A Scalar Acoustic Equation for Gases, Liquids, and Solids, Including Viscoelastic Media

The work deals with a mathematical model for real-time acoustic monitoring of material parameters of media in multi-state viscoelastic engineering systems continuously operating in irregular external environments (e.g., wind turbines in cold climate areas, aircrafts, etc.). This monitoring is a high-reliability time-critical task. The work consistently derives a scalar wave PDE of the Stokes type for the non-equilibrium part (NEP) of the average normal stress in a medium. The explicit expression for the NEP of the corresponding pressure and the solution-adequateness condition are also obtained. The derived Stokes-type wave equation includes the stress relaxation time and is applicable to gases, liquids, and solids.


Introduction
One of the applications of the acoustic-sensing technology is monitoring of material parameters of engineering systems continuously operating in irregular external environments. This type of the operation indicates that the monitoring must be regular (e.g., periodic) and in the real-time mode. Many problems in this area deal with the systems that are multi-state and viscoelastic in the following sense. A system comprises at least two spatial domains, each of which is occupied with an isotropic medium that is spatially homogeneous at equilibrium and for all of the aforementioned entries in solids ( [8], the equation in the article "0138"). This feature allows [8] better focusing the models on sharp practical applications.
The work is arranged as follows. Section 2 summarizes the basic facts on the Cauchy stress matrix and the key component of it, scalar and deviatoric stresses. The main result derived in the work is presented in Section 3 that also discusses the novelty of it and its connection to the related results of other authors. Section 4 concludes the work. The detailed derivation of the main result is carried out in Appendix A. It applies selected representations associated with the coupling of Eulerian and Lagrangian coordinates outlined in Appendix B.

Preliminaries: Scalar and Deviatoric Stresses
Acoustic signals present the spatiotemporal deviation, S ∆ , of the non-equilibrium part (NEP) of the Cauchy stress matrix, S , of the medium from its equilibrium version S , The terms denoted with the sign "overline" are specified in the remark below. Remark 2.1. As is well known, physical quantities at equilibrium are independent of time. The present work considers the media only such that, at equilibrium, they are independent of space as well. Consequently, the equilibrium versions of physical quantities do not depend on space either. These versions are denoted with the sign "overline" applied to the notation of the corresponding quantity (e.g., see (2.1)).
 One usually represents matrix S in the form of two components, I is the identity 3 3 × -matrix, and ( ) tr ⋅ is the trace of a matrix. As (2.3) shows, P is a scalar variable and, thus, matrix PI in (2.2) presents the scalar stress.
The diagonal and off-diagonal entries of matrix S are known as the scalar normal and shear stresses, respectively. Since (2.3) determines P as the arithmetic mean of the total normal stresses, P is called the average normal stress (ANS).
As follows from (2.2) and (2.3), matrix Z is traceless, i.e. such that ( ) is called the deviatoric stress. Also, this stress is zero at equilibrium, i.e. 0 Z = . (2.5) The relaxation of deviatoric stress Z to its equilibrium value (2.5) is usually described in terms of the stress relaxation time, say, θ, and according to asymptotic representation d d Z t Z θ = − . The stress relaxation exists in any material medium, in gases, liquids, and solids, no matter if the medium is spatially non-homogeneous or spatially homogeneous (e.g., see both [3], (28.19) and [3], (28.22) for the case of fluids).
In an isotropic medium, deviatoric stress Z explicitly depends on shear modulus G of the medium, no matter if the latter is a solid (e.g., [5], (1.43)) or a fluid (e.g., [9], [10], p. 655). In the case of a fluid, this dependence is presented implicitly, by means of the explicit dependence on shear viscosity of the medium G µ θ = . For the sake of simplicity, we also use θ in the expression K η θ = for volume viscosity η where K is the bulk modulus of the medium. Deviatoric stress Z does not depend on bulk modulus K . The equilibrium versions of the mentioned equality is Value K and the equilibrium value ρ of the mass density of the medium ρ determine parameter This parameter is sometimes called the speed of bulk waves. We note that ANS P completely determines not only scalar stress PI but also the entire stress S at equilibrium with expression respectively.  Some of the above relations are used in the derivation of the main result of the present work (see Appendix A).

The Stokes-Type Wave PDE for the Non-Equilibrium Part of the Average Normal Stress
As is shown in Appendix A, under the assumptions listed in Table 1, a closed description for the NEP P ∆ of ANS P is PDE The derivation of (3.1) also provides the corresponding description for the NEP p ∆ of pressure p (see (A.1.11)), which is any of the following two relations Equation ( In comparison with common wave PDE 2 includes an extra term, the one with stress relaxation time θ . This is a damping term. It, however, does not result from the choice of one or another damping model. On the contrary, it is consistently derived from the basic laws for continuum media (see Appendix A). The damping term represents the internal, viscous friction in the medium.  Table 1. Assumptions used in the derivation of acoustic equation system (3.2), (A.3.6). 1 The medium is spatially homogeneous and isotropic at equilibrium.

2
Elastic properties of the medium can be treated in terms of linear elasticity.

3
The medium is close to the equilibrium state sufficiently in order to neglect deviatoric component (2.4) of the total stress (2.2) (see also (2.5)).

4
The medium is assumed to be isothermal. As is well known (e.g., [16], p. 617), acoustical vibrations are almost always so rapid that there is no time for conduction to remove the heat developed and equalize the temperatures. The contractions and expansions take place adiabatically, i.e. without loss of heat. In spite of that, the above assumption on the isothermalness is used. The reason is avoiding the need in description of the spatiotemporal evolution of the temperature in the medium and, thereby, keeping the complexity of the model at a reasonable level. This in particular means that the aforementioned heat is neglected.

