nonlocal maxwellian theory of sound propagation in fluid-saturated rigid-framed porous media

Wave Motion 50 (2013) 1016–1035

Contents lists available at SciVerse ScienceDirect

Wave Motion

journal homepage: www.elsevier.com/locate/wavemoti

Nonlocal Maxwellian theory of sound propagation influid-saturated rigid-framed porous mediaDenis Lafarge, Navid Nemati ∗Laboratoire d’Acoustique de l’Université du Maine, UMR 6613, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France

h i g h l i g h t s

• We propose a macroscopic nonlocal theory of sound propagation in rigid-framed porous media saturated with a viscothermal fluid.• This theory takes not only temporal dispersion into account, but also spatial dispersion.• An alternative procedure for homogenization is expressed, taking advantage of an acoustics–electromagnetics analogy.• No explicit scale separation of type asymptotic approach is required to perform the upscaling procedure.

a r t i c l e i n f o

Article history:Received 14 September 2012Received in revised form 13 April 2013Accepted 16 April 2013Available online 2 May 2013

Keywords:Spatial dispersionMetamaterialsPorous mediaHomogenizationElectromagnetic analogyViscothermal fluid

a b s t r a c t

Following a deep electromagnetic–acoustic analogy and making use of an overlookedthermodynamic concept of acoustic part of the energy current density, which respectivelyshed light on the limitations of the near-equilibrium fluid-mechanics equations and thestill elusive thermodynamics of electromagnetic fields in matter, we develop a newnonperturbative theory of longitudinal macroscopic acoustic wave propagation allowingfor both temporal and spatial dispersion. In this manner, a definitive answer is suppliedto the long-standing theoretical question of how the microgeometries of fluid-saturatedrigid-framedporousmaterials determine themacroscopic acoustic properties of the latters,within Navier–Stokes–Fourier linear physics.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

The recent focus onmetamaterials in electromagnetics and acoustics –materials whose response is crucially determinedby the geometrical arrangement of their constituents – has highlighted the intricate nature of the relationship that exists, ingeneral, betweenmicrogeometry andmacroscopicwave properties. A precise understanding, in the full range of geometries,would be highly desirable in view of the possible implementation of novel concepts and ideas appearing in a growingmetamaterial literature. In particular, for the class of fluid-saturated porous materials with motionless solid frame andconnected fluid phase being the foundation for acoustic analyses,1 the importance of a theoretical clarification of the generalgeometry/acoustics relationship must be evident to all, as it determines in part our power of designing materials of all kind,exhibiting desired acoustical properties.

∗ Corresponding author.E-mail addresses: [email protected] (D. Lafarge), [email protected] (N. Nemati).

1 The solid being motionless either because of large mass, rigidity, or both.

0165-2125/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.wavemoti.2013.04.007

http://dx.doi.org/10.1016/j.wavemoti.2013.04.007

http://www.elsevier.com/locate/wavemoti

http://www.elsevier.com/locate/wavemoti

http://crossmark.dyndns.org/dialog/?doi=10.1016/j.wavemoti.2013.04.007&domain=pdf

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.wavemoti.2013.04.007

D. Lafarge, N. Nemati / Wave Motion 50 (2013) 1016–1035 1017

The purpose of the present paper is to provide a theoretical nonlocal framework allowing such clarification to be made,for the propagation of macroscopic longitudinal waves, assuming for simplicity macroscopic homogeneity and isotropy, orotherwise, propagation along a symmetry axis.2

The traditional approach to this problem assumes long-wavelengths, i.e., large scale separation between typicalmacroscopic wavelengths and typical pore sizes. Relying upon the two-scale asymptotic homogenization method [1–8],it leads to considering that, to the leading order, the media react locally and can be described in terms of a frequency-dependent effective density accounting for inertial and viscous effects, and a frequency-dependent effective compressibilityaccounting for thermal effects [9–12]: spatial dispersion only appear as a second order approximation, meaningful whenthe wavelengths reduce sufficiently. In the literature the corresponding theory is often referred to as the equivalent-fluidtheory (see, e.g., [12]). It is generally found very successful. Although, this traditional homogenization approach also leads toBiot’s theory when generalized to allow for the frame elasticity [13,8], and consequently describes in a satisfactory mannerthe acoustic properties of many more porous materials, multilayered or not, utilized in noise control [14], it neverthelessexpresses a perturbative approach of spatial dispersion phenomena, which cannot provide the physical solution in the fullrange of geometries. Despite its undeniable success and wide acceptance, it is missing an important part of the nonlocalwave physics. The inclusion of frame rigidity is beyond the scope of the present study, and not relevant for the main results.Keeping working with rigid-framed materials, the following general considerations make apparent limitations, of physicalnature, of the two-scale asymptotic homogenization method.

Comparing the structure of the macroscopic equivalent-fluid theory with that of the macroscopic electromagnetictheory, it appears that the effective density may be viewed as the acoustic counterpart of the effective electric permittivity,whereas the effective compressibility may be viewed as the acoustic counterpart of the effective magnetic permittivity.Now, recent studies on electromagnetic metamaterials have shown that for suitably microstructured materials, spatialdispersion effects are not necessarily small corrections meaningful in the short-wavelength limit: strong spatial dispersioneffects at long-wavelengths can be found [15], implying that the permittivities should be considered, in general, as nonlocaloperators described in terms of wavenumber-dependent as well as frequency-dependent kernels. Thus, the time-variableand spatially-variable polarizations responses induced by the presence of the waves, appear to depend not only on thetemporal variations of the fields, but also – and in essential manner for some microstructures – on their spatial variations.The novel metamaterial-type of behaviors appear intimately related to the latter dependences, which may lead to strongeffects even at long wavelengths.

The same should be true in acoustics: the effective density and compressibility should be viewed in general as nonlocaloperators described in terms of wavenumber-dependent as well as frequency-dependent kernels. In fact, for materialswith embedded structures in the form of Helmholtz resonators, resonant behaviors can appear at long-wavelengths [16],which are missed by the classical equivalent-fluid: at long wavelengths and in first approximation, this description alwaysrepresents the density and compressibility functions in terms of distributions of relaxation times—purely damped terms, seee.g. [17] or [10, Appendix A] and [11, Appendix C], which cannot produce any Helmholtz’s resonant behavior. The origin ofthis failure is that long-wavelength resonances are nothing but strong long-wavelength spatial dispersion effects—absentby definition in the classical two-scale homogenization approach.

