gusev, J. Acoust. Soc. Am. 117 (2005) 1850–1857. [11] V. Gusev, Wave Motion 42 (2005) 97–108. [12] V. Gusev, W. Lauriks, J. Thoen, J. Acoust. Soc. Am. 103 (1998) 3216–3226. [13] H. Sato, M.C. Fehler, Seismic Wave Propagation and Scattering in the Heterogeneous Earth, AIP Press, Springer, NY, 1998. [14] O.V. Rudenko, V.A. Robsman, Doklady Phys. 47 (2002) 443–446. [15] M.B. Moffett, R.H. Mellen, J. Sound Vibr. 76 (1981) 295–297. [16] H.C. Woodsum, J. Sound Vib. 76 (1981) 297–298. [17] O.V. Rudenko, Sov. Phys. Acoust. 29 (1983) 234–236. [18] V.E. Gusev, Moscow Univ. Phys. Bull. 39 (1984) 29–34.