ISSN 10637710, Acoustical Physics, 2011, Vol. 57, No. 5, pp. 616–619. © Pleiades Publishing, Ltd., 2011. Original Russian Text © V.E. Nazarov, A.V. Radostin, 2011, published in Akusticheskii Zhurnal, 2011, Vol. 57, No. 5, pp. 596–599.
NONLINEAR ACOUSTICS
Amplitude Modulation of Sound by Sound in WaterSaturated River Sand V. E. Nazarov and A. V. Radostin Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603950 Russia email:
[email protected]nnov.ru Received October 16, 2010
Abstract—We present a description and the results of a laboratory experiment on observing amplitude mod ulation of a weak continuous highfrequency wave under the action of a sequence of strong lowfrequency pulses in a statically loaded grainy medium—watersaturated river sand. We present characteristic oscillo grams of the obtained acoustic signals, testifying to the manifestation of dissipative nonlinearity of river sand; as well, we observed both the effect of damping and gain of sound by sound. In the first case, this is connected with an increase in the dissipation of the medium under the action of a powerful lowfrequency pulse, and in the second case, with its decrease. Keywords: nonlinear waves, dissipative nonlinearity, modulation of sound by sound. DOI: 10.1134/S1063771011040178
The discovery of “unusual” nonlinear effects in the propagation of elastic waves in geophysical structures, to which river sand pertains, constitutes one of the urgent problems in modern nonlinear acoustics [1], and studies of nonlinear acoustic effects in river sand in controlled laboratory conditions makes it possible to simulate nonlinear wave processes in seismoacous tics and geophysics. Laboratory experiments have shown that many microinhomogeneous solidstate media (rock, certain metals, soils, river sand, etc.) possess anomalously high acoustic nonlinearity, not only reactive (elastic) and hysteresis, but also dissipa tive (inelastic). Dissipative nonlinearity manifests itself in the fact that the coefficient of dissipation of a medium is dependent on the amplitude of an acoustic wave. As a result, during propagation of two waves of different frequency—powerful lowfrequency (LF) and weak highfrequency (HF)—in a medium with dissipative nonlinearity, the effects of damping or gain of sound by sound are observed. In an amplitude modulated powerful LF wave, as a result of modula tion of the coefficient of dissipation of the medium, amplitude modulation of a weak, probe HF wave will occur, i.e., transfer of amplitude modulation from the powerful wave to the weak, probe wave. Such a dynamic (or modulation) effect of gain of sound by sound has been observed in natural seismoacoustic experimenst on surface waves in wet sandy earth [2]. With the goal of discovering this effect in controlled and regulated conditions, we conducted a laboratory experiment on observing amplitude modulation of a weak relatively HF wave under the action of a periodic sequence of strong LF pulses in a grainy medium—
watersaturated (or wet) river sand. Note that in usual homogeneous media, e.g., in water, in silicate and organic class, solid metals (steel, molybdenum, tita nium, etc.), such an effect is not observed. Figure 1 shows a diagram of the experiment. Water saturated river sand was located in a thinwalled vessel with a diameter of 16 cm and a height of 22 cm. The −2 mean size a of sand grains was about 2 × 10 cm. In creating the watersaturated medium, dry sand first was poured into water and then, as it evaporated, water was added to the sand. To obtain stable and repeatable acoustic measurement results, static pressure was created in the sand using a system of weights 1 with mass M.
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5
1
1 2
4 3
Fig. 1. Diagram of experiment: 1, system of weights; 2, plate; 3 and 4, LF and HF emitters; 5, piezoaccelerometer.
AMPLITUDE MODULATION OF SOUND BY SOUND IN WATERSATURATED
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X 1
Y 1
X 2
–32.000 ms
18.000 ms
68.000 ms
Fig. 2. Oscillograms of LF pulses (bottom) and HF initially continuous wave (top) received by piezoaccelerometer 5 in damping of sound by sound.