To see that the resonances are closely related to spatial dispersion, it suffices to observe that the fluid is not goingin and out of a resonator, without simultaneously producing by mass conservation, spatial variations in the macroscopicwavefield. But spatial dispersion, by definition, expresses the dependence of the macroscopic properties of matter on thespatial variations of the fields [18, p. 360], hence the necessary connection between resonances and spatial dispersion.

The new nonperturbative physical theory we develop here, does not rely on asymptotic homogenization techniques,by nature not suitable to address the full physics of our wave propagation problem. We follow instead, as our heuristicguide, a deep formal analogy postulated to exist between the structure of macroscopic electromagnetic wave propagationtheory on one hand, and the structure of the wanted macroscopic acoustic theory on the other hand. This analogy, whichhighlights both the limits of our current understanding of the fluid-mechanics equations, and of the thermodynamics ofelectromagnetic fields in matter, will accordingly not be written in full; nevertheless, within Navier–Stokes–Fourier physicsit will successfully lead us to the definition of the wanted new, fully nonlocal, macroscopic acoustic theory of longitudinalacoustic waves in fluid-saturated rigid-framed porous materials.

Once the general concepts of temporal and spatial dispersion are borrowed from electromagnetics, the question is todistribute the right – temporal and spatial dispersion – effects in the right acoustic ‘polarizations’; one resembling electricpolarization and corresponding to the ‘density’ permittivity, the other resembling magnetization and corresponding tothe ‘compressibility’ permittivity, the distribution being nonunique. The main idea of the nonperturbative homogenizationperformed in this paper is that, when the distribution is performed in the right physical manner, permittivities are obtainedwhich are independently determinable by means of simple action–response problems. It turns out that the right physicalway of doing the distribution, and working out the solutions of the action–response problems, is the expression of anoverlooked fundamental thermodynamic concept henceforth referred to as the acoustic part of the energy current density.Within Navier–Stokes–Fourier longitudinal wave physics we shall see that a usable expression exists for this concept [19].In the absence of a similar thermodynamic concept of an ‘electromagnetic part of the energy current density’, the method

2 These material symmetries could be removed without changing the general principles of the present nonlocal physical framework of solution.

1018 D. Lafarge, N. Nemati / Wave Motion 50 (2013) 1016–1035

is so far not directly transferable to macroscopic electromagnetics itself; but it highlights, we believe, the still incompletestage of thermodynamic development of the latter.

The present macroscopic theory is statistical in nature and developed in principle for fluid-saturated rigid-framedmediawhich are homogeneous in an ensemble-averaged sense. Because it will include spatial dispersion without making anyperturbative simplification or introducing any explicit scale-separation condition, it will allow now not only for strongspatial dispersion at long-wavelengths, but also for arbitrary short wavelengths to be described. In random media it willdescribe the propagation of the so-called coherent waves; in periodic media it will provide the propagation constantsand define the impedances of the so-called Bloch modes. It does so, in both cases, without limitations on frequencies,wavelengths, and geometries. This ismarkedly in contrast with the variants of conventional homogenization [20–22], whichare limited in these respects.

The paper is organized as follows. Mechanical wave propagation in a viscothermal fluid is first revisited in Section 2.Recasting the equations in electromagnetic form andmaking use of the appropriate thermodynamic concept of ‘the acousticpart of the energy current density’, the longitudinal wave motions are shown to derive from an equivalent nonlocal densityand compressibility described by frequency- and wavenumber-dependent kernels. These kernels are found to be directlyrelated to the solutions of two independent action–response problems. On the one hand, the density kernel reflects thenonlocal response of the fluid subjected to a time variable and spatially variable external bulk force. On the other hand, thecompressibility kernel reflects the nonlocal response of the fluid subjected to a time variable and spatially variable externalbulk rate of heat supply.

These fundamental observations are next directly used in Sections 3 and 4, when the viscothermal fluid is permeatinga macroscopically homogeneous porous structure. Generalizing the electromagnetic recasting of the equations andthe aforementioned thermodynamic identification as well, we conjecture that to describe macroscopic longitudinalwave propagation, there are macroscopic density and bulk modulus operators reflecting the macroscopically-averagedpermeating fluid responses to the same external actions as in Section 2. This results in establishing definite recipes forthe first-principles computation of the operators from microgeometry. In Section 5, we consider briefly how this theorycan enable us predicting the correct propagation constants and impedances, in macroscopically homogeneous media, andalso, the propagation constants and impedances of Bloch modes in periodic media. To conclude, the main points of the newtheory are summarized. Future perspectives are briefly considered.

2. Electromagnetic recasting of the acoustic equations in viscothermal fluids

The analogy which we invoke here, between acoustics and macroscopic electromagnetic wave theory, is a fruitfulexample of Maxwell’s real analogies in nature nicely described by Torrance [23, pp. 6–7].3 We use it as a general guideto introduce, in our acoustic problems, the notions of temporal and spatial dispersion well discussed e.g. in Landau andLifshitz [18] and Agranovich and Ginzburg [24]. In this section we introduce the analogy for the case of small-amplitudemechanical wave propagation in a viscothermal fluid. The fact that this mechanical wave propagation is describable in away very similar to that of small-amplitude electromagnetic wave propagation in matter, at the macroscopic level, is notsurprising: wave propagation in the fluid is already a macroscopic wave phenomenon in the sense that a ‘fluid particle’actually represents the behavior of a huge – macroscopic – ensemble of molecules.

In the proper recasting of the fluid-mechanics equations in electromagnetic form, however, some difficulties arise.Working within the ordinary Navier–Stokes–Fourier linearized equations, it will be seen that the reformulated equationsare singular, unless the viscous shear waves are excluded. For this reason, the generalizations in Sections 3–5, concern onlythose motions which can be induced in the material by the free-fluid longitudinal type of motion, and not the free-fluidshear type of motion. This will not be a significant reduction in scope of the theory, since the motions induced in our rigidframematerial by shearwaves, can be expected to be of very limited significance compared to those induced by longitudinalwaves, and more precisely by the so-called acoustic waves.

