## Abstract

Acceleration and temperature rate waves in lossless Green and Naghdi gases are investigated. The exact equations of motion are also derived and then simplified under the finite-amplitude approximation. Bounds are established for the theory-specific coupling parameter, as well as several other quantities, and results are compared/contrasted with those for classical perfect gases.

## 1. Introduction

An acoustic acceleration wave is defined as a propagating singular surface, i.e. wavefront, across which at least one of the first derivatives of the pressure, density or the velocity component normal to the surface suffers a jump discontinuity (or jump for short) (see Chen 1973; McCarthy 1975). Thomas (1957) investigated the growth and decay of ‘sonic discontinuities’, i.e. acceleration waves, in ideal gases (see also Elcrat 1977). Coleman & Gurtin (1967) conjectured that blow-up of the acceleration wave's amplitude implied the formation of a shock wave. They noted, however, that a rigorous mathematical proof of their supposition was lacking. Lindsay & Straughan (1978) examined the evolutionary behaviour of acceleration waves of arbitrary shape in a perfect fluid. Subsequently, Greenspan & Nadim (1993) examined the blow-up of spherical acceleration waves in the context of sonoluminescence. Lin & Szeri (2001) studied acceleration wave blow-up in the presence of entropy gradients. Jordan & Christov (2005), who considered acoustic acceleration waves under the Lighthill–Westervelt equation, have provided numerical support for the ‘shock-conjecture’ of Coleman & Gurtin (1967). More recently, Jordan (2005) and Ciarletta & Straughan (in press) have presented studies of acceleration waves in the context of nonlinear poroacoustic propagation. It should also be noted that the study of acceleration waves has been, and remains, a topic of considerable interest not only in acoustics, but in the general field of continuum mechanics as well (e.g. Ames 1970; Chen 1973; McCarthy 1975; Straughan 1986; Müller & Ruggeri 1993; Saccomandi 1994; Ostoja-Starzewski & Trebicki 1999; Quintanilla & Straughan 2004; Jordan & Puri 2005 and references therein).

Green & Naghdi (1995) proposed a theory for a new type of thermoviscous fluids, which we will refer to here as Green & Naghdi (GN) fluids, whose thermal properties are more general than those of ordinary fluids. In their formulation, dissipationless flows with undamped second sound heat transport are possible. These novel features follow, in part, from the scalar quantity , which GN have termed the *thermal displacement*, that can be defined by , where is the absolute temperature and a superposed dot denotes the material derivative. In particular, their formulation may offer a single phase theory of helium II,1 which would be a major improvement over the current ‘two fluid’ model developed in the 1940s.

Although GN published their theory over a decade ago, there appear to be very few works in the continuum mechanics literature devoted to analysing and/or uncovering its predictions. In fact, we are not aware of any works that address the topic of acoustic propagation in GN gases. Hence, the primary aims of this communication are as follows: (i) derive the exact, nonlinear equations of motion governing acoustic propagation in a lossless GN gas, which we assume behaves as a perfect gas2; (ii) carry out an analytical study of acoustic acceleration waves in the context of the fully nonlinear GN theory and (iii) compare/contrast our findings with these for a classical perfect gas.

## 2. Balance laws, equation of state and basic assumptions

Consider a lossless, compressible GN fluid that is initially in its equilibrium state. As given by GN (1995), the equations of continuity, momentum and energy for such a fluid under homentropic flow are(2.1)(2.2)(2.3)Here, is the mass density; is the thermodynamic pressure; , which we term the GN parameter, is a theory-specific constant which carries (SI) units of ; , where ; the homentropic assumption means that the specific entropy *η* is everywhere equal to its (constant) equilibrium value , thus implying , for all (Thompson 1972; Ockendon & Ockendon 2004); and we recall that a superposed dot denotes the material derivative. What's more, all body forces and thermal sources have been neglected and by equilibrium state we mean the unperturbed, quiescent state in which, along with , , , and , where and are positive constants and is a vector whose components are constants.

In addition to equations (2.1)–(2.3), an equation of state must also be specified. For simplicity, let us now also assume that our GN fluid obeys the (homentropic) equation of state for a perfect gas, namely,(2.4)where is the adiabatic index and the constants denote the specific heats at constant pressure and volume, respectively.

