## Abstract

We apply a phenomenological theory of continua put forth by Rubin, Rosenau and Gottlieb in 1995 to an important class of compressible media. Regarding the material characteristic length coefficient, *α*, not as constant, but instead as a quadratic function of the velocity gradient, we carry out an in-depth analysis of one-dimensional acoustic travelling waves in inviscid, non-thermally conducting fluids. Analytical and numerical methods are employed to study the resulting waveforms, a special case of which exhibits compact support. In particular, a phase plane analysis is performed; simplified approximate/asymptotic expressions are presented; and a weakly nonlinear, KdV-like model that admits compact travelling wave solutions (TWSs), but which is not of the class *K*(*m*,*n*), is derived and analysed. Most significantly, our formulation allows for compact, pulse-type, acoustic waveforms in both gases and liquids.

## 1. Introduction

Theories of continua that possess an intrinsic length scale have been around for over a century now; see the treatise by Truesdell & Noll (1992) for details on the contributions of early authors—in particular, those of the Cosserat brothers to the theory of polar materials. Today, the literature of continuum mechanics is replete with theories describing the mechanics of such media, which are often categorized as *generalized continua* (Maugin & Metrikine 2010). Indeed, the past two decades have witnessed something of a minor renaissance in attempts to formulate tractable models of realistic continua. To give just a few examples, Dunn & Rajagopal (1995) presented an in-depth examination of fluids of differential type, a well-known special case of which is second-grade fluids (Dunn & Fosdick 1974; Schowalter 1978). Quintanilla & Straughan (2005) and Jordan & Straughan (2006) considered flow phenomena in incompressible and compressible, respectively, Green–Naghdi fluids. Generalizations of the original *α*-model closure scheme for the Navier–Stokes equations have been put forth by, among others, Zhao & Mohseni (2005) and Fried & Gurtin (2008); see also Fried & Gurtin (2011). And, more recently, Margolin (2009) presented the finite-scale version of the compressible, one-dimensional Navier–Stokes equations.

Of the theories of generalized continua that have been put forth in recent years, one of the most promising is a phenomenological approach developed by Rubin *et al.* (1995). The theory proposed by these authors, which we henceforth refer to as RRG, seeks to model dispersive effects caused by the introduction of a medium's characteristic length, which under RRG is regarded as an inherent material property. Among its many appealing features, RRG is both simple in its formulation and widely applicable. Under RRG, only the Helmholtz free energy and the Cauchy stress are modified; however, these modifications are achieved by *adding* perburtative terms, which must satisfy certain constraint equations, to the constitutive relations of the former and latter.

Just over a decade after it was introduced, Destrade & Saccomandi (2006) showed that RRG is a special case of the *theory of simple materials*, simple materials being a general class of continua for which the Cauchy stress is determined by the entire history of the deformation gradient. The implication of this finding, as these authors point out, is that it is possible to generalize RRG so as to describe the mechanics of an extremely diverse range of continua, from nonlinear elasticity to fluid turbulence. To highlight this, Destrade & Saccomandi (2006) went on to demonstrate that RRG is capable of modelling *nonlinear* dispersive effects as well; in particular, material dispersion described by constitutive relations under which compact structures are possible, i.e. solitary waves with compact support, similar to those recently found in conventional solid mechanics (Destrade & Saccomandi 2008; Destrade *et al.* 2009).

The main aim of this study is to extend the work of Rubin *et al.* (1995) and Destrade & Saccomandi (2006) by applying RRG to an important class of compressible media, specifically, lossless (i.e. inviscid, non-thermally conducting) gases. As such, the present investigation can be regarded as a generalization of those of Bhat & Fetecau (2006) and Keiffer *et al.* (2011), both of whom considered one-dimensional acoustic travelling waves under the simplest possible Lagrangian-averaged Euler-α (LAE-α) model of lossless compressible fluids. Here too, the introduction of a characteristic length, denoted in the former and latter (and herein) by *α*, yields a dispersive generalization of Euler's equations. However, unlike those carried out in Bhat & Fetecau (2006) and Keiffer *et al.* (2011), which were based on the assumption of constant^{1} *α*, in this study, a more general (i.e. nonlinear), constitutive relation for this quantity is posited—one that is, however, admissible under RRG.