The singular nature of the reformulation should not be viewed as a physical failure of the analogy, but rather, as a physicallimitation of the range of validity of the classical Navier–Stokes–Fourier equations. Indeed, in writing the momentum andenergy balance laws, three important thermodynamic simplifications aremade: (i) the fluid obeys a caloric equation of staterequiring that the specific internal energy is a function of two independent thermodynamic variables—e.g. specific entropyand specific volume, (ii) it obeys Newton–Stokes’s constitutive law (an ansatz giving the deviatoric part of the stress tensordensity in terms of the spatial derivatives of the velocity, via two viscosity constants), and (iii) it obeys Fourier’s constitutivelaw (an ansatz giving the heat flux density in terms of the spatial derivatives of temperature, via one thermal conductionconstant). As a result, Maxwell’s thermal stresses predicted by the kinetic theory of gas are not included [25–27]. This near-equilibrium thermodynamic framework is not adapted to describe the correct behavior of waves at very short times andwavelengths [28]. Shear and longitudinal waves are viewed as completely unrelated type of motions, which is not the real

3 Especially relevant to characterize the construction philosophy of the foregoing macroscopic acoustics, are, in these pages, the mention of Maxwell’sway of thinking in terms of dynamic relations, his use of patterns of behavior determined in one set of phenomena to assist him in imagining and workingout the patterns inherent in another set of phenomena, and Faraday’s own way of proceeding from the whole to the parts.


physical situation.4 Another well-known unphysical feature of this description is that it leads to partial differential parabolicequations implying boundless speed responses for temperature and vortical signals. It is not surprising, therefore, that theNavier–Stokes–Fourier near-equilibrium thermodynamic simplifications are at the root of difficulties encountered withthe electromagnetic reformulation. Addressing these difficulties would require working in the more general frameworkof extended irreversible thermodynamics and generalized hydrodynamics. But this is not our concern here. Instead, wedescribe how, by leaving apart the shearmotions, we can arrive at a usable notion of ‘acoustic part of energy current density’,and at the related existence of action–response problems allowing to separately determine the two acoustics susceptibilities.

2.1. Navier–Stokes–Fourier equations and classical hydrodynamic modes

Let us now start by recalling the set of linearized Navier–Stokes–Fourier equations [29–31]:

ρ0∂v∂t

= −∇p − η∇ × (∇ × v) +

4η3

+ ζ

∇(∇ · v) (1a)

1ρ0

∂ρ ′

∂t+ ∇ · v = 0 (1b)

γχ0p =ρ ′

ρ0+ β0τ (1c)

ρ0cp∂τ

∂t= β0T0

∂p∂t

+ κ∇2τ (1d)

where the wave variables v, ρ ′, p, τ , are the fluid velocity, excess density, thermodynamic excess pressure, and excesstemperature, respectively, and the fluid constants ρ0, η, ζ , γ , χ0, β0, cp, T0, κ , represent the ambient density, first viscosity,second viscosity, ratio of heat coefficients cp/cv , adiabatic compressibility, thermal expansion coefficient, specific heatcoefficient at constant pressure, ambient temperature, and thermal conduction coefficient, respectively.

Solutions to Eqs. (1) can be analyzed in terms of hydrodynamic modes or normal modes varying like v = v0ei(k·x−ωt)

etc., with constant amplitudes. With five independent first order-in-time equations of motion,5 for a given real value of thewavevector k, there are five complex-frequency normal modes [32]:

1. two longitudinal ‘first type’ modes v0 ∥ k (the pressure or sound modes; the duality arising from the two oppositedirections of propagation)

2. one longitudinal ‘second type’ mode v0 ∥ k (the heat or entropic mode)3. two transverse ‘third type’modes v0 ⊥ k (the viscous or vorticalmodes; the duality arising from the two possible j = 2, 3

polarizations).

Indeed, the substitution of fields of the type v = v0ei(k·x−ωt) etc., with constant amplitudes in Eqs. (1), is equivalent to passingover to the Fourier transform. The resulting homogeneous system is either satisfied by purely transversal motions k ·v0 = 0carrying no pressure, temperature, or condensation, with frequency eigenvalues

ωj = −iη

ρ0k2, j = 2, 3 (2)

or by purely longitudinal motions k × v0 = 0, with frequency eigenvalues satisfying Kirchhoff–Langevin’s dispersionequation [29,30], obtained by setting to zero the determinant

− ω2+

c2a − iω

κ

ρ0cv+

4η3 + ζ

ρ0

k2 −

κ

ρ0cv iω

c2i − iω

4η3 + ζ

ρ0

k4 = 0 (3)

with ca the adiabatic sound speed defined by c2a ≡ 1/ρ0χ0, and ci the isothermal sound speed defined by c2i ≡ c2a /γ . Forsufficiently small k (long wavelengths), the solutions of (3) can be expanded in powers of k. The first type of modes havefrequency eigenvalues [32]

ω = ±cak − iΓ k2 ∓ O(k3) − iO(k4) (4)

4 Indeed, as mentioned in [24, p. 45], in most problems of electrodynamics and optics of anisotropic media, a division of the field into a transverse and alongitudinal component is not possible, physically. But in the presence of spatial dispersion, the permittivities become tensors, even in isotropic media: adistinctive direction is generated by the wave vector [18, p. 360]—see the acoustic equations (27) and (31) which illustrate this. In this sense, the mediumexhibits a sort of anisotropy, and the division of the field into a transverse and a longitudinal component is no longer a physical division. Thus in fluidacoustics also, we expect that the division makes no sense, when proper account is made of spatial dispersion. This will concern the very short times orvery short wavelengths ‘Lamé–Frenkel’ physics, briefly mentioned later on.5 Three Eqs. (1a), Eq. (1b) with ρ ′ substituted from (1c), and Eq. (1d).


with Γ =1

2ρ0

4η3 + ζ + κ( 1

cv−

1cp

)the classical sound damping constant, whereas the purely damped second type of

mode has frequency eigenvalue [32]

ω = −iκ

ρ0cpk2 + iO(k4). (5)

The fact that the modes are either purely longitudinal, carrying no vorticity (∇ × v = 0), or purely transversal, carryingno pressure, temperature, or condensation, is an effect of our restricted Navier–Stokes–Fourier thermodynamic framework.Because of this complete separation between vortical and compressional–dilatational motions, any possible coincidence ofthe frequencies (5) and (2) will not correspond to the case of multiple essential roots considered in [24]; hence the abovefive modes constitute a complete system of modes.