Weights were situated on a thin organic glass plate 2 located on the surface of the sand and covering nearly all of section S of the vessel so that the vertical compo nent of static pressure P0 in the sand was nearly con stant over the height and was determined by the expression P0 ≅ Mg S Ⰷ ρgh, g is acceleration of gravity and ρ is the density of wet sand. Experiments were conducted at a static pressure of P0 ≅ 6.2 × 103 Pa. Two acoustic emitters (LF 3 and HF 4) were located on one axis and at depths of h1 = 13 cm and h2 = 10.5 cm from the surface of sand, respectively. With the aid of these, vertically propagating (from bottom to top) lon gitudinal LF pulses were excited with a carrier fre quency of F = 4 kHz, a duration of T = 10 ms, and a repetition period of T0 = 35 ms, as well as a continuous HF wave with a frequency of f = 18 kHz. The diame ter of the emitters was 8 and 4 cm, respectively. To record the acoustic waves that passed through the sand, piezoaccelerometer 5 was used, which reacted to the vertical component of acceleration; it was attached to the center of plate 2, on which weights were situ ated. Signals from piezoaccelerometer 5 via a system of filters reached a dualchannel digital oscilloscope spectral analyzer (Hewlett Packard 54520 A), where the oscillograms of LF and HF waves were displayed. At the selected static pressure, the pulse propagation rate in watersaturated sand was C0 ≅ 2.9 × 10 4 cm/s. Wave length λ 1,2 of LF and HF acoustic waves in sand was −2 nearly 7.2 and 1.6 cm and the grain size was ~2 × 10 cm, so that the minimal ratio min{λ1,2 a} ≈ 80 . As a result, in this experiment, river sand can be considered as a microinhomogeneous solid medium [3]. As experimental studies on the effects of selfaction of an acoustic wave have shown [4], the nonlinear (elastic and inelastic) properties of weakly consoli dated grainy media are to a certain extent random. In ACOUSTICAL PHYSICS
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many aspects, this is related to the random and irregu lar packing of a large number of grains (sand grains) differing in shape and size, as well as with the presence of liquid and gas between them, which determine the nonlinear dynamic elastic and inelastic stresses at the contact of these grains. At a relatively small static pres sure P0 , the packing of grains can change quite rapidly to relatively weak mechanical interactions (shocks) on the walls of the vessel containing the sand. After sev eral shocks, the microscopic configuration of weakly stressed grains, which mainly determine the acoustic nonlinearity of the grainy medium, change. Owing to this, its nonlinear properties also change, which leads to essentially (qualitatively and quantitatively) differ ent manifestations of the effect of amplitude modula tion of sound by sound in such a medium. In multiple experiments with watersaturated river sand (after mechanical actions on the vessel—rela tively weak shocks and “refilling” of water in it), we observe the effects of amplitude modulation of a weak continuous HF wave under the action of a periodic sequence of powerful LF pulses; as well, we observed both the damping and gain of sound by sound. In the first case, these effects are related to the increase in dissipation of the medium under the action of the powerful LF wave, and in the second, with its decrease. Figures 2 and 3 show two characteristic oscillograms for these cases of LF pulses and the HF wave received by the piezoaccelerometer. From this it is seen that during the action of the LF pulse, the amplitude of the HF wave changes; namely, it decreases by almost a factor of 10 (Fig. 2) or increases by more than a factor of 2 (Fig. 3). The relatively small pulling of the amplitude modulated HF wavefronts in Figures 2 and 3 is related to the transition processes in the system of filters used in receiving and separating LF and HF waves. Damping of sound by sound took place after a sufficiently prolonged time (more than a
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NAZAROV, RADOSTIN X 1
Y 1
Y 2
X 2
–19.200 ms
5.800 ms
30.800 ms
Fig. 3. Oscillograms of LF pulses (bottom) and HF initially continuous wave (top) received by piezoaccelerometeer 5 in gain of sound by sound.