To this end, this study is organized as follows. In §2, we introduce the system of nonlinear equations that describes acoustic propagation in lossless perfect gases under RRG. In §3, we investigate the travelling wave solutions (TWSs) admitted by this system using analytical methods and numerical simulations. Then, in §4, a weakly nonlinear version of our model, which is KdV-like in form and applicable to both gases *and* liquids, is derived and its TWS determined. Our presentation then concludes with §5, wherein our major findings are discussed and some remaining open questions are posed.

### Remark 1.1

As Destrade & Saccomandi (2006) have pointed out, in the case of incompressible media, there are a number of interesting similarities between RRG and the theory of second-grade fluids (Dunn & Fosdick 1974).

## 2. Basic equations

By way of background, ** X** denotes the position of a material point in the reference configuration and

**denotes the position of the same material point in the present configuration at time**

*x**t*. The deformation and velocity gradients are defined as

**=∂**

*F***/∂**

*x***and , respectively, where, for future reference,**

*X***=∂**

*u***/∂**

*X**t*denotes the velocity vector. As alluded to earlier, under RRG, the constitutive equation for

**, where the tensor**

*T***represents the Cauchy (i.e. total) stress, is expressed as 2.1In this study, the first term of this sum is the usual one for inviscid fluids, namely**

*T*

*T*^{(1)}=−℘

**, where**

*I***is the identity tensor and ℘ is the thermodynamic pressure; the second (i.e. perburtative) term, which captures the effects of material dispersion, is given by 2.2Here,**

*I***and**

*a***denote the acceleration vector and the symmetric part of**

*D***, respectively;**

*L**ϱ*is the mass density; thermodynamic consistency requires that

*α*

^{2}(≥0) be a function of the invariant tr(

*D*^{2}) (Rubin

*et al.*1995); and it should be noted that (2.2) can be rewritten as where

*A*_{1}=2

**and**

*D*

*A*_{2}=d

*A*_{1}/d

*t*+

*A*_{1}

**+**

*L*

*L*^{T}

*A*_{1}are the first two Rivlin–Ericksen tensors (Destrade & Saccomandi 2006).

In the case of planar propagation perpendicular to and along the *x*-axis, to which we henceforth confine our attention, the velocity vector assumes the simple form ** u**=

*u*(

*x*,

*t*)

*e*_{1}, while

*ϱ*=

*ϱ*(

*x*,

*t*) and ℘=℘(

*x*,

*t*). Thus, while the continuity equation becomes 2.3exactly as it would in this flow geometry under classical gas dynamics theory, the momentum equation, having acquired the contribution of

*T*^{(2)}, takes on the less familiar form 2.4where, in deriving (2.4), the absence of all body forces was assumed. Like its classical (i.e. Euler equations) counterpart, however, which (2.4) reduces to when

*α*≡0, the model proposed here is inviscid in the sense that the coefficients of shear and bulk viscosity have both been neglected.

Because we are working in the context of compressible flow, an equation of state, i.e. the relationship that exists between the thermodynamic variables present, must be specified. Having assumed that the propagation medium behaves like a lossless gas, we naturally employ the polytropic relation
2.5which is, of course, the equation of state for a *perfect gas* under homentropic flow (Thompson 1972). In (2.5), is a positive constant, the constant denotes the adiabatic index, and ℘_{0} and *ϱ*_{0} denote the (assumed constant) equilibrium state values of ℘ and *ϱ*. Additionally, we note for future reference that *κ* is related to *c*_{0}, the adiabatic sound speed, via the relation , where is the speed of sound in the undisturbed gas.

To close our system of equations, a constitutive relation for *α* is required. Given that *α*^{2} is a function of (*u*_{x})^{2} in the present context, as well as the need for simplicity of presentation, we assume the quadratic form
2.6and regard *α*_{0,1} as constants. Here, *α*_{0,1} must be such that *α*^{2}≥0, where we observe that *α*_{0,1} carries SI units of metre and (*metre*×*second*)^{2}, respectively.