2.2. Affinities with electromagnetics: interchange of symmetry

We easily verify that the Navier–Stokes–Fourier linearized Eqs. (1), are equivalent to the following set of equations∂bij∂t

= −12

vi;j + vj;i

(6a)

(ρ0)ij∂vj

∂t= −(χ0

−1)ijklbkl;j + ji (6b)

(ρ0)ij = ρ0δij, (χ0−1)ijkl = χ−1

0 δijδkl (6c)

ji = (χ0−1)ijklbkl;j − δijp;j +

η

vi;j + vj;i −

23δijvk;k

+ ζ δijvk;k

;j

(6d)

δijp − (χ0−1)ijklbkl = −(γ − 1)δijp + χ−1

0 δijβ0τ (6e)

ρ0cpδij∂τ

∂t= β0T0δij

∂p∂t

+ κδijδklτ;k;l (6f)

where δij is the Kronecker symbol, vi are the components of the velocity v, and the spatial derivative with respect to xi isdenoted by ;i. Eq. (6a) can be viewed as an acoustic equation–definition analogous to the electromagnetic macroscopic fieldequation–definition

∂Bij

∂t= Ei;j − Ej;i. (7)

Indeed, in the same way as Eq. (7) is a consequence of the definitions Ei ≡ −∂Ai/∂t and Bij ≡ Aj;i − Ai;j of the electric andmagnetic tensor fields, in terms of the macroscopic 3-potential Ai (written in the temporal or Weyl’s gauge A0 ≡ 0 in themedium rest-frame), Eq. (6a) is a consequence of the definitions vi ≡ ∂ui/∂t and bij ≡ −

12

ui;j + uj;i

of the velocity and

opposite strain, in terms of the 3-vector macroscopic fluid displacement ui. The invoked correspondence Ai → ui betweenelectromagnetic potential and acoustic displacement field, is the same as what has been considered, e.g. in [33], for a corre-spondence between electromagnetics and linear elasticity. The difference between electromagnetic and acoustic equationsexpresses in the way the fields Ei and Bij, or vi and bij, are derived from Ai, or ui: by using ‘antisymmetry’, or ‘symmetry’. Weargue below that this symmetry interchange, will transform the whole pattern of macroscopic electromagnetic equationsin a coherent pattern of reformulated ‘acoustic equations’.

The Eq. (6b) is intentionally written in a form suitable to be compared with the corresponding macroscopicelectromagnetic equation6

(ϵ0)ij∂Ej∂t

= (µ−10 )ijklBkl;j − ⟨j⟩i (8)

written for an isotropic medium

(ϵ0)ij = ϵ0δij, (µ−10 )ijkl =

12µ−1

0 (δikδjl − δilδjk) (9)

with ϵ0 and µ0 two scalar constants which characterize the electric and magnetic permittivities of the medium, and ⟨j⟩ian induced macroscopic density current, which appears in response to the time and space variations of the fields, and isnon-zero because of lossy, thus non instantaneous, polarization response processes.7

In a similar manner the ji in Eq. (6b) is to be interpreted as an induced macroscopic density of force,8 which appears inresponse to the temporal and spatial variations of the fields, and is non-zero because of lossy, thus non instantaneous,

6 We consider small-amplitude electromagnetic wave propagation in an unbounded homogeneous medium in a rest-state of thermodynamicequilibrium, devoid of charge, current, or magnetization.7 Any instant reaction in ⟨j⟩i is not associated with losses. It should be considered as already incorporated in ‘ϵ0 ’ and ‘µ0 ’.8 As regards the notation, ji replacing ⟨j⟩i , recall that the wave propagation in the fluid is already a macroscopic phenomenon: thus the fluid-mechanics

vector j is, here also, some average ⟨⟩ of quantities of the microtheory – kinetic-theory – level.


‘polarization’ response processes. By virtue of the symmetry interchange, and assumed isotropy, the quantities (ρ0)ij

and (χ−10 )ijkl replacing (ϵ0)ij and (µ−1

0 )ijkl should be written in the most general case, in the familiar form in elasticitytheory [34]

(ρ0)ij = ρ0δij, (χ−10 )ijkl = χ−1

0 δijδkl + µ0

δikδjl + δilδjk −

23δijδkl

. (10)

We can identify ρ0 and χ−10 with the fluid density and adiabatic bulk modulus, and name µ0 the ‘Lamé–Frenkel’ shear

modulus ormodulus of rigidity.The reason for this denomination is that this modulus allows for the propagation of undamped shear waves at very short

times—so short that the induced force ji has no time to be built. Furthermore, such behavior will be consistent with Frenkel’s(1925) long overlooked idea [35, p. 189] stating that a fluid behaves like a solid at very short times. In this connectionwe notethat, recently, a phonon theory of thermodynamics of liquids has been developed using Frenkel’s theoretical framework ofthe phonon states in a liquid. It has shown remarkable agreement for the calculations of heat capacity coefficients of differentliquids in a wide range of temperature and pressure [36].

Now, Eqs. (6) which express in another form Eqs. (1) show that the Navier–Stokes–Fourier fluid-mechanics thermo-dynamic framework is a singular, insufficiently general one: comparing the second equation (10) with (6c) we see that iterroneously sets to zero the Lamé–Frenkel constantµ0. We shall nevertheless try tomake abstraction of this singularity andsee how far we can pursue our electromagnetic–acoustic analogy in this unsatisfactory and near-equilibrium framework.9

The remaining Eqs. (6d)–(6f) will tell us, to some extent corresponding to the simplifications made, how the inducedforce density ji comes from some acoustic ‘polarization processes’ completely determined by the time and spatial variationsof the fields vi and bij. For this analysis we adopt the electromagnetic viewpoint of the Lorentz decomposition of the inducedmacroscopic current density into a time derivative and a spatial derivative term10

⟨j⟩i =∂Pi∂t

+ Mij;j (11)

withMij an antisymmetric tensor by virtue of the characteristic antisymmetry encountered in electromagnetics. The current⟨j⟩i appears in response to the time variations and the space variations of the fields, respectively defining the temporaldispersion and spatial dispersion effects [18]. The use of the timederivative and spatialderivatives tomake apparent differentpolarization fields Pi and Mij is natural: previously the different fields Ei and Bij were also introduced from the componentsof the 3-potential, by means of time derivatives and spatial derivatives. The polarization fields Pi and Mij will be relatedto these fields Ei and Bij by constitutive nonlocal laws manifesting temporal and spatial dispersion effects (often, but notnecessarily, very small).

Here, we introduce in the same manner a formal decomposition of the induced force density

ji =∂pi∂t

+ mij;j (12)

with, this time, a symmetricmij to account for the characteristic symmetry encountered in acoustics. As in electromagnetics,the ‘polarization fields’ pi and mij will be related to the fields vi and bij by constitutive nonlocal relations. Now, it can beobserved that the Eq. (12) is by itself not sufficient to specify the polarizations pi andmij: we have two quantities to describeone vector. Obviously, we need an additional condition to unambiguously set which part of ji is to be taken into account bythe term ∂pi/∂t and which one by the term mij;j. The same ambiguity arises in electromagnetics also, for the definition ofthe fieldsMij and Pi. We defer to Section 2.4 a mention of the way it is usually resolved.

