## Abstract

A model for acoustic waves in a porous medium is investigated. Due to the use of lighter materials in modern buildings and noise concerns in the environment, such models for poroacoustic waves are of much interest to the building industry. The model has been investigated in some detail by P. M. Jordan. Here we present a rational continuum thermodynamic derivation of the Jordan model. We then present results for the amplitude of an acceleration wave making no approximations whatsoever.

## 1. Introduction

Noise prevention is a major modern environmental problem. Modern buildings are constructed with much lighter porous materials and typically have thinner walls. Hence, there is a great need to study the acoustic properties of porous materials including the nature of the solid matrix and the gas filling the pores. Measurements are being made of acoustic properties of many materials, such as aluminium foams, e.g. Maysenhölder *et al*. (2004), and models to fit properties have been devised, e.g. Wilson (1997). In particular, in seismic zones such as the region around Avellino, near Salerno, brick manufacturers typically attempt to increase the porosity (gas volume/total volume) to make the brick lighter. However, this usually has the effect that sound propagation through the brick is amplified and the brick itself becomes more brittle thereby making it less strong when subject to earth movement. In an attempt to create lighter bricks but retain strength, engineering laboratories in the Salerno region are experimenting with filling porous materials used in brick design with small pieces of chemical, which when heated infuse into the brick and remain trapped in the pores in gaseous form. There is interest in investigating the acoustic properties associated with various gases infused this way into the brick. We believe acoustic propagation in such materials will be a nonlinear phenomenon and so any theory which allows us to accurately predict the behaviour of a sound wave in a porous material is welcome.

As stated above, there is a need for accurate theoretical modelling of acoustic wave propagation in porous media. A recent simple nonlinear model has been proposed in a very interesting paper of Jordan (2005). Jordan effectively uses a classical perfect fluid model, but adds a term in the momentum equation, which is proportional to the velocity, a Darcy like term (cf. Nield & Bejan (1999) for an account of Darcy's law). Fellah *et al*. (2003) indicate how transport properties in air saturated porous media may be measured and Fellah & Depollier (2000) show that the equations for mass conservation and momentum in a perfect fluid, in a certain low frequency approximation lead to a linear system of equations equivalent to the Jordan–Darcy model (in a linearized form).

Here we generalize the work of Jordan (2005) and study his Darcy model using acceleration waves. An acceleration wave is a two-dimensional singular surface in a three-dimensional body across which the acceleration suffers a finite discontinuity. The study of acceleration waves and related analyses of shock phenomena have proved very useful in recent investigations of wave motion in various dispersive media (e.g. Fu & Scott 1988, 1991; Jordan & Puri 1999, 2005; Ostoja-Starzewski & Trebicki 1999; Jordan *et al*. 2000; Jordan 2004, 2005; Jordan & Feuillade 2004; Puri & Jordan 2004; Quintanilla & Straughan 2004; Christov & Jordan 2005; Jordan & Christov 2005; Su *et al*. 2005). Since we are able to obtain exactly the wave amplitude with no approximations, even though we deal with a completely nonlinear theory, we believe this is a main reason why acceleration waves are especially useful.

Precisely, this paper does two things. Jordan's (2005) Darcy model is presented, like that of Fellah & Depollier (2000), in an *ad hoc* manner. We begin by showing how the Jordan–Darcy model follows in a rational way from the continuum thermodynamic theory of Eringen (1994). Jordan (2005) develops an interesting nonlinear acceleration wave analysis, but this is based on an approximate theory, which follows from his model by neglecting terms of , where *ϵ* is the Mach number of the flow. We show that one can develop a full nonlinear analysis with no approximation whatsoever. Moreover, a precise evolutionary behaviour is predicted for the amplitude of an acceleration wave. This demonstrates that the use of porous materials for constructing modern buildings (horizontal structures and walls) guarantees a large effect on attenuation of acoustic waves. Due to the interest in acoustic theory in building materials we believe our results are of practical value.

## 2. The Jordan–Darcy model

In this section, we show how Jordan's (2005) equations (2.1)–(2.4) follow from the continuum thermodynamic theory for flow in an elastic solid as developed by Eringen (1994).

If we let be the partial densities of fluid and solid in a mixture of a fluid and an elastic solid, let be the fluid and solid velocities, and let be the fluid pressure and solid partial stress tensor, then for an isentropic flow (entropy constant, and we assume the fluid is a gas) Eringen's (1994) model consists of his eqns (2.12) and (2.15) for conservation of mass and balance of momentum,(2.1)(2.2)(2.3)(2.4)In these equations standard indicial notation is assumed, e.g. _{,t} and _{,i}. The functions and are given by Eringen (1994) eqns (3.9) and (3.23),where is the total density and the Helmholtz free energy , with being the deformation tensorIf the elastic matrix is fixed then and equations (2.1) and (2.3) reduce to Jordan's (2005) model. We might observe that to be consistent with Jordan (2005), equation (2.2), , where *K* and *Χ* are the permeability and porosity of the medium and *μ* is the dynamic viscosity of the gas. It is also worth observing that equations (2.1)–(2.4) yield a more general model which will allow the study of poroacoustic waves taking into account also the movement of the elastic matrix. We shall study this in a future paper.

