Introduction

Acoustics is a branch of continuum mechanics. Therefore, equations of hydrodynamics and theory of elasticity are employed to describe acoustic oscillations and waves in gases, liquids, and solids. The total system of equations consists of the equation of motion (Newton's second law), continuity equation, thermal-transport equation, and dynamic equation of state [1-6]. The first three equations of the system are universal and are, essentially, identical for every media. Acoustic properties of the particular medium, indeed, are engraved on its equation of state and, in general, different media are described by different equations of state.

All of the equations of continuum mechanics are nonlinear. Therefore, no exact solution to the system exists. In this connection an approximate approach is employed to describe wave processes in acoustics and the total system is simplified by small-parameter expansion to derive the wave equation. For liquids and gases this parameter is three-dimensional compression, , , where and are the perturbed and steady-state density of the medium; in the case of homogeneous solids this one is strain (longitudinal and shear). In a description of acoustic waves it can be assumed that no heat exchange occurs between the rarefaction and compression parts of medium during half of a wave period; the absorbed energy of the wave changes the equilibrium state of the medium weakly and its movement is close to adiabatic; in addition, dissipative processes due to viscosity and heat conductivity are linear.

There are two equivalent approaches when describing the movement of continuum media [1, 2]. The first one, Eurelian, is employed in hydrodynamics; it describes the movement of medium particles by fixed space coordinates, (), and time, . In the second one, Lagrangian, the independent variables are initial coordinates, , of a particle in a certain fixed instant of time, ; with time the particle moves in space and running coordinates are the functions of the initial coordinates (and time ): , where are vector components of a displacement of the particle in regard to its initial position. (It is notable that both of the approaches were proposed by Euler). Lagrangian coordinates are more applicable to describe wave processes in solids (particularly in the case of one-dimension problems). In linear approximation Eulerian and Lagrangian approaches are identical. However, if nonlinearity is taken into account, the corresponding equations in Eulerian and Lagrangian coordinates become different. Therefore, a derived solution in moving Lagrangian coordinates should be transformed into that in fixed Eulerian coordinates.

I.1 Nonlinearity of Gases and Liquids

In gases and liquids, longitudinal acoustic waves (compression and rarefaction) propagate. In these waves, particles of medium make oscillations along a direction of wave propagation. The description of nonlinear acoustic waves in ideal gases and liquids is founded on the Taylor expansion of the adiabatic equation of state, , in terms of small three-dimensional compression, , where and are pressure and density, is entropy. In the quadratic approximation this can be written as:

where is pressure at and is adiabatic sound velocity.

The equation of state for gases has the Poisson form: , where is the adiabatic exponent, and are the capacities per unit mass of the gases at constant pressure and volume, respectively. The nonlinearity of ideal gas is related to its heating and cooling at adiabatically fast compression and expansion under the action of the acoustic wave. The sound velocity in the gas is determined as , where is absolute temperature, is the gas constant, is molecular weight, and . It is worth noting that since , the equation of state for gases is always nonlinear. For air () at temperature 20 °C and atmospheric pressure , the adiabatic exponent and the sound velocity are equal to and .

For liquids the analogous equation of state is used, so-called Tate's empirical formula, , where and are intrinsic pressure and exponent; these constants are weakly dependent on the temperature and can be measured by experiment. (For many liquids the pressure, , is about Pa and the value of in the range from 4 (as for liquid nitrogen) to 12 (as for mercury). For water the values of the constants are Pa, , m/s.) The expressions for and in the case of liquids are the same as for gases with and substituted instead of and . Nonlinear properties of gases and liquids can be characterized by the nondimensional parameter ; the form of this parameter is chosen in such a way to make easy the passage to the limit case of linear media, when , that is, corresponds to . Since , then liquids are "more nonlinear" than gases, . It also should be noted that nonlinearity of liquids is stipulated by the interaction of molecules.

I.2 Nonlinearity of Homogeneous Solids

Unlike gases and liquids, in solids there can be not only longitudinal but also shear elastic stresses for which . Therefore, in solids, shear (or transverse) waves, as well as longitudinal acoustic waves of compression and rarefaction, are possible. In these waves, the medium particles make oscillations in directions perpendicular to that of the propagation of a wave.

It is customary to describe propagation and interaction of acoustic waves in solids within the framework of the classical five-constant theory of elasticity [1, 3, 5-7]. This theory, being essentially mathematical, determines the nonlinear (in the quadratic approximation) equation of state (i.e., the dependence of the elastic stress tensor, , on the derivative, , of the components of the displacement vector, , with respect to Lagrangian coordinates, ) for ideal elastic isotropic media under adiabatic deformation:

where is the internal energy of solid, is a strain tensor, and , .

In cubic approximation, with respect to the internal energy, , is determined as a Taylor expansion in terms of the strain tensor invariants , , and :

In this expansion the solid is assumed to be in equilibrium state, hence , . Introducing in Equation I.3 the notations , , , , yields:

where and are the uniform compression and shear moduli, , , and are the Landau moduli; all of these are determined experimentally and their quantity-five-gave the name to the five-constant theory. Clearly, all of the elasticity moduli -, , , , and -correspond to their adiabatic values. Additionally, owing to the infinitesimal thermal expansion coefficient of solids, the adiabatic and isothermal values of the moduli , , , and differ insignificantly, while these values of the shear modulus, , are the same [1, 5].

Essentially, total lack of sound velocity dispersion (up to hypersound) is an inherent feature of homogeneous media, hence their linear () and nonlinear elasticity moduli are independent of the frequency of the acoustic wave. [It is worth mentioning that for a description of the elastic properties of anisotropic solids-monocrystals-many more independent constants are required; in the general case (in the quadratic approximation), the number is greater than two hundred. Nevertheless, accounting for symmetries reduces this value abruptly; for instance, in the case of cubic crystals it is necessary to introduce no more than three linear and eight nonlinear elasticity moduli [1, 3, 5]. Thus, in spite of differences in chemical composition and structure, all monocrystals are described by the same matrix equation of state. The number and values of independent coefficients in this equation are determined by symmetry of the crystal and by the potential interaction of neighboring atoms].

Often, other pairs of independent moduli are used for characterization of linear properties of isotropic solids: Lamé coefficients and , and Young's modulus, , and Poisson's ratio, . Young's modulus determines the relationship between longitudinal stress, , and strain, , in the rod (), whereas Poisson's ratio determines the relationship between strains of lateral contraction, , and axial tension, . From thermodynamic relationships it follows that , , and Poisson's ratio can vary from to ; for homogeneous media its value belongs to the range , therefore also . The extreme case corresponds to going from solid to liquid () and, in turn, the materials with are called water-like materials. The Murnaghan moduli (, , and ) are sometimes used instead the Landau moduli (, , and ) [8, 9]; they are simply related by the expressions , , and [1, 3, 5].

Substitution of Equation I.2 into Equation I.4 yields the equation of state for homogeneous perfectly elastic solids:

It can be seen from this equation that the dependence contains a geometric nonlinearity, which is related to the nonlinearity of strain tensor and a physical (or material) one (the terms with moduli , , and ), so that even in the case of , it remains nonlinear.

In spite of a certain "heaviness", Equation I.5 is a rather simple algebraic expression, determining the single valued relationship between and . For longitudinal stress, , and strains, , this equation has completely simple form, which can be received from the Taylor expansion of the continuously differentiable, that is, the analytical, function with respect to small strain ; assuming it can be written...