2.3. Electromagnetic recasting: formal structure of the equations

For the time being, and irrespective of the aforementioned ambiguity, let us pursue in a formal manner the EM recastingof the equations. Substituting (12) in (6b) and introducing the two new acoustic ‘Maxwell’ density fields di and hij

11

di = (ρ0)ijvj − pi, hij = (χ−10 )ijklbkl − mij (13)

the macroscopic acoustic linearized equations now write in the following ‘Maxwellian’ form.

9 Obviously, the ‘Lamé–Frenkel’ very-short time or very-short wavelength physics lies outside the realm of near-equilibrium thermodynamics.10 Lorentz in his theory of electrons [37] was the first to consider a similar decomposition. He introduced a separation between ‘bound charges’ and ‘freecharges’ and applied Eq. (11) to the bound charges motions only. Thereby, Lorentz’s density fields Hij and Di were defined to incorporate the effects ofbound charge motions only, and this definition was generally adopted in subsequent literature. But a clear distinction between bound charge and freecharge motions cannot – and need not – be made. What is truly important is that the abstract ‘polarization’ fields Pi and Mij , whatever their origin –including also the non-classical spin contributions – necessarily are related at themacroscopic classical level to the fields Ei and Bij by constitutive nonlocaloperator relations.11 We name them density fields because an analysis of their variance reveals that they are fields of weight W = −1—see Weinberg [38, p. 98] for thisnotion—exactly like Maxwell fields Di and Hij—see [39, p. 30].


field equations:

∂bij∂t

= −12

vi;j + vj;i

,

∂di∂t

= −hij;j (14)

constitutive relations:

di(t, x) = ρijvj(t, x) =

t

−∞

dt ′

dx′ρij(t − t ′, x − x′)vj(t ′, x′) (15a)

hij(t, x) = (χ−1)ijklbkl(t, x) =

t

−∞

dt ′

dx′(χ−1)ijkl(t − t ′, x − x′)bkl(t ′, x′). (15b)

Clearly, the field di would define an ‘acoustic momentum density’, and the field hij, an ‘acoustic stress density’. Theoperators have to be difference-kernels operators to respect time and space homogeneity, and the relations (15) are themost general linear oneswhich can bewritten on account of the fact that the fields vi and bij are not completely independent,but related by the first field equation (14) (see Landau and Lifshitz [18, p. 359]). These equations appear to be a ‘symmetric’version of the electromagnetic equations which result from the formal introduction of the following Maxwell fields

Di = (ϵ0)ijEj + Pi, Hij = (µ−10 )ijklBkl − Mij (16)

and which write in the following Maxwellian form.field equations:

∂Bij

∂t= Ei;j − Ej;i,

∂Di

∂t= Hij;j (17)


Di(t, x) = ϵijEj(t, x) =

t

−∞

dt ′

dx′ϵij(t − t ′, x − x′)Ej(t ′, x′) (18a)

Hij(t, x) = (µ−1)ijklBkl(t, x) =

t

−∞

dt ′

dx′(µ−1)ijkl(t − t ′, x − x′)Bkl(t ′, x′). (18b)

2.4. A key notion: ‘the acoustic part of the energy current density’; generalized susceptibilities

Now, because of the ambiguity of the Lorentz decomposition, the fields Pi, pi, and Mij,mij, and thus also the fields Di, di,and Hij, hij, as well as the operators ϵij, ρij, and (µ−1)ijkl, (χ

−1)ijkl, are, as yet, not completely determined. Indeed, thereis a difference of nature, between the Lorentz true tensor fields Ei, Bij, and vi, bij, which can be conceived as the directmacroscopic average of some underlyingmicrotheory quantities (this is explicit in the electromagnetic case: Ei = ⟨ei⟩, Bij =

⟨bij⟩, where ei and bij are the so-calledMaxwell–Lorentz fields and ⟨ ⟩ themacroscopic averaging operation), and theMaxwelldensity fields Pi,Di,Mij,Hij, and pi, di,mij, hij, which are not expressed as the directmacroscopic average of some underlyingmicrotheory quantities.

We suggest that the determination of all these ‘Maxwell’ quantities should be made at once, by requiring the‘Heaviside–Poynting’ or ‘Umov’ products

Si = HijEj, si = hijvj (19)

to be thermodynamic quantities representing respectively an ‘electromagnetic part of the energy current density’ or an‘acoustic part of the energy current density’. In this way, the operators would belong to the class of quantities referred to asgeneralized susceptibilities, and could be determined by independent action–response problems. The fact that the operatorshave all characteristics of generalized susceptibilities if we set the conditions (19), can be described a priori as follows, takingthe example of the acoustic case.

If an hypothesized ‘acoustic part of the energy current density’ si exists as a thermodynamic entity, it allows defining anotion of ‘acoustic energy’ per unit volume of material w, by setting ∂w/∂t = −si;i. Then using (19) and the macroscopicMaxwell field equations (14), it follows that the time variation of w is given by ∂w/∂t = (∂dj/∂t)vj + (∂bji/∂t)hij. Byvirtue of this relation, hij would be given by an equation hij ≡ δw/δbij formally similar in its philosophy, to the definitionof thermodynamic pressure p ≡ − (∂ε/∂υ)s, for a fluid obeying a caloric equation of state ε = ε(s, υ), with ε the specificinternal energy per unit mass, s the specific entropy, and υ = 1/ρ the specific volume.

Now, following Landau and Lifshitz [40, p. 393] and [18, p. 359], we note that, if we choose the components (taken at allspatial points x) of the fields vj and hij, to be the two sets of variables xa, a ≡ (j, x), and yb, b ≡ (ij, x), which describe themacroscopic response of the system, then the corresponding generalized forces fa and gb12 will be the components of the

12 These forces are defined such that

∂w∂t dx =

xa

∂ fa∂t da +

yb

∂gb∂t db.


fields dj and bij respectively. It follows that the coefficients αaa′(t − t ′) or βbb′(t − t ′) in the action–response relations ofgeneralized-susceptibility type xa(t) =

αaa′(t − t ′)fa′(t ′)da′dt ′ and yb(t) =

βbb′(t − t ′)gb′(t ′)db′dt ′, will be respectively

the components of the tensors kernels ρ−1ji (t − t ′, x − x′) and (χ−1)ijkl(t − t ′, x − x′), in the relations of type vj = ρ−1

ji diand hij = χ−1

ijkl bkl. This establishes the general character of susceptibility functions of the two permittivities defined throughEq. (19).