5
There are no chemical reactions in the medium. 6 There are no body forces in the medium. 8 The medium is close to the equilibrium state sufficiently in order to replace η in (A.3.2) with its equilibrium value η .

9
The medium is close to the equilibrium state sufficiently in order to neglect velocity v in expression (A.3) thereby reducing it to (B.5).
in theory of inviscid solids (see Remarks A.1.1 and A.2.1). Thus, the derived model is suitable for gases, liquids, and solids.  We also note a connection of PDE (3.1) to a special wave equation that was introduced in 1845. Formally, PDE (3.1) for NEP P ∆ of ANS P is identical to the Stokes wave PDE [7] for the velocity potential in a viscous fluid. Thus, (3.1) is one of the Stokes-type wave PDEs.
The Stokes-type wave PDEs for different variables are used in acoustic of viscoelastic solids since long ago. For example, Section 1 discusses the well-known Stokes-type wave PDE system for the displacement vector in a solid. Paper [11] analyzes propagation of plane and spherical waves in viscoelastic solids with the help of the normalized Stokes-type wave PDE, which is mathematically equivalent to (3.1). Book [12] discusses transient waves in gases, liquids, and solids in connection with applications to viscoelastic-solid acoustics in seismology. The entire modeling in this book is based on the Stokes wave PDE but treats its variable in a broader sense, as a generating function (e.g., [12], (31) on p. 36). The in-depth discussion in ( [12], the chapter "Epilogue") emphasizes a number of the advantages of the Stokes-type models.

Concluding Remarks
The present work considers material media, which are isothermal, spatially homogeneous and isotropic at equilibrium, with elastic properties treatable in terms of linear elasticity, and can be gaseous, liquid, or solid. The chemical reactions and body forces in the media are neglected.
Under the assumptions listed in Table 1, the work consistently derives a scalar wave PDE (3.1) for the NEP of the ANS in the medium. Normal stress in any medium turns up in almost all situations, dynamic or not. Equation

Appendix A. Derivation of a Scalar PDE for the Non-Equilibrium Part of the Average Normal Stress
The purpose of this section is derivation of a description for the stress NEP S ∆ (see (2.1) or (2.9)), which would include stress relaxation time θ . The derivation follows the line formulated in Section 1 and, therefore, is implemented in terms of viscous-fluid mechanics.
There are two approaches in continuum mechanics to modeling the space-time phenomena, Lagrangian and Eulerian (e.g., [13], Sections 2.1 and 2.2). They are formally different but equivalent in the sense of mechanics. Eulerian coordinates are t and spatial vector x. Lagrangian coordinates are t and spatial vector y discussed in Appendix B.
Models for linear inviscid solids are based on Lagrangian approach (e.g., [14], Section 1.8 on p. 142-143). They include both equilibrium elastic moduli K and G but do not include stress relaxation time θ .
Models for viscous fluids are based on Eulerian approach. They include volume and shear viscosities η and µ , and, thus (see Section 2), not only both elastic moduli K and G but also stress relaxation time θ . Therefore, the derivation applies Eulerian approach.
According to Eulerian approach, a spatial point moving along a determinate trajectory is described with ODE d d x ∈  is the vector of the point position, and the vector of the point velocity 3 v R ∈ depends not only on t but also on x , The total time derivative of a scalar variable, which depends on time and space, is, in view of (A.1), expressed as follows where column vector is the gradient with respect to the entries of vector x . Consequntly, T ∇ is the corresponding divergence. Note the last term on the right-hand side of (A.3) is because of the x -dependence in (A.2).
In view of Assumption 3 in Table 1, relation (2.9) is reduced to (2.11) that switches attention from S ∆ to P ∆ . The equation for the latter is derived below.
In view of Assumption 4 in Table 1, we confine ourselves with the equations for the laws of conservation of mass and momentum in the medium. They are formulated in terms of ρ and the (volumetric) density v ρ of the momentum vector. Under Assumptions 2-6 in Table 1, the equations are of the following form (e.g., [13], We consider the quasi-equilibrium versions of these equations. These are the topics of Appendixes A.1 and A.2, respectively. In each of the two cases, the related inviscid/elastic-solid representations are indicated.

A.1. Quasi-Equilibrium Version of the Mass Conservation Law
The present section derives the quasi-equilibrium version of the mass conservation law (A.5). Quantity  We also note that Equation (A.1.14) can be rewritten as the expression for T v ∇ , This equation is the quasi-equilibrium version of Equation (A.5), which is used below.

A.2. Quasi-Equilibrium Version of the Momentum Conservation Law
The present section derives the quasi-equilibrium version of the momentum conservation law (A.6). Owing to (A.5), equation (A.6) can be rewritten as d d v t P ρ = −∇ or, equivalently, as where P ∆ is determined with (2.10). Under Assumption 7 in Table 1, the momentum conservation law (A.6) is equivalent to its quasi-equilibrium version d d . 2) is equivalent to the well-known equation of inviscid solid mechanics (e.g., [5], (1.15)) that, in the scalar-stress case (2.11), is of the following form 2) includes terms p ∆ and P ∆ , respectively. These terms are mutually coupled because of the well-known Stokes relation introduced 170 years ago (e.g., [14], (12) Proof. The proof is based on inequality , which follows from the hypothesis of the proposition, and the representation for the solution of ODE (3.2) with initial value ( )

Appendix B. Auxiliary Summary on the Interrelation of Eulerian and Lagrangian Coordinates
As noted in Appendix A, the position of a spatial point in Eulerian coordinates is described with ODE (see (A.1