Since the flow is homentropic, it follows that , where *h* is the specific enthalpy. Making this substitution and then taking the curl of the momentum equation, it is not difficult to establish that , for all , since the flow was initially without vorticity (e.g. Thompson 1972; GN 2005). Consequently, , where *ϕ* is the scalar velocity (or acoustic) potential, and we find, after collecting all terms on the LHS, that equation (2.2) reduces to(2.5)Next, we use , which of course is equivalent to *h*=*c*_{p}*θ*, along with , the perfect gas law, to obtain the following thermodynamic relations (e.g. Naugolnykh & Ostrovsky 1998):(2.6)where denotes the sound speed in the perturbed gas. Using these relations, equations (2.1) and (2.5) can be re-expressed as(2.7)

(2.8)

Clearly, equation (2.8) can be integrated directly. It is also clear that the expression inside the is, at most, equal to some function of time, which we can take to be a constant with no loss in generality. Calling this constant and observing that the equality just described must also hold in the equilibrium state, we find that , where denotes the sound speed in the undisturbed gas and . Consequently, we can solve for to obtain(2.9)Using equation (2.9), can be eliminated from equation (2.7) to yield, after some simplification, the following partial differential equation (PDE):(2.10)Here, we observe that letting reduces equation (2.10) to the corresponding (exact) equation for a classical perfect gas (see Bisplinghoff *et al*. 1955; Chester 1991; Hamilton & Morfey 1997). Employing the relations given in equation (2.6) once more, we find that equation (2.3) can be recast as , which through the use of equation (2.9) can also be expressed as(2.11)This coupled pair of PDEs, equations (2.10) and (2.11), exactly describes the lossless propagation of acoustic waves in perfect GN gases. The dependent variables are *ϕ* and *α*.

In the one-dimensional case, assuming propagation along the *x*-axis, equations (2.10) and (2.11), respectively, reduce to(2.12)(2.13)where , and the constant denotes the value of in the equilibrium state. It is clear that even in the one-dimensional case, these two PDEs appear extremely difficult, if not impossible, to extract exact solutions from. We have presented them here to help highlight the differences between the theory presented by GN (1995) and that of classical perfect gases in the context of nonlinear acoustics. (See Ockendon & Ockendon (2001) for a study of the classical, i.e. , case of equation (2.12).) Equations (2.10) and (2.11) are also intended to serve as a possible starting-off point for future research into acoustic phenomena in GN gases.

## 3. Acceleration wave analysis

### (a) Hadamard's lemma and jump amplitude equation

While we could study the propagation of an acceleration wave of arbitrary (smooth) shape as a two-dimensional surface in three-dimensional space, the technical details of the differential geometry involved are likely to obscure the essential physics we seek to uncover. Hence, let us now restrict our attention to the one-dimensional case and consider propagation along the *x*-axis. To this end, we return to equations (2.1)–(2.3) and recast them as the scalar PDEs(3.1)(3.2)(3.3)where we note that follows from equation (2.4). Observing that equation (3.3) can be used to eliminate from (3.2), we find that the latter reduces to(3.4)

Imagine a smooth planar surface propagating along the *x*-axis of a Cartesian coordinate system in a region filled with a GN perfect gas. Let the speed of with respect to the gas immediately ahead of it be . Suppose that for but that at least one of their first derivatives, say , suffers a jump on crossing ; i.e. the surface is an acceleration wave. Mathematically, this means that , but that , where the amplitude of the jump in the function across is defined here as(3.5)Additionally, are assumed to exist and a ‘+’ superscript corresponds to , the region into which is advancing, while a ‘−’ superscript corresponds to , the region behind . Let us further suppose that , but that , and are all non-zero. Also, let and on each side of , i.e. in both , for all . Hence, given the value of at time *t*=0, we set ourselves the task of determining the behaviour of for all *t*>0.