For completeness we now recall the Jordan–Darcy model, i.e. equations (2.1) and (2.3) with . We put , and . The equations are(2.5)(2.6)where and . We do point out that the Jordan–Darcy model applies when the fluid saturating the porous medium is either a liquid or a gas.

## 3. Wave amplitudes

In the work of Jordan (2005), an acceleration wave analysis is performed but by neglecting terms in (2.5) and (2.6), where *ϵ* is the Mach number of the flow. We now develop a complete analysis for the full system of equations (2.5) and (2.6). The theory of acceleration waves is now well known and documented in detail in the research review article of Chen (1973). For system (2.5) and (2.6) we define an acceleration wave to be a singular surface across which the velocity and the density *ρ* are continuous in both and *t*, while their first and higher derivatives in both and *t*, in general, possess finite discontinuities.

We could develop a general acceleration wave analysis in three-dimensions for system (2.5) and (2.6), i.e. where a two-dimensional surface is the acceleration wave moving through a three-dimensional porous body. However, the essential physics is captured by considering a plane wave moving through a three-dimensional body and in this case we can restrict attention to equations (2.5) and (2.6) in the one-dimensional scenario. In the full three-dimensional situation the differential geometry involved, cf. the calculations in elasticity in Chen (1973), is likely to obscure the essential physics we wish to highlight. In one-dimension with and a wave moving along the *x*-axis, equations (2.5) and (2.6) are(3.1)(3.2)

If + denotes the region ahead of and − the region behind the wave moving in the + direction, then the amplitudes and of the acceleration wave are defined as(3.3)and(3.4)Since is a singular surface in three-space, orthogonal to the *x*-axis, we refer to and as the limits of on from the left and right, respectively. Therefore, *A* and *B* are functions of *t*, but this *t* depends also on *x* since it is defined by itself, cf. Chen (1973). From equations (3.1) and (3.2) one now shows,where *V* is the wavespeed, so that(3.5)(Note that so we need no distinction.)

The amplitude equation is calculated from (3.1) and (3.2) by differentiating with respect to *x* and then taking the jump. This is a routine calculation (cf. the method in Chen (1973)). We simply quote the final equation(3.6)where is the intrinsic derivative (i.e. the derivative at the wavefront),(3.7)and(3.8)where we have put .

The solution to equation (3.6) is(3.9)This represents the evolutionary behaviour of the wave amplitude of an acceleration wave moving into a porous medium.

We now specialize to the case where the fluid is at rest and density is constant in the porous medium ahead of the wave, then and . Of course, . The wave amplitude is still given by equation (3.9), but now(3.10)It is of interest to compare solution (3.9) to the corresponding solution when no porous medium is present. In this case, *b*=0 and the solution to (3.6) is(3.11)Our findings now do corroborate the very interesting results of Jordan (2005), although we stress no approximations have been made. (The interesting results we refer to concern the effect the Darcy term in (2.6) has on preventing finite time blow up of the wave amplitude and its effect on attenuation of an acoustic wave.)

If the wave is expansive. The wave in a gas (i.e. no porous medium) has an amplitude which decays like in time. The effect of the porous medium is clearly seen from equation (3.9). The wave amplitude is more rapidly damped out, indeed, exponentially fast.

When the wave is compressive. The wave amplitude in a gas *always* blows up in a finite time, i.e. from (3.11), as . This is believed to be associated with shock wave formation, cf. Fu & Scott (1991). Such formation is of much importance in acoustics. However, the attenuation of the wave amplitude due to the porous medium is clearly seen from equation (3.9). Instead of the amplitude always blowing up the presence of the porous medium prevents blow up unless the initial amplitude is sufficiently large. To clarify this point we include a graph of the solution (3.11) and (3.9) in a representative case, namely when . In figure 1 we see the manner in which blows up for . When *b*=1 the amplitude stays constant and for *b*>1 it decays as is seen in the graph with .

To investigate this more deeply we consider the case where , for positive constants. Then,(3.12)For the wave amplitude always blows up when . However, from solution (3.9) we see that when the porous medium is present there will be no blow up if(3.13)To consider the restriction of inequality (3.13) further we take values of , , which are given by Nield & Bejan (1999), p. 5, as representative values for brick and we use *μ*=1.78×10^{−4} gm cm^{−1} s^{−1}, *ρ*=1.225×10^{−3} gm cm^{−3}, which are values appropriate to air at 15 °C, see Batchelor (1967), p. 594. Then inequality (3.13) shows there is no blow up if |*A*(0)|<1.90298×10^{7} s^{−1}…2.47123×10^{9} s^{−1}. Clearly the porous medium has a very large effect on attenuation of an acoustic wave.

We conclude by noting that solution (3.9) applies to any porous medium and any gas. This should be useful when considering acoustic waves in a general porous medium.

## Acknowledgments

We are indebted to three anonymous referees for their careful reading of the manuscript and their helpful comments which have led to marked improvements in this article.

## Footnotes

- Received March 17, 2006.
- Accepted April 5, 2006.

- © 2006 The Royal Society