### Remark 2.1

Our constitutive relation for ** T**, the Cauchy stress, is reminiscent of one such relation proposed by Man & Sun (1987), and labelled ‘model (II)’ by these authors, to describe the flow of polycrystalline ice. It should be noted, however, that model (II), which finds application in the study of glacier flow (Man & Sun 1987), applies only to

*incompressible*power-law media of grade 2.

### Remark 2.2

It is noteworthy that *T*^{(2)} is the most general isotropic, gyroscopic tensor one can construct using *A*_{1} and *A*_{2}, where we observe that *tr*(*T*^{(2)}** D**)=0; again, see Destrade & Saccomandi (2006).

### Remark 2.3

Setting *α*_{1}=0 reduces (2.4) to the momentum equation of the LAE−*α* model proposed by Bhat & Fetecau (2006), with the product here corresponding to the parameter ‘*α*’ of Bhat & Fetecau (2006).

## 3. Travelling wave reduction: exact theory

### (a) Ansatzs and ordinary differential equations

Introducing the right-running travelling wave ansatzs *u*:=*U*(*ζ*) and *ρ*:=*ϱ*(*ζ*), where *ζ*=*x*−*ct* is the wave variable and the constant *c*(>0) denotes the speed of the assumed travelling waveform, and substituting into (2.3) and (2.4) results in, after then integrating the former once, the system
and
where the resulting constant of integration was determined by imposing the equilibrium state conditions *U*=0 and *ρ*=*ϱ*_{0} and a prime denotes d/d*ζ*. Using the former to eliminate *ρ* from the latter and then integrating the result yields
3.1where the second constant of integration is equal to ℘_{0}. If we now recast (3.1) in the form
where
then it is not difficult to show that
3.2where the third constant of integration will be determined shortly.

Recalling now the expression for *α*^{2} given in (2.6), the former becomes
3.3where follows on setting *U* to its equilibrium state value. Introducing the dimensionless variables *V* =*U*/*c* and *ξ*=*ζ*(*c*^{2}*α*_{1})^{−1/4}, where henceforth a prime denotes d/d*ξ* and *α*_{1}>0 is assumed, (3.3) is reduced to
3.4where
3.5

Using the fact that (3.4), a first-order ordinary differential equation (ODE), is of degree two in (*V* ^{′})^{2}, we can re-express it in factorized form as
3.6where we have set for convenience. Thus, the general solution of (3.4) can be constructed from those of the following two (first-degree) equations:
3.7and where we observe that *Q*(*V*,0)=[*Π*(*V*)]^{1/4}.

### (b) Phase plane analysis

Clearly, the equilibria of (3.7) correspond to the zeros of *Π*(*V*). Let us, therefore, expand under the assumption *V* <1. Equation (3.4) then becomes, after simplifying,
3.8From (3.8), it is clear both that 1>*γσ*^{2} (⇒*c*>*c*_{0}) is a necessary requirement for the existence of travelling waves and that is a double root of *Π*(*V*)=0, where we recall that *V* =0 corresponds to the equilibrium state of the gas.

Henceforth, assuming 1>*γσ*^{2} is always satisfied, we observe that (3.7) admits a second equilibria, which we denote as , where *V* _{1}∈(0,1) is a single root of *Π*(*V*)=0.

Further restricting our focus to *V* ∈[0,*V* _{1}], so as to ensure that *V* is bounded, we observe from (3.7) that every value of *V* ∈(0,*V* _{1}) has associated with it *two* distinct slopes, which are of equal magnitude but opposite sign. From this, we conclude that the integral curves we seek assume the form of symmetric *pulses*, whose axis of symmetry is any member of the one-parameter family of vertical lines *ξ*=*ξ*_{0}.

Taking up now the questions of uniqueness and stability, we begin by noting that
3.9which for *a*_{0}=0 reduces to
3.10Using these expressions, it is then possible to show that
3.11and
3.12

Thus, uniqueness can be ensured *only* at , but only when *a*_{0}>0; otherwise, uniqueness is *never* ensured at either equilibria. In figure 1, which is based on the value of *γ* for diatomic gases (e.g. O_{2} and N_{2}), the phase plane results just presented are clearly illustrated. Figure 1 also makes clear that, when *a*_{0}>0, is stable and unstable for , respectively, whereas the stability of *V* =*V* _{1} is indeterminate, behaviour identical to that of the corresponding equilibria of the KdV's soliton solution.