The first step in the process is employing Hadamard's lemma (e.g. Chen 1973; Bland 1988),(3.6)where denotes the (one-dimensional) displacement derivative (i.e. gives the time-rate-of-change measured by an observer travelling with ) and is the velocity of measured by an observer at rest, along with the assumptions , to obtain the jump relations(3.7)Second, we take the jumps of equations (3.1) and (3.4), which is permissible since they hold on both sides of . After employing the formula for the jump of a product, , and simplifying, we get the two additional jump relations(3.8)Using equations (3.7) and (3.8)_{1}, we can now express the jumps in , , and in terms of the jump in . This yields, after simplifying and setting ,(3.9)Here, we observe that since *U*>0 is assumed, and by definition, it follows that .

Our next step is to determine *V*. For this, we set the determinant of the coefficient matrix of the above system of four jump equations, i.e. those given in equations (3.7) and (3.8), to zero. This leads to the propagation condition and consequently the solutions(3.10)where we note that the ‘+’ and ‘−’ cases refer to *downstream* and *upstream* waves, respectively (Whitham 1974). Furthermore, we henceforth assume that the GN parameter admits the upper bound3(3.11)so that is a real number, and is therefore physically meaningful.

Although we omit the remaining details, it is a somewhat lengthy, but relatively straightforward, process using Hadamard's lemma and equations (3.1), (3.4) and (3.9), along with equation (3.19)_{2} (see §3*c*), to derive the jump amplitude equation,(3.12)where(3.13)Also, for definiteness, we have assumed that , which corresponds to selecting the downstream (i.e. ‘+’) case in equation (3.10), and that . Thus, it follows that *λ* is strictly positive.

### (b) Evolution of acceleration wave amplitude

Since equation (3.11) is an ordinary differential equation of the Bernoulli type, its exact solution, which has been exhaustively studied (e.g. Chen 1973), can be readily determined. Omitting the details, we express it here as(3.14)where denotes the value of at time *t*=0.

To investigate the evolution of , we introduce the quantity (Chen 1973)(3.15)which is known as the *critical initial amplitude*. Now, if *μ* and *λ* are integrable on every sub-interval of and , then the following hold true:

If , then .

If , then , where is the (unique) value of

*t*known as the*breakdown time*.

Here, we observe that contained in case (II) are the conditions for finite-time blow-up, which of course *cannot* occur when .

### (c) Temperature rate waves

In the present one-dimensional context, implies that(3.16)Taking jumps of these expressions and then employing the jump product formula, it is not difficult to establish the temperature jump relations(3.17)where we have also used the assumptions . Similarly, taking jumps of equation (3.3) yields(3.18)With the aid of equations (3.6) and (3.9)_{3}, we can now use equation (3.18) to express the jumps in the second derivatives of *α* in terms of . This gives us(3.19)where we have again used the assumptions . On substituting equations (3.19) into (3.16) and simplifying, we find that(3.20)Thus, provided , we see that acceleration waves imply *temperature rate waves* (Chen 1969), i.e. propagating jumps in and/or , under GN theory.

While it does not presently appear to be possible, given current technology, future experimentalists equipped with more sensitive instruments and/or measuring techniques might be able to test for temperature rate waves in gases. Indeed, thermal waves, also known as ‘second sound’, have already been observed in a number of mundane materials such as processed meat; see Mitra *et al*. (1995).

### (d) Special case results

If the gas ahead of is in its equilibrium state, then and are constants. Consequently, ; , where(3.21)and equation (3.14) reduces to(3.22)In addition, inequality (3.11) becomes(3.23)from which we can establish that(3.24)and we recall that *m*>0 and *γ*>1.

Thus, according to equation (3.22), the evolution of can qualitatively be described as follows:

If , then is expansive, for all , and from below as .

If , then is compressive, but is defined only for and , where can now be given explicitly as(3.25)

Furthermore, we observe that equations (3.22) and (3.25) reduce to their classical counterparts, wherein (e.g. Jordan & Christov 2005), when is zero. Also, from equations (3.20) we see that and are both zero, as they are in the classical case, when is zero.

Let us continue to assume that the gas ahead of is in its equilibrium state. If we once again make use of both the perfect gas law and the GN defining relation , then we can recast equation (3.10) as(3.26)From this last result we can easily establish the equilibrium state inequality(3.27)where is the specific gas constant (Thompson 1972) and the constants and , respectively, denote the values of *θ* and in the equilibrium state.