day) after filling of water, and the gain of sound by sound, as a rule, was observed initially, with relatively large water saturation of sand. Note that Figs. 2 and 3 show in essence two cases, close to the limiting case, manifestations of modulation effects, namely, a large damping and gain of sound by sound. As well, in the majority of observations, “intermediate” oscillograms occurred, where the amplitude modulation of sound by sound was less than in Figs. 2 and 3, and during the gain–damping transition, modulation effects were not observed. As the experiment was being conducted, its geom etry did not change; therefore, the discovered modula tion effects are not caused by geometry. The conclu sion on the nonlinearity of river sand follows from nonfulfillment of the superposition principle for LF and HF acoustic waves. In accordance with this prin ciple, waves in linear media do not interact, and they propagate linearly and independently. From the pre sented oscillograms, it is seen that in watersaturated river sand, the powerful LF pulse influences the con dition of propagation of the weak HF wave: it changes the amplitude of the weak probe HF wave and, corre spondingly, the absorption of the medium. Therefore, here, the superposition principle is not fulfilled and river sand is a nonlinear medium, one possessing dis sipative nonlinearity depending on its water satura tion. The discovered nonlinear effects can be explained using the following phenomenological equation for the state of the medium: (1) σ(ε, ε) = E ε + αρ ⎢⎣1 + β ε ⎥⎦ ε, where σ, ε, and ε are the longitudinal dynamic stress, deformation, and deformation rate; E = ρ C 02 is the elasticity modulus; α is the linear dissipation coeffi cient; and β and m are the parameter and index of the m
degree of dissipative nonlinearity, m > ⎯1. Here, sum mand E ε determines the linear elasticity of the m medium and summands αρε and αρβ ε ε describe its linear viscous dissipation and dissipative nonlinear ity. The latter is responsible for the increase (at β > 0 ) or the decrease (at β < 0) of the wave’s absorption coefficient with an increase in its deformation ampli tude. In hydrodynamics, media possessing similar properties are called nonNewtonian (or Binghamian) [5]. Calculations show that in such a medium, the amplitude of an HF wave at distance L from the emit ter will be modulated according to the law ⎛ αβω2 L ⎡ Γ[(m + 1) 2]⎤ m ⎞ ε 0 П(t ) ⎟ , U (t ) = exp ⎜ − 1/2 3 ⎢ ⎥ ⎝ 2π C 0 ⎣Γ[(m + 2) 2]⎦ ⎠
where ε 0 and П(t ) are the deformation amplitude and the rectangular modulation function (or envelope) of LF pulses, ω = 2π f . From this expression it follows that at small amplitudes ε 0 of deformation of LF α β ω2L ⎡ Γ[(m + 1) 2]⎤ m ε 0 Ⰶ 1, the pulses, when 12 3 2π C0 ⎢⎣Γ[(m + 2) 2]⎥⎦ medium behaves almost linearly and the weak HF wave propagates without experiencing the effect of the strong wave. With increasing amplitude ε 0 , the nonlin earity of the medium becomes less noticeable, its dis sipation changes, and the amplitude of the HF wave also changes. It is this behavior of the weak HF wave with increasing amplitude of the LF pulse that was observed in experiment with watersaturated river sand, and equation of state (1) reflects precisely this property and behavior of the medium. For nonNew tonian media (water, glass, etc.), the dissipative non linearity parameter β = 0 and U (t ) = 1, therefore, simi lar modulation effects in them are not observed. ACOUSTICAL PHYSICS
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AMPLITUDE MODULATION OF SOUND BY SOUND IN WATERSATURATED
Thus, the results of the laboratory experiment on the amplitude modulation of sound by sound is evi dence that wet river sand possesses dissipative acoustic nonlinearity depending on the degree of water satura tion; its mechanism is related to the manifestation of nonlinear dynamic friction at the boundaries of con tacting sand grains. The presence of dissipative non linearity substantially broadens the “spectrum” of wave processes in similar media, which can be used for their seismoacoustic diagnostics.
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REFERENCES 1. L. A. Ostrovskii and O. V. Rudenko, Akust. Zh. 56, 724 (2009) [Acoust. Phys. 56, 715 (2009)]. 2. A. L. Bagmet, V. E. Nazarov, A. V. Nikolaev, A. P. Rezni chenko, and A. M. Polikarpov, Dokl. Akad. Nauk 346, 390 (1996). 3. M. A. Isakovich, General Acoustics (Nauka, Moscow, 1973) [in Russian]. 4. V. E. Nazarov, A. B. Kolpakov, and A. V. Radostin, Akust. Zh. 56, 82 (2010) [Acoust. Phys. 56, 77 (2010)].
ACKNOWLEDGMENTS The study was financially supported by the Russian Fund for Basic Research.
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5. W. L. Wilkinson, NonNewtonian Fluids. Fluid Mechan ics, Mixing and Heat Transfer (Pergamon, London, 1960; Mir, Moscow, 1964).