### (c) Insight from Weierstrass's theorem

If we examine (3.7) using Weierstrass's theorem (Destrade *et al.* 2007), then it is a relatively simple matter to recognize the exceptional nature of the special case *a*_{0}=0, which as noted yields a degenerate version of our model earlier. When an energy integral (in a generalized sense) exists for a second-order ODE, Weierstrass's theorem allows us to understand whether the motion of our system is periodic or non-periodic simply by inspecting the zeros of the potential energy. In classical mechanics, because of the specification of the kinetic energy, we may have only a quadratic term in the first integral, i.e. (3.4) here. In classical mechanics, therefore, the simple zeros of the potential energy are *inversion* points (i.e. the motion reverses course after reaching them), whereas its double zero(s) are asymptotic points (an ‘infinite’ time is required to reach them).

This is exactly what happens under (3.7) when *a*_{0}≠0. On the other hand, because (3.4) contains a fourth degree term, when *a*_{0}=0 it is possible to reach the double zero (i.e. ) of *Π*(*V*) at a *finite* value of *ξ*. The consequences of this is, of course, failure to satisfy the *Lipschitz condition* (Burden & Faires 1993), leading to a possible lack of uniqueness, and therefore to the possibility of compact solutions. Thus, for *a*_{0}≠0 the travelling waveforms always exhibit infinite ‘tails’; for *a*_{0}=0, however, there is the possibility of compact structures. As we shall soon see, as , the solutions become more and more localized, tending to what is, in fact, a travelling wave in the form of a compact pulse.

Figure 2 shows four integral curves of (3.7); they are presented to illustrate the effect of . All were generated numerically for *σ*=(0.5)*γ*^{−1/2}, with *γ*(=1.4) corresponding to diatomic gases, using Euler's method (Burden & Faires 1993), where |Δ*ξ*|=3.125×10^{−6}, the smallest step size employed, was required for the case *a*_{0}=0.05. Here, the travelling wave condition *σ*<*γ*^{−1/2} is satisfied and, to avoid getting our finite difference scheme ‘stuck’ on the *envelope curve* (Kaplan 1958) *V* =*V* _{1},^{2} we have taken the wavefront condition as *V* (0)=(0.99999)*V* _{1}, where *V* _{1} was determined numerically using the `FindRoot[ ]` routine provided in the software package Mathematica (v. 5.2). From figure 2, it is clear that these pulses, which are spread-out over the entire *ξ*-axis for *a*_{0}>0, become more and more localized as .

### (d) Approximate and asymptotic results

While determining an exact, closed-form expression for the general solution of (3.7) appears to be out of the question, given the complicated nature of *Π*(*V*), deriving asymptotic and approximate representations for *V* , on the other hand, is not. Here, we present a number of such results, our purpose being to provide analytically tractable expressions from which insight into the physics captured by our model might be extracted.

To this end, we begin by giving a rather simple approximation for the value of the equilibrium point *V* _{1}. On replacing the LHS of (3.8) with zero and neglecting terms of on the right-hand side, it becomes a relatively straightforward task to show that the resulting (quartic) polynomial equation admits a single positive root, denoted here by , which approximates the positive root of *Π*(*V*)=0, i.e. *V* _{1}. Specifically, provided 0<*c*−*c*_{0}≪1, where
3.13Here, we observe that as , which does so, to lowest order, like
3.14

With regards to the large-|*ξ*| behaviour of *V* for the case *a*_{0}>0, it can be established that
3.15which is valid for . This result, which shows that the large-|*ξ*| behaviour of *V* for *a*_{0}>0 is qualitatively the same as that of the KdV's soliton solution, was obtained by first expanding *Q* (see (3.7)) for small-*V* , neglecting terms of and then integrating the resulting (linear) ODE subject to *V* (0)=*V* _{1}.

