Handbook of Contemporary Acoustics and Its Applications

The following sections are included:

• Introduction
• Preface
• About the Author
• Contents

A fluid that does not involve any irreversible process due to viscosity, thermal conduction, relaxation, etc., is considered an ideal or inviscid fluid. Unlike a viscous fluid, the force acting on any surface of an ideal fluid with any curvature is always normal to the surface, independent of the motion of the fluid. In most cases, when the frequency of the acoustic wave is not too high (or its corresponding wave length is much greater than the atomic scale) and the area of interest is not too close to a solid boundary, most fluids such as water and air can be approximately considered ideal fluids. Fluids such as air and water have mass and elasticity, the former represents the inertial and the latter makes fluid oppose to be compressed and tend to return to its equilibrium state.

Sources of the acoustic wave are also called transducers, which can convert the other forms of energy (vibration energy, electromagnetic energy, thermal energy, etc.) to acoustic energy. They can be divided into the volume and surface sources from their geometry or the characteristic of the acoustic field of their radiation. The surface source usually is the vibrating surface of a piezoelectric solid. The acoustic field generated by the source may be treated as a boundary condition problem mathematically. When an acoustic wave propagating in a region with a sound speed encounters another region of a medium with a different sound speed whose dimensions are relatively smaller than the wavelength of the sound wave, the trajectory of the incident wave departs its original direction, this phenomenon is called the scattering. The resulting total wave due to the scattering subtracting the incident wave is called the scattered wave.

In the previous chapters, we discussed the excitation and propagation in an inviscid ideal fluid which has no energy loss of any kind associated with the propagation. In practice, the fluid in which the acoustic wave propagates is not ideal; we have to consider the irreversible processes such as viscous, thermal conduction and relaxation, etc. accompanying with the propagation of the sound. During the propagation of acoustic waves in such a viscid medium, the “ordered” acoustic energy is converted to “disordered” thermal energy when the acoustic energy is absorbed by the medium, consequently its amplitude decays with increasing travel distance. In addition to the aforementioned irreversible acoustic energy absorption, the acoustic wave amplitude also attenuates due to the spreading of acoustic wavefront (spherical waves) or scattering from small particles (floating dusts, air bubbles in water, water-droplets of fog, etc.), the wave propagation direction consequently is partially altered. Thus, in the original direction, the acoustic wave energy is attenuated.

Due to the reflection of the walls, the acoustic field in a finite-size cavity may form a standing wave; it is also called an eigen-mode of an acoustic cavity. If the shape of the cavity is irregular, the acoustic field is quite complex, we must seek an approximation method to study this acoustic field. The choice of the approximation method to be adopted is determined by the dimensions of the cavity with respect to the wavelength. When (low frequency), where V is the volume of the cavity; λ is the wavelength, this case can be considered as a uniform acoustic field in the cavity, in other words, the cavity may be considered as an oscillator vibrating in phase; when (high frequency), we may use the geometric acoustics to study the acoustic field; when (mid-frequency), the eigen-mode method should be used, i.e. several eigen-modes could co-exist in the cavity by appropriate excitations.

In the previous chapters, we assumed the following: (1) when an acoustic wave propagates in a medium, the medium particles vibrate with velocity amplitudes much smaller than the propagation velocity; (2) the vibration displacement amplitude of a particle is much smaller than the wavelength of the acoustic wave; (3) the density variation amplitude is much smaller than the equilibrium density. Based on these conditions, we were able to linearize the nonlinear fluid dynamic equation to obtain a linear wave equation. The theory of acoustics based on the aforementioned assumptions is called the linear acoustics. However, the fluid dynamics itself, on which the acoustics is based on, is nonlinear. If the above assumptions are no longer valid in some cases, we must keep the nonlinear terms in the fluid dynamic equations, this is the origin of the nonlinear acoustics. The wave equation which is developed to describe the nonlinear acoustic wave propagation is called the nonlinear wave equation. The most evident difference between the linear and nonlinear acoustic wave equations is the superposition principle which works so well in the linear acoustics is no longer valid in nonlinear acoustics. For example, the frequency spectrum analysis, Fourier analysis and Green's theorem which are widely used in linear acoustics, are no longer applicable in nonlinear acoustics.