In the electromagnetic case the ambiguity is usually not resolved in this manner. It is resolved by a petitio principii:following e.g. Landau and Lifshitz [18, p. 359], or Melrose and McPhedran [41, p. 72], one usually considers that there is nopoint in introducing a non-trivial Hij field in the presence of spatial dispersion, and hence sets by definition Mij = 0 andHij = µ−1

0 Bij.We regard this definition as an expedient justified by the usual smallness of thewave-inducedmagnetic effects(relativistic effects proportional to β2

= v2/c2, v the electron velocity in the atom, c the speed of light [18, p. 269]). In fact,the thermodynamics of electromagnetic fields in matter remains elusive. These substitutions will not lead to any incorrectdescription of the medium propagation constants, but would have disadvantages for the description of impedances if themagnetic effects were not small corrections. The foregoing acoustic developments will help clarifying these points.

2.5. Singular EM recasting using Schoch’s identification

Assuming the validity of the above ‘Heaviside–Poynting’ or ‘Umov’ idea (19), we can do nothingwith it in the electromag-netic case, as the necessary thermodynamic concepts and quantities are, so far, entirely missing. But acoustics is an innatethermodynamic theory and there, we can determine the acoustic permittivities in the restricted Navier–Stokes–Fourierthermodynamic framework, making use of Schoch’s profound suggestion stating that the expression

si = pvi (20)

where p is the thermodynamic excess pressure, should precisely give this postulated ‘acoustic part of the energy currentdensity’—even in the presence of the viscous and thermal conduction losses [19]. Equating (19) and (20) yields

hij ≡ pδij. (21)

The expression for the polarizationmij then follows from the second equations (13) and (21) as

mij ≡ (χ0−1)ijklbkl − δijp. (22)

Then substituting the above expression in (12) and using (6d), we have

pi ≡

η

ai;j + aj;i −

23δijak;k

+ ζ δijak;k

;j

(23)

for polarization pi, and a definition of the field di then follows from the first equation (13).As such, separation between viscous and thermal effects is made: only the former are involved in pi, whereas only the

latter are involved in mij. The polarization field pi represents an irreversible viscous momentum, thus by the first equation(13) the acoustic momentum di appears as a ‘reversible’ momentum given by the total momentum minus the irreversiblepart. The polarization fieldmij represents a thermal stress. To demonstrate the thermal nature ofmij, we observe that, owingto (6e), the abovemij rewrites

mij = (γ − 1)δijp − χ−10 δijβ0τ . (24)

This expression vanishes when there is no thermal conduction. In fact, if κ = 0, Eq. (6f) shows that the temperature andpressure should be related by the (adiabatic) relation ρ0cpτ = β0T0p. Substituting this relation in (24) and using the generalthermodynamic identity [42]

γ − 1 =T0β2

0

ρ0χ0cp(25)

it is verified that the expression (24) vanishes.The above expressions are remarkably simple and natural, however, we should remember that they are written in a

too narrow thermodynamic framework. In particular, while the starting idea (19) of the existence of an ‘acoustic part ofthe energy current density’ s should be general, its implementation by means of (20), reveals (through the absence of off-diagonal components, see (21)) the insufficiencies of the Navier–Stokes–Fourier framework. We believe that within a largerthermodynamic framework, as no complete separation between viscous and thermal effects would arise, a well-formed(not only hydrostatic) hij would appear, consistent with the presence of nonzero ‘Lamé–Frenkel’ terms. These, could also benamed ‘Maxwell’ terms, as they would represent a general elaboration of Maxwell’s early phenomenological attempts tounify solid and fluid behaviors, representing in particular the total strain due to a given shear stress, as the sum of an elasticsolid part, and a fluid viscous part [26].


For the time being, because of the absence of these terms, the operator ρij is not a well-formed susceptibility: it is not aninvertible operator. Similarly for the operator χ−1

ijkl ; it is not invertible, in spite of its notation. As we have mentioned, withinthe Navier–Stokes–Fourier model equations, the shear motions produce nonzero fields vi and bij, but vanishing fields di andhij. Thus, no reversible momentum d is transported by the viscous shear waves; and no acoustic stresses hij are generated.The first shows the singular nature of operator ρij; and the second the singular nature of operator χ−1

ijkl .Having these limitations in mind, we shall not try to directly express the postulated kernels functions in the original

space. Singularities and convergence problems can be avoided by working in Fourier space. Thus reasoning on the Fourieramplitudes ρij(ω, k) and (χ−1)ijkl(ω, k) such that

ρij(t − t ′, x − x′) =

dω2π

dk(2π)3

ρij(ω, k)ei[k.(x−x′)−ω(t−t ′)]

(χ−1)ijkl(t − t ′, x − x′) =

dω2π

dk(2π)3

(χ−1)ijkl(ω, k)ei[k.(x−x′)−ω(t−t ′)] (26)

and passing over to the Fourier transform of the original space equations, using Eqs. (21) and (14)–(15) with Eqs. (6),straightforward calculations give

ρij(ω, k) = ρt(ω, k)

δij −kikjk2

+ ρl(ω, k)

kikjk2

(27)

with t and l referring to ‘transverse’ and ‘longitudinal’ [18, p. 360] and

ρt(ω, k) = ρ0

1 +

η

ρ0

k2

−iω

, ρl(ω, k) = ρ0

1 +

4η3 + ζ

ρ0

k2

−iω

(28)

and

χ−1ijkl (ω, k) = χ−1(ω, k)δijδkl (29)

with

χ−1(ω, k) = χ−10

1 −

γ − 1γ

κρ0cv

k2

−iω +κ

ρ0cvk2

(30)

employing the thermodynamic identity (25). The Fourier kernel ρij(ω, k) (27) is the general form given in Landau andLifshitz [18] for a symmetric tensor in an isotropic medium (possessing in addition center symmetry). The transversekernel is determined by the shear viscosity; the longitudinal kernel by the bulk viscosity. The Fourier kernel χ−1

ijkl (ω, k)is determined by the thermal conduction coefficient.

Because of the assumptions (i)–(iii) in Section 2, there are missing terms in (28), which would result in a non convergentintegral in (26). There are similarlymissing terms in (30)with the same result.Moreover, there are a number ofmissing termsalready in (29): in a sufficiently complete thermodynamic framework the tensorwould be found in the general form allowedby the symmetry. Since with center symmetry and isotropy the only available tensors are δij and ki, and the expressionmustbe symmetric over the first and second pair of indexes, and, as a generalized susceptibility, must also be subject to thegeneral principle of symmetry of kinetic coefficients [40, p. 380], which requires (χ−1)ijkl(ω, k) = (χ−1)klij(ω, −k), see [18,pp. 359–360], this most general form would be

χ−1ijkl (ω, k) = χ−1(ω, k)δijδkl + µ(ω, k)

δikδjl + δilδjk −

23δijδkl

+ a(ω, k)

δik

kjklk2

+ δjkkiklk2

+ δilkjkkk2

+ δjlkikkk2

+ b(ω, k)

δij

kkklk2

+ δklkikjk2

+ c(ω, k)

kikjkkklk4

(31)

with some functions χ−1, µ, a, b and c of ω and k. Here, because of the Navier–Stokes–Fourier simplifications, only the firstterm χ−1(ω, k)δijδkl is present, in its incomplete version (30).13Although the expressions (27)–(30) are thus singular andincomplete in several respects, let us check that they consistently encode the Navier–Stokes–Fourier wave physics.