Of special interest, as noted in §3*b*, is the degenerate case *α*_{0}=0, which henceforth will be the main focus of our efforts. On neglecting terms of , and recalling our assumption *γσ*^{2}<1, the *a*_{0}=0 special case of (3.8) can be integrated and yields the following implicit, piecewise-defined, approximation:
3.16which approximates the *composite integral curves* of (3.7). Here, _{2}*F*_{1} denotes the Gauss hypergeometric series; the positive constants *ξ*_{c} and are given by
3.17and we observe that (3.16) satisfies the condition as . In contrast, neglecting instead the terms of in the *α*_{0}=0 case of (3.8), and once again integrating subject to *V* (0)=*V* _{1}, we obtain a more revealing, explicit, small-*V* approximation, namely the parabolic profiles
3.18

## 4. The case *α*_{0}=0 under the weakly nonlinear paradigm

While they are exact, the governing equations of our model are rather complicated, so much so that they cannot be solved to yield closed-form solutions, even under the assumption of travelling waves (recall §3*d*). Thus, in order that further progress might be achieved, we now invoke the assumptions and arguments of weakly nonlinear acoustics. Under this paradigm, we seek to synthesize ‘small, but finite-amplitude’ (i.e. approximate) versions of our governing equations into a *single*, weakly nonlinear equation of motion, where, by weakly nonlinear, we mean a PDE from which the terms of quadratic and higher order in the Mach number have been neglected.

To this end, we set aside (2.5) and in its place take (Makarov & Ochmann 1996)
4.1which we note is the quadratic Taylor series approximation of the general (barotropic) equation of state ℘=℘(*ϱ*). Here, *s*=(*ϱ*−*ϱ*_{e})/*ϱ*_{e} denotes the *condensation*, where |*s*|≪1 is assumed; *β*(>1) is known as the *coefficient of nonlinearity* (Beyer 1997); *c*_{e}(>0) denotes the speed of sound in the undisturbed fluid; and it is assumed that the equilibrium values of all quantities, which are those appended by an ‘*e*’ subscript, as well as the values of all material parameters, are constant.

### Remark 4.1

It is noteworthy that (4.1) is valid for both gases *and* liquids, provided of course that fluctuations in the density about its equilibrium state value are sufficiently small.

### Remark 4.2

In the case of gases whose behaviour is approximately that of perfect gases (recall (2.5)), *ϱ*_{e}, ℘_{e} and *c*_{e} correspond to *ϱ*_{0}, ℘_{0} and *c*_{0}, respectively, and the coefficient of nonlinearity is given by *β*=(*γ*+1)/2 (Makarov & Ochmann 1996; Beyer 1997).

### (a) Bidirectional equation of motion

We begin our derivation by eliminating ℘ from (2.4) using (4.1), after which the former becomes
4.2where we assume henceforth that *α*^{2} is given by the *α*_{0}=0,*α*_{1}>0 special case of (2.6).

Next, we introduce the velocity potential *ϕ*=*ϕ*(*x*,*t*), where *u*=*ϕ*_{x} by virtue of the fact that the irrotationality condition ∇×** u**=0 is identically satisfied here, along with the following dimensionless variables:
4.3where the positive constants

*U*

_{c}and

*L*denote, respectively, a characteristic speed and length. After also replacing

*ϱ*with

*ϱ*

_{e}(1+

*s*) in both (2.3) and (4.2), the former and latter PDEs become, after some manipulation, 4.4and 4.5respectively. Here,

*ϵ*=

*U*

_{c}/

*c*

_{e}is the Mach number, where

*ϵ*≪1, and are henceforth assumed, in accordance with the weakly nonlinear paradigm; we have set

*a*

_{1}:=2

*α*

_{1}

*U*

^{2}

_{c}

*L*

^{−4}, where we also assume ; and all diamond superscripts have been suppressed for typographical convenience.

Dividing (4.5) by (1+*s*) and then expanding in a binomial series, recalling that , yields, after re-arranging terms and simplifying,
4.6

On eliminating *s*_{x} using the relation which follows from the compressible-unsteady version of Bernoulli's theorem, and then neglecting terms of and , (4.6) becomes, after applying the operator ∂_{t} to both sides,
4.7which immediately integrates to
4.8where the resulting function of integration has been set to zero.