The main physical effects of a finite amplitude acoustic wave include: (1) Acoustic radiation force/pressure: When an acoustic beam is incident on the surface of an object, this object experiences a radiation force/pressure which is proportional to square of the acoustic pressure. Generally speaking, for a moderate acoustic intensity, the radiation force is smaller than the weight of an object. When the intensity is high enough, the acoustic radiation force can be greater than the weight of the object, thus, the object can be in a floating state. This technique is called the acoustic levitation. Application of acoustic radiation force in vivo can distort the tissue shape inside a human body generating low frequency shear wave, associate with imagine technique, the elastic shear wave moduli of tissue can be measured, this technique has been developed as a new imaging modality — ultrasound elastic shear wave imaging (Walker et al., 2000; Greenleaf et al., 2003); (2) acoustic streaming: In a finite amplitude acoustic field of a fluid, each “fluid element” has a time-independent (DC) component of velocity in addition to a sinusoidal (to and fro) oscillatory motion; this DC velocity is called acoustic streaming. Generally speaking, acoustic streaming can be divided into two categories according to the dimension of the fluid system (h) in relation to the wavelength (λ) of the acoustic wave. When h ≫ λ, it is called Eckart streaming while δ ≪ h ≪ λ, it belongs to Rayleigh streaming, where δ is called the viscous boundary layer thickness , μ is shear viscosity, ρ is the density of the fluid) and f is frequency of the acoustic field. Eckart streaming can take place either in a traveling wave or a standing wave; the streaming velocity in a standing wave is usually weaker since the opposite steady stress in a standing wave may partially reduce the streaming velocity. (3) Acoustic cavitation: “Acoustic cavitation” refers to activities associated with gas bubbles, pockets and cavities under excitations of acoustic waves. There are two types of acoustic cavitation, “inertial” and “non-inertial”. Inertial cavitation, formerly called “transient” cavitation, occurs if the acoustic pressure amplitude is sufficiently high and above a threshold level. Under this condition the bubbles will first grow in volume, and then implode violently. The accompanying high temperature and pressure and formation of free radicles in situ is the foundation of “ultrasound chemistry” and sonoluminescence. Noninertial cavitation, formerly called “stable” cavitation, occurs when bubbles in a liquid is forced to oscillate with a relatively small to moderate amplitude (bubble pulsation), when the external acoustic pressure amplitude is not too high. (4) Thermal effects: Since an ultrasound beam can be focused to a small tissue volume in the order of the cubic of wavelength inside a human body, it is possible to use an extracorporeal (“outside-the-body”) source of ultrasound for therapeutic purpose (Wu, 2006). If sufficient acoustic energy is concentrated within the focal volume and absorbed by the soft tissue, the temperature in that region may be raised to a level (≥65°C) at which the irreversible change of tumor-tissue takes place, resulting in coagulation necrosis. This type of medical modality is now called “HIFU” (high intensity focused ultrasound). Since HIFU would not damage overlying and surrounding vital structures, it is a truly noninvasive modality.

In this chapter, we will concentrate on the elastic wave such as Lamb waves and Rayleigh surface waves (SAWs) propagating in a solid plate. We will not discuss the basic strain and stress relationship, readers may refer to other references such as “Acoustic fields and waves in solids by Auld (1990). When a plate of finite thickness bordered with an otherwise spatially infinite and homogeneous fluids at its both sides part of energy of Lamb waves in the plate is coupled into the liquid; this type of disturbances in the plate is called leaky Lamb waves. The dispersion equations of leaky Lamb waves for an isotropic plate were first derived by Schoch (1952). A Leaky Lamb wave in a plate has been extensively used in various applications. We will discuss its applications in bio-sensing, particle sorting (acoustic sieve) and vibrational energy harvesting. We will also discuss the current applications of Rayleigh surface waves, such as interactions of Rayleigh surface waves with graphene film and their sensing applications.

In earlier chapters, we have dealt with the acoustic wave propagation in a continuous medium based on the classical continuum mechanics (CCM), which considers materials to be continuous. However, some bio- and nano-materials are not genuinely continuous, there are scaling factors that can be used to relate to their discrete property. The relatively newly developed Doublet Mechanics (DM) (also called Nanomechanics) is a new elastic theory dealing with materials that possess discrete microstructures (Ferrari et al., 1997). In DM, the fundamental microscopic unit of a material can be represented by a bundle of spheres or nodes separated by a finite distance (called inter-nodal distance) (Ferrari et al., 1997). Nodes in DM could be cells in biological cases or molecules of a solid. Nodes are either in contact with each other or separated by adhesive layers. In DM, the mechanical strain developed in a bundle due to its distortion is a function of the inter-nodal distance. Depending on the order and the discreteness of the system, it can be expressed as a multi-scale Mth-order Taylor expansion of the inter-nodal distances at its equilibrium configuration. It should be noted that the order (M) of the Taylor expansion should not be related to the degree of accuracy of this presentation in a common mathematic sense. In fact, the order of M represents the degree of the discreteness of a system; the higher the Mth-order a Taylor expansion includes, the more discrete nature of the system DM represents (Ferrari et al. 1997). When M = 1, i.e. only the first-order of the Taylor expansion is used; DM mechanics reduces to the COM case, i.e. the system described by DM becomes a system that does not have a discrete nature at all. In this case, the results obtained from DM completely agree with those predicted by CCM. When M = ∞, on the other hand, it describes a solid of single crystal which has a regular and periodic lattice structure. When ∞ > M > 1, discreteness of systems represented by DM are between continuous (nongranular) and periodically discrete. A recent success of DM in an ultrasonic application to characterization of elastic properties of tissue at relatively low frequency (<10MHz) shows that DM is quite versatile and powerful in description of wave propagation in a partially discrete and granular materials such as biological soft tissue (Liu & Ferrari, 2002).

The following sections is included:

• References

The following sections are included:

• Appendix A: Acoustic Relevant Parameters
• Appendix B: Mathematic Background Review
• Appendix C: Strain–Stress Relation in Cylindrical and Spherical Coordinates
• Appendix D: Operational Formulas of Tensors
• Appendix E: Some Integral Theorems
• Appendix F: Some Useful Thermodynamics Formulas

The following sections is included:

• Index