Experimental study on subharmonic and ultraharmonic acoustic waves in water-saturated sandy sediment

When a primary acoustic wave with a finite amplitude propagates in a medium, the nonlinear acoustic waves such as subharmonic, ultraharmonic, second harmonic, and higher harmonic acoustic waves can be generated not only due to nonlinearity of a medium but also due to finite amplitude of a primary acoustic field (Ekimov et al., 1996; Solodov et al., 2002; Korshak et al., 2002; Bazhenova et al., 2005). If two primary acoustic waves of different frequencies propagate in a medium, nonlinear acoustic waves at the sum and difference frequencies can be also generated in the medium (Sutin et al., 1998; Ostrovsky et al., 2003). The generation of the subharmonic $(f0∕2)$ and the ultraharmonic $(3f0∕2,5f0∕2,7f0∕2,…)$ acoustic waves is very interesting because they can be generated in the medium only when the acoustic pressure of the primary acoustic wave exceeds a certain threshold value.

The subharmonic and the ultraharmonic acoustic waves can provide important information with other nonlinear acoustic waves in order to detect cracks and defects in solid media (Ekimov et al., 1996; Solodov et al., 2002; Korshak et al., 2002). Ekimov et al. (1996) showed that these nonlinear acoustic waves could be applied to the diagnosis of ice cover in a natural fresh water lake. Recently, Solodov et al. (2002) and Korshak et al. (2002) showed the generation of both nonlinear acoustic waves due to cracks in laminated solid samples. The subharmonic and the ultraharmonic acoustic waves can also provide important information related to resonance which is used to detect bubbles in bubbly granular media. However, the investigation of both nonlinear acoustic waves in this granular media has not been performed because the primary acoustic wave is heavily attenuated. Furthermore, even in water-saturated granular media, the behavior of both nonlinear acoustic waves has not been investigated; this may provide background information for the study on the subharmonic and the ultraharmonic phenomena in bubbly granular media.

The objectives of this paper are, first, to experimentally investigate the generation of the subharmonic and the first ultraharmonic acoustic waves in water-saturated sandy sediment (a water-saturated granular medium) and, second, to experimentally investigate the acoustic characteristics such as the dependence on the driving acoustic pressure and on the receiving distance. Most likely mechanisms of the subharmonic and the ultraharmonic generation in water-saturated sandy sediment are also discussed here.

An anechoic water tank of volume $500×280×300mm3$ was prepared to pack water-saturated sandy sediment, which was filled with fresh water. The water temperature in the tank was maintained in the range $11°C$ and $15°C$ for all measurements. To avoid any inclusion of small bubbles within water-saturated sandy sediment, the sediment was slowly packed through a large sieve installed in the anechoic water tank. The porosity of water-saturated sandy sediment and the density of sand grains were $40.8±1.3%$ and $2559±52kg∕m3$⁠, respectively. The diameters of sand grains were between 250 and $500μm$⁠.

Figure 1 shows a schematic diagram of the experimental setup to measure the subharmonic and the first ultraharmonic acoustic waves in water-saturated sandy sediment. A transducer with a diameter of $80mm$ was used to transmit the signals and was buried in water-saturated sandy sediment. The driving frequency was $76kHz$⁠, which was the first major resonance frequency of the transducer. The signals transmitted through water-saturated sandy sediment were sinusoidal tone burst signals with a pulse duration of $1ms$ and repetition time of $100ms$⁠. The tone burst driving method was selected to get a continuous wave condition with a very high driving acoustic pressure. The driving acoustic pressure was defined as the acoustic pressure measured at a distance of $1mm$ from the transducer in water-saturated sandy sediment. It increased from 346 to $507kPa$ for the subharmonic and the first ultraharmonic acoustic waves in sediment, respectively.

An arbitrary waveform generator (Agilent 33250A) and a power amplifier (Amplifier Research AR 75A 250) were used to drive the transducer. The transmitted signals in water-saturated sandy sediment were received by a hydrophone (B&K 8103). The hydrophone had an omni-directional receiving beam pattern of receiving sensitivity $−211.3dB$ re $1V∕μPa$ within $±2dB$ between $1Hz$ and $150kHz$⁠. Movement of the hydrophone in sediment was controlled by a positioning system. To minimize any variation in the structural composition of the sediment for all measurements, the hydrophone was always moved towards, rather than away from, the transducer as shown in Fig. 1. The received signals were acquired using a $500MHz$ digital storage oscilloscope (LeCroy LT342) and stored on a computer for off-line analysis.