Finally, on eliminating *s*_{t} using (4.4), followed by the elimination of *s* and *s*_{x} in the resulting expression using once again the relation , we obtain, after neglecting terms of and simplifying, a single, weakly nonlinear equation of motion in terms of the velocity potential, specifically,
4.9

### Remark 4.3

Taking the limit , (4.9) reduces to
4.10which is the special case of the Blackstock–Lesser–Seebass–Crighton model corresponding to classical lossless fluids; see Jordan *et al.* (2012) and the references therein.

### (b) Right-running approximation: a travelling wave solution with mild discontinuities

If we now divide (4.9) by [1−2*ϵ*(*β*−1)*ϕ*_{t}], which can never be zero, expand each occurrence of the reciprocal of this quantity in a binomial series (recall *ϵ*≪1) and then simplify and neglect terms , our equation of motion becomes
4.11which in the limit reduces to the lossless version of Kuznetsov's equation; again, see Jordan *et al.* (2012) and the references therein.

Because we have confined our attention to only right-running waves, let us now replace^{3} the wave operator and the ‘small’ term (*ϕ*_{t})^{2} in (4.11) with 2∂_{t}(∂_{t}+∂_{x}) and (*ϕ*_{x})^{2}, respectively. Consequently, we get
4.12which after integrating with respect to *t* and then differentiating with respect to *x* can be written as
4.13where we have used the fact that *u*=*ϕ*_{x}. Now replacing ∂_{t} on the RHS with −∂_{x}, i.e. employing the linear impedance approximation once again, and then introducing the change of variables x=*x*−*t* and *t*=*t*, (4.13) is finally reduced to the evolution equation
4.14

If we now set *u*(x,*t*)=*g*(*η*), where *η*:=x−{*t* is a new wave variable and {(>0) the corresponding wave speed, then (4.14) becomes, after two integrations,
4.15where the resulting constants of integration have both been set to zero. On integrating this ODE, mindful of the fact that the Lipschitz condition is *not* satisfied at *g*={0,3{/*ϵβ*}, it is readily established that the following is an exact TWS of (4.14):
4.16where *Γ*(⋅) denotes the Gamma function and
4.17

In figure 3, we have plotted the TWS given in (4.16) for three different gases under normal conditions (i.e. 20^{°}C and 1 atm); see p. 640 of Thompson (1972), recalling the relation *β*=(*γ*+1)/2. Clearly, for *a*_{1}, *ϵ* and { fixed, the closer *β*(>1) is to unity, the *greater* the peak value (*g*(0)) and maximum width (2*η*_{c}) of the pulse are—facts that can also be inferred directly from (4.16) and (4.17), respectively. This means that in the case of gases, and occur for monatomic gases (i.e. ), examples of which include He, Ne and Ar; see p. 80 of Thompson (1972). This, then, suggests that experimentalist should seek these waveforms in gases for which *β*(>1) is very close to unity, i.e. in gases composed of molecules possessing a great many degrees of freedom; see §2.5 of Thompson (1972).

Another interesting feature of this TWS is the following:
4.18Here, the amplitude of the jump in a function across the plane *η*=*z* is defined as
4.19where are assumed to exist. Thus, while , ; and the fact that [[*g*^{′′}]](∓*η*_{c})≠0 means that the *g* versus *η* solution profile exhibits a *mild discontinuity* (Coleman & Gurtin 1967), of the lowest possible order, at each endpoint, in contrast to the shock and acceleration wavefronts normally encountered under classical gas dynamics theory.

### Remark 4.4

Deriving a unidirectional equation of motion (i.e. using the linear impedance approximation) has destroyed the invariance of the weakly nonlinear version of our model under superposed rigid body motions (SRBM), a phenomenon also encountered in water wave theory when deriving the KdV equation from Boussinesq's. That is, while the bidirectional equations (4.9) and (4.11) are both invariant under SRBM, which, in the present (one-dimensional) setting, implies invariance under the transformations , , (4.13) is not; and, of course, (4.14) is not invariant under , .