13 It may be noticed that, owing to (25) and (30), the deviation of this term from its adiabatic value χ−10 δijδkl is proportional to β2

0 . Liquids, characterizedby very small values of the thermal expansion coefficient β0 , are in this respect comparable to the materials for which the magnetic effects are relativisticcorrections determined by the small parameter β2

= v2/c2 .


2.6. A check on the EM recasting

Substituting the fields vi(t, x), etc., varying like v0iei(k·x−iωt), etc., in Eqs. (14)–(15b), or equivalently passing over to theFourier transform, we find the polarization relations leading to the following homogeneous system for v0

ω2ρij(ω, k) −12

(χ−1)ikjl(ω, k) + (χ−1)iljk(ω, k)

kkkl

v0j = 0. (32)

Using the above equation and expressions (27)–(30) yieldsω2ρtδij +

ω2

k2(ρl − ρt) − χ−1

kikj

v0j = 0. (33)

This can be satisfied in two different ways. When the velocity is purely transversal, and thus kiv0 i = 0. Then, by using(33) and (28), we obtain: 1+ ηk2/(−iωρ0) = ρt = 0; which is the dispersion equation (2) describing purely damped shearwaves. We see that the operator ρij is singular. Its so-called operator kernel is nonzero Ker(ρ) = 0, and made of all purevortical motions: Ker(ρ) = {v| v = ∇ × ψ}.

On the other hand, when the velocity is purely longitudinal, and consequently v0i = v0k−1ki. Putting in (33) an amplitudev0 i satisfying this condition, we get: −χ−1

+ (ρl −ρt)ω2/k2 = 0. Putting in it the expressions (28) and (30) of the functions

ρl(ω, k), ρt(ω, k), and χ−1(ω, k), yields the Kirchhoff–Langevin’s dispersion equation (3), previously found to describe theso-called acoustic and purely damped entropic modes. As such, it is verified that the entirety of Navier–Stokes–Fourierphysics is encoded in the EM singular recasting (14)–(15) and (27)–(30).

2.7. Acoustic illustration of the customary definition of Maxwell fields in macroscopic electromagnetics

Applying in acoustics the customary point of view used in electromagnetics, we would set by definition mij = 0, hij =

χ−10 bij, and obtain within Navier–Stokes–Fourier physics the expressions

ρij(ω, k) = ρt(ω, k)

δij −kikjk2

+ ρl(ω, k)

kikjk2

(34)

with

ρt(ω, k) = ρ0

1 +

η

ρ0

k2

−iω

1 −

γ − 1γ

κρ0cv

k2

−iω +κ

ρ0cvk2

−1

,

ρl(ω, k) = ρ0

1 +

4η3 + ζ

ρ0

k2

−iω

1 −

γ − 1γ

κρ0cv

k2

−iω +κ

ρ0cvk2

−1

(35)

and

χ−1ijkl (ω, k) = χ−1

0 δijδkl. (36)

The separation between viscous and thermal effects is no longer made in these expressions. However, once inserted inEq. (32) they yield exactly the same shear, acoustic, and entropic wavenumbers as before.14 Defined in this manner, theoperator ρij no longer appears as a generalized susceptibility and there is no basis to expect that it should be exactlydeterminable by an action–response problem. In what follows, the ‘Umov–Schoch’ definition is selected. For the case oflongitudinal waves it will successfully leads to stating well-defined action–response problems, whose solutions will allowthe computation of the susceptibilities.We note that for liquids the coefficient β0 is very small and γ −1, proportional to β2

0 ,can be set to zero in first approximation; the definitions in Sections 2.7 and 2.5 then give similar results for the operators.

2.8. Nonsingular EM recasting of longitudinal motions using Schoch’s identification

The singularities which make the susceptibilities non invertible, disappear when considering only the longitudinalmotions. These are described by the equations

ρ0∂v∂t

= −∇p +

4η3

+ ζ

∇(∇ · v) (37a)

1ρ0

∂ρ ′

∂t+ ∇ · v = 0 (37b)

14 This conception affects the impedance factors zijk in the relations, hij = zijkvk .


γχ0p =ρ ′

ρ0+ β0τ (37c)

ρ0cp∂τ

∂t= β0T0

∂p∂t

+ κ∇2τ . (37d)

Here, the displacement 3-vector u is still taken as an acoustic counterpart of the macroscopic electromagnetic 3-potentialA, but we use it only to construct the velocity v = ∂u/∂t and a scalar field b = −∇ · u.

The above equations are equivalent to the following set of equations

∂b∂t

= −vi;i (38a)

ρ0∂vi

∂t= −χ−1

0 b;i + ji (38b)

ji = χ−10 b;i − p;i +

4η3

+ ζ

vk;k;i (38c)

p − χ−10 b = −(γ − 1)p + χ−1

0 β0τ (38d)

ρ0cp∂τ

∂t= β0T0

∂p∂t

+ κτ;k;k. (38e)

Eq. (38a)maybe viewed as an equation–definitionwhich expresses the above definitions of v and b in terms ofu. The variableb then acquires themeaning of the condensation ρ ′/ρ0 only as a side effect of the other Eqs. (38); which thenmake Eq. (38a)expressing the equation of continuity. This subtlety should be borne in mind, as it helps explain the physical position of thedifferent equations in the foregoing electromagnetic recasting. By proceeding as before and now writing ji = ∂pi/∂t + m;ifor the decomposition (12), the following ‘Maxwellian’ acoustic equations are suggested.

field equations:

∂b∂t

+ ∇ · v = 0,∂d∂t

= −∇h (39)


d = ρv, h = χ−1b (40)where ρ and χ are two constitutive linear difference-kernels scalar operators in which the fluid physical longitudinal wavesproperties will be encoded. Here again the first equation (39) is viewed as a definition, which will become the equation ofcontinuity only on account of the other equations.