Figure 2 shows the frequency spectra of signals transmitted through pure water and water-saturated sandy sediment at maximum driving acoustic pressure and a distance of $100mm$ from the transducer in the anechoic tank. As shown in Fig. 2(b), the subharmonic $(38kHz)$ and the ultraharmonic (114, 190, and $266kHz$⁠) acoustic waves were generated due to the nonlinearity of the water-saturated sandy sediment at the primary frequency of $76kHz$⁠. The acoustic pressure level of the subharmonic acoustic wave in sediment was about $24dB$ higher over the background noise level, while the levels of the ultraharmonic acoustic waves were between 9 and $20dB$ higher.

Figure 3 shows the acoustic pressure variations of the primary, the subharmonic, and the first ultraharmonic acoustic waves as a function of the driving acoustic pressure at the primary frequency of $76kHz$⁠, at a distance of $100mm$ from the transducer in water-saturated sandy sediment. As shown in Fig. 3, the acoustic pressure of the primary acoustic wave was linearly increased as the driving acoustic pressure increased, while the acoustic pressure of the subharmonic acoustic wave was gradually and exponentially increased. In Fig. 3(b), the acoustic pressure of the first ultraharmonic acoustic wave was also gradually and exponentially increased as the driving acoustic pressure increased up to $484kPa$⁠. However, it approached a saturation value as the driving acoustic pressure became greater than $484kPa$⁠.

Figure 4 shows the acoustic pressure level variations of the primary, the subharmonic, and the first ultraharmonic acoustic waves as a function of receiving distance. The solid, the dashed, and the dot-dashed lines indicate exponential fitting lines for each data, respectively. The acoustic pressure levels for the subharmonic and the first ultraharmonic acoustic waves more rapidly decreased than that for the primary acoustic wave as the receiving distance increased.

To confirm the subharmonic and the ultraharmonic phenomena at another primary frequency in water-saturated sandy sediment, another transducer (RESON TC2122, $180mm$ in diameter) with a resonance frequency of $33kHz$ was used as the driving transducer and was driven at a maximum acoustic pressure of $246kPa$⁠. Figure 5 shows the frequency spectra of the signals transmitted through pure water and water-saturated sandy sediment at a distance of $100mm$⁠. As shown in Fig. 5(b), the subharmonic $(16.5kHz)$ and the ultraharmonic (49.5, 82.5, and $115.5kHz$⁠) acoustic waves were generated due to the nonlinearity of the water-saturated sandy sediment. The acoustic pressure levels of the subharmonic and the ultraharmonic acoustic waves in sediment were about $17dB$ higher than the background noise level.

The driving acoustic pressure of $346kPa$ in Fig. 3(b) is the minimum acoustic pressure to generate the subharmonic and the first ultraharmonic acoustic waves in water-saturated sandy sediment. Therefore, the driving acoustic pressure of $346kPa$ can be considered as the threshold acoustic pressure to generate the nonlinear acoustic waves when the primary acoustic wave propagates in water-saturated sandy sediment at a frequency of $76kHz$⁠. Generally, it is well known that the acoustic pressures of the subharmonic and the first ultraharmonic acoustic waves saturate as the driving acoustic pressure increases (Lostberg et al., 1996; Shankar et al., 1999). In this study, the saturation phase of the acoustic pressure was only observed for the first ultraharmonic acoustic wave as shown in Fig. 3(b). For the subharmonic acoustic wave in Fig. 3(b), the saturation phase of the acoustic pressure might be observed at a driving acoustic pressure greater than $507kPa$⁠. However, this could not be confirmed because of the limitation of the transmitting response of the transducer.