### (c) Conserved quantities

By performing a standard energy analysis (Straughan 2004), we multiply (4.14) by *u* and then integrate over x∈(−*η*_{c},+*η*_{c}). Using then integration by parts and enforcing the endpoint conditions as , it is readily established that
4.20Similarly, rewriting (4.14) as 6*u*_{t}=−∂_{x}{3*ϵβu*^{2}+*a*_{1}[(*u*_{x})^{3}]_{x}}, and once again integrating over x∈(−*η*_{c},+*η*_{c}), it can also be established that
4.21Thus, just as they are under the KdV equation, the energy (*E*) and the momentum (*M*) are constants of motion, i.e. they are both conserved quantities under (4.14). Of course, it is possible that, in addition to the former and latter, (4.14) could admit other conserved quantities.

## 5. Discussion

By applying RRG to the case of acoustic propagation in lossless fluids, we have been able to investigate the role played by nonlinear dispersion in the generation of travelling waves in the form of pulses. This was made possible, in part, by the fact that RRG is not based on the assumption of a kinematical microstructure, meaning that higher-order gradient effects are not introduced as they are in other models. Thus, while *α* is a function of the velocity gradient, the manner in which the effects of material dispersion are accounted for under RRG does *not*, unlike other theories of generalized continua (see Quintanilla & Straughan (2005) and the references therein), create the need for additional boundary conditions.

We established that, much like in conventional solid mechanics (Destrade & Saccomandi 2008; Destrade *et al.* 2009), compact travelling waves are associated with a particular (i.e. degenerate) form of nonlinear dispersive effect. In a rigorous manner, we then illustrated the behaviour and mathematical features of these ‘acoustic compact waves’ via the use of straightforward, but extremely transparent, analytical and computational methods. To the best of our knowledge, these are the first compact, nonlinear, pulse-type travelling waves found in the context of fluid-acoustics.

We also considered RRG under the weakly nonlinear acoustic approximation. In this framework, we derived, assuming also unidirectional propagation, a KdV-like equation, (4.14), that appears to be a previously unknown acoustic model. In particular, (4.14) we observe that is *not* a member of the class *K*(*m*,*n*) (Rosenau & Hyman 1993; Rosenau 2005), is characterized by a nonlinear, third-order dispersive term, along with the usual quadratic nonlinearity in the convective term. This is interesting in light of the fact that (16) of Destrade & Saccomandi (2006), which describes transverse waves in incompressible solids, is the corresponding (i.e. degenerate) variant of the *modified* KdV equation (Miura 1968), a PDE which exhibits a cubic convective nonlinearity.

Lastly, while we were able to show that (4.14) admits a TWS in the form of a compact pulse, and that energy and momentum are both conserved under this PDE, the following important questions, which arose out of the findings presented in §4*b*,*c*, were not taken up here, and thus appear to remain open: (i) Does (4.16) represent a *compacton* (Rosenau & Hyman 1993; Rosenau 2005)? (ii) What (if any) other quantities are conserved under (4.14)? And (iii), is there a Miura-like transformation (Miura 1968) connecting (4.14) of the present article to (16) of Destrade & Saccomandi (2006)?

## Acknowledgements

The authors thank Dr Len G. Margolin for his constructive comments and suggestions, and Dr Josette P. Fabre for her careful proofreading of an earlier version of this article. P.M.J. was supported by ONR funding. G.S. acknowledges GNFM of INdAM and by PRIN 2009 ‘Matematica e meccanica dei sistemi biologici e dei tessuti molli.’ All figures were generated using the software package MATHEMATICA (v. 5.2).

## Footnotes

↵1 It should be noted that not all α-model schemes assume constant

*α*(Zhao & Mohseni 2005).↵2 Actually, when

*a*_{0}=0,*both*are tangent to the integral curves of the ODEs in (3.7); i.e. each equilibria is reached at a*finite*value of*ξ*when*a*_{0}=0.↵3 Based on the

*linear impedance approximation*, which in the present setting assumes the form*ϕ*_{x}≃−*ϕ*_{t}; see, e.g., Crighton's (1979) derivation of Burgers' equation from the thermoviscous version of (4.10).

- Received May 28, 2012.
- Accepted June 13, 2012.

- This journal is © 2012 The Royal Society