## 4. Finite-amplitude approximation

Introducing the following dimensionless variables,(4.1)where the positive constants , and , respectively, denote a characteristic speed, length and absolute temperature, we recast equations (2.12) and (2.13) in non-dimensional form as(4.2)(4.3)where is the Mach number, , , and all primes have been omitted but are understood. Now, multiplying equation (4.2) by and then expanding in a binomial series, assuming is sufficiently small, yields(4.4)where we have assumed . If terms of order are neglected, then, after simplifying, we have(4.5)where and we note that .

In a similar way, the finite-amplitude approximation reduces equation (4.3) to(4.6)Solving for subject to the condition yields(4.7)On substituting equation (4.7) into (4.5), we obtain the single equation of motion(4.8)Rearranging terms and then carrying out the indicated differentiation with respect to *t* of the last term on the RHS results in(4.9)In keeping with the finite-amplitude approximation, i.e. neglecting terms of , equation (4.9) is reduced to(4.10)which is the lossless version of Kuznetsov's equation, the PDE governing acoustic propagation in classical perfect gases under finite-amplitude theory (e.g. Coulouvrat 1992; Naugolnykh & Ostrovsky 1998).

## 5. Summary

In this communication we have, in the context of the fully nonlinear theory, derived the exact equations of motion for acoustic propagation in lossless, GN perfect gases. In addition, we have considered the propagation of acoustic acceleration waves and temperature rate waves in such gases, again under the fully nonlinear theory, and have obtained exact expressions for the wave speed, amplitude, and breakdown time, along with establishing bounds on a number of quantities. We have also examined this theory under the finite-amplitude approximation and noted special cases. Based on an analysis of these findings, we report the following:

unlike classical gasdynamics, GN theory predicts that the acoustic and temperature fields can influence each other even in the case of homentropic flow (see equations (2.10) and (2.11));

since physical relevance requires to be real-valued, it follows that

*m*, the GN parameter, should satisfy inequality (3.11);provided that

*m*satisfies inequality (3.11), the relative speed*U*predicted by GN theory is always less than or equal to that of the classical case, with strict equality holding only when is zero (see equation (3.10));in the acceleration wave context, the classical and GN theories give identical results when is zero (see equations (3.8), (3.10) and (3.20));

if (resp. ), then is compressive (resp. expansive) for all ; however, if , then is again compressive, but it is defined only for (see §3

*b*);under GN theory, an initial jump in a first derivative of

*u*or induces propagating jumps in their first derivatives, i.e. acceleration waves,*and*, if , those of*θ*as well, i.e. temperature rate waves; however, if , then, as with the classical theory, acceleration waves alone are predicted (see §3*c*);when the gas ahead of is in its equilibrium state, GN theory predicts a value of , the breakdown time, that is less than or equal to that of the classical value, with strict equality holding only if (see §3

*d*);under the finite-amplitude approximation, the effects of coupling with the thermal displacement, being only , are immaterial; i.e. with regards to acoustic propagation, GN perfect gases and classical perfect gases are indistinguishable under the finite-amplitude approximation (see §4); and

one possible experimental test that could be used to establish the existence of a GN gas involves measuring the speed of ; in a GN gas with , the relative speed

*U*would be*less*than that predicted by classical gasdynamics (see also item (iii) above).

## Acknowledgments

This research was initiated while P.M.J. was a visiting Fellow (Mathematics) at Grey College, Durham University, Durham, UK from 19 January to 20 February 2004. P.M.J. was supported by ONR/NRL funding (PE 061153N).

## Footnotes

↵While we hope to investigate the theory's applicability to helium II in a future work, the focus of the present paper is strictly limited to gases.

↵By which we mean in the sense of Thompson (1972, p. 79). That is, an ideal gas for which the specific heats, and therefore their ratio, are constants.

↵Of course, we would expect for ordinary gases under normal conditions.

- Received November 24, 2005.
- Accepted May 11, 2006.

- © 2006 The Royal Society