In Fig. 4, the subharmonic and the first ultraharmonic acoustic waves exhibited more fluctuations in pressure level than the primary acoustic wave as a function of receiving distance. Since the source of the primary acoustic wave in water-saturated sandy sediment was obviously the transducer, the primary acoustic wave coherently radiates from the vibrating plane of the transducer to the water-saturated sandy sediment. Therefore, the acoustic pressure level fluctuation of primary acoustic wave was hardly observed in sediment as shown in Fig. 4(a). However, since the nonlinear sources of the subharmonic and the first ultraharmonic acoustic waves could be randomly distributed in the nonlinear acoustic interaction zone of the water-saturated sandy sediment, the nonlinear acoustic waves might exhibit both coherent and incoherent properties. In this case, the acoustic pressure levels of both nonlinear acoustic waves might fluctuate as shown in Fig. 4(b). Since high driving acoustic pressure amplitude at the transducer was required to generate the subharmonic and the first ultraharmonic acoustic waves in water-saturated sandy sediment, the rapid decreases of the acoustic pressure levels of both nonlinear acoustic waves in Fig. 4(b) might indicate that the nonlinear sources for the generation of the subharmonic and the ultraharmonic acoustic waves were mainly distributed in the vicinity of the transducer.

In water-saturated granular media, structural inhomogeneities, such as defects and discontinuities, are considered to be parametric resonance sources which generate the nonlinear acoustic waves. Specific investigation for structural inhomogeneities in granular media has been performed by Ostrovsky et al. (2000). They considered the granular media as media with soft and hard phases. The soft phases occupy small volumes in the media and they produce strong deformations for acoustic waves with finite amplitude, whereas the hard phases produce significantly less deformations. Since these strong deformations in granular media give rise to strong nonlinear acoustic responses, the deformations may be considered to be structural inhomogeneities. The contacts among individual grains in granular media are much softer than the grains. Therefore, the concentration of stress due to external pressure at the contact boundaries of individual grains may cause strong nonlinearity as a result of deformations in the media. Deformations at the contact boundaries of individual grains can ultimately be considered as deformations of pores in granular media, because the contact boundary surfaces between individual grains are parts of the pore surface in the media. Since the transducer in Fig. 1 was placed in water-saturated sandy sediment, the contacts on the boundary between the transducer and the sediment might be also considered as the nonlinear sources for the generation of the subharmonic and the ultraharmonic acoustic waves. However, it could not explain the subharmonic and the ultraharmonic phenomena observed through water-saturated sandy sediment slab separated from the transducer in water. Therefore, the subharmonic and the ultraharmonic phenomena in this study might be caused by various mechanisms based on the contacts between individual sediment grains and the contacts on the boundary between the transducer and the sediment.

The subharmonic and the ultraharmonic phenomena were first observed in water by Korpel and Adler (1965). Generally, the observations of these nonlinear phenomena are difficult in water because the nonlinear parameter of water is very small, around 3.5 (Beyer, 1998). However, they showed that the nonlinear phenomena could be observed in water if a standing wave pattern was made in the primary acoustic wave field. The theoretical approaches for this system have been performed by Adler and Breazeale (1970) and Hughes (1977). The subharmonic and the ultraharmonic phenomena can also be observed in parametric resonance systems, such as bubbly water and solid media with cracks, defects, and discontinuities (Ekimov et al., 1996; Lostberg et al., 1996; Shankar et al., 1999; Solodov et al., 2002; Korshak et al., 2002). The Rayleigh-Plesset type equation (Lostberg et al., 1996; Shankar et al., 1999) for the motion of the bubbles is widely known as the nonlinear oscillation equation which predicts the subharmonic and the ultraharmonic phenomena in bubbly water. In solid media, such theoretical model equations are not yet well known, except nonlinear oscillation equations, such as Duffing (Fyrillas and Szeri, 1998) and nonlinear Mathieu (Boston, 1971) equations. They can be used for a phenomenological understanding of the subharmonic and the ultraharmonic phenomena in solid media. They have been practically used for understanding nonlinear phenomena in physical oscillation systems (Hayashi et al., 1960; Fyrillas and Szeri, 1998).

Nonlinear acoustic waves were observed in water-saturated sandy sediment at the subharmonic and the ultraharmonic frequencies; they resulted from nonlinearity of the sediment. Such nonlinearity might be generated from the contacts between individual sediment grains and the contacts on the boundary between the transducer and the sediment. These nonlinear acoustic waves strongly depended on driving acoustic pressure. The experimental results in this study show that the subharmonic and the ultraharmonic phenomena in water-saturated sandy sediment can be significantly observed when the driving acoustic pressure is greater than a threshold value. These phenomena show close resemblance to those produced in bubbly water.

The authors would like to thank Dr. Alexander M. Sutin at ARTANN Laboratories, Dr. Igor N. Didenkulov at Institute of Applied Physics in Russia, and anonymous reviewers for their valuable comments. This work was supported by the Agency for Defense Development, Republic of Korea.