Then requiring that the acoustic Poynting vector ss = hv (41)

coincides with Schoch’s definition of the acoustic energy current density (20), it follows that the field h identically matchesthe thermodynamic excess pressure field p : h ≡ p. The nonlocal Maxwellian acoustic equations then take the form

∂b∂t

+ ∇ · v = 0,∂d∂t

= −∇p (42)

with

d(t, x) =

t

−∞

dt ′

dx′ρ(t − t ′, x − x′)v(t ′, x′) (43a)

p(t, x) =

t

−∞

dt ′

dx′χ−1(t − t ′, x − x′)b(t ′, x′). (43b)

Contrary to the case before, the equalization of (41) and (20) (instead of (19) and (20)) matches two scalars h and p: nosingularity then arises and the corresponding operators are now invertible.15

Passing over to the Fourier transform, it is easy to determine the Fourier amplitudes of the kernels. Using (37) and(42)–(43), gives

ρ(ω, k) = ρ(ω, k) = ρ0

1 +

4η3 + ζ

ρ0

k2

−iω

(44)

χ−1(ω, k) = χ−1(ω, k) = χ−10

1 −

γ − 1γ

κρ0cv

k2

−iω +κ

ρ0cvk2

. (45)

15 Convergence problems in original space still persist, however, because of the missing terms due to the too-narrow thermodynamic framework.


The density is now given by the previously obtained longitudinal core ρl, and the bulk modulus by the previously obtainedfunction χ−1. The operators ρ and χ−1 are well-formed regarding their kernel (Ker) which is nowmade of the null field. Thefact that the correct longitudinal Navier–Stokes–Fourier wave physics [32] is encoded in the expressions (44)–(45), can beeasily checked by writing the dispersion equation obtained by equating to zero the determinant of the transformed system(42)–(43)

ρ(ω, k)χ(ω, k)ω2= k2. (46)

Substituting in the above equation the expressions (44)–(45), we get back Kirchhoff–Langevin’s dispersion equation (3) forlongitudinal waves.

We note that, although the integrals defining the kernel functions in original space are still not convergent, it is nowconsistent to invert the well-formed operators in Fourier space. This suggests that the Fourier kernels of these operatorscould be directly determined in terms of the solution of some appropriate action–response problems. We show next, thatthis is indeed the case: subjecting the medium to the action of suitable source terms varying like ei(k·x−ωt), will permit theindependent determination of the Fourier kernels functions ρ(ω, k) and χ−1(ω, k).

2.9. The two independent, fundamental action–response procedures

We now state the fundamental two important action–response procedures, which will be generalized later on, inSection 4, to arrive at the wanted new nonlocal macroscopic theory. Let e = k−1k be the direction along which we study thesound propagation, and x = x · e be the coordinate along this direction. It will be shown that the two functions ρ(ω, k) andχ−1(ω, k) are related to the solutions of two independent action–response problems obtained by putting, respectively, afictitious harmonic pressure termP (t, x) = P0ei(ke·x−ωt) in the Navier–Stokes equation (37a), or the Fourier equation (37d).

Firstly, adding the potential bulk force f = −∇P = −ikP e to the right-hand side of Eq. (37a) belonging to the equationsystem (37a)–(37d), and writing the fields as

v(t, x) = v0ei(ke·x−ωt), ρ ′(t, x)/ρ0 = b(t, x) = b0ei(ke·x−ωt)

p(t, x) = p0ei(ke·x−ωt), τ (t, x) = τ0ei(ke·x−ωt) (47)

we can easily get the complex response amplitudes constants v0, b0, p0, τ0. We then observe that the same expression as in(44) for ρ(ω, k) is obtained through the equation

− ρ(ω, k)iωv0 = −ike(p0 + P0). (48)

In this problem, the response pressure amplitude p0 is added to the fictitious driving pressure amplitude P0 to represent asort of total effective pressure field amplitude h0. This establishes a direct relation between the Fourier coefficient ρ(ω, k)(44) of the operator density, and the response of the fluid subjected to an external harmonic bulk potential force.

Secondly, putting the bulk rate of heat supply Q = β0T0∂P/∂t = −iωβ0T0P in the right-hand side of Eq. (37d) belongingto the equation system (37a)–(37d), and writing the fields as before, we get the response amplitudes, v0, b′

0, p0, τ0—witha prime on the condensation for later convenience. We then observe that the same expression as in (45) for χ−1(ω, k) isobtained through the equation

p0 + P0 = χ−1(ω, k)(b′

0 + γχ0P0). (49)

In this problem, the response pressure amplitude p0 is again added to the fictitious driving pressure amplitude P0 torepresent a sort of total effective pressure field amplitude h0, and the term γχ0P0 is added to the response condensationamplitude b′

0, in order to represent a sort of total field amplitude b0, related to h0 by the second constitutive relation (40).The interpretation of the corresponding splitting of b0 in two terms is that b′

0 is the nonisothermal response part whereasγχ0P0 is the complementary isothermal part; γχ0 being the isothermal compressibility. This establishes in turn a directrelation between the Fourier coefficientχ−1(ω, k) (45) of the operator bulkmodulus, and the response of the fluid subjectedto an external harmonic bulk rate of heat supply.

Before generalizing these two fundamental action–response procedures to the new nonlocal macroscopic theory appliedto porous media, we will see how the macroscopic quantities are to be defined, and express the macroscopic Maxwellianequations including the field equations and constitutive relations. Finally, we shall also introduce the crucial notion ofmacroscopic ‘acoustic part of energy current density’.

3. Generalization to fluid-saturated rigid-framed porous media

Let us consider, now, that the viscothermal fluid pervading the connected network of pores of a rigid-framedhomogeneous porous material, is set to vibrate following an incident longitudinal small-amplitude sound wave. The aim isto describe the sound propagation in this complexmedium, at themacroscopic levelwhere the latter appears homogeneous.

As stated in general terms, this problem recalls that of Lorentz’s theory of electrons, related to linear electromagneticwave propagation in matter [43]. In the same way as Lorentz could not hope to follow in its course each electron, here we


do not intend to analyze the detail of the wave propagation at the pore level. Following Lorentz [43, p. 133], we remark thatit is often not the microlevel wavefield that can make itself felt in the experiments, but only the resultant effect producedby some macroscopic averaging. A macroscopic description of the sound propagation in the medium will be possible if wefix from the outset our attention not on the pore level irregularities, but on some relevant mean values. We proceed now toclarify it through some definitions.

3.1. Basic definitions

The porous medium occupies the whole space and is assumed to be macroscopically homogeneous in an ensemble-average sense. Ensemble-averaging is here, a more adapted averaging procedure than Lorentz’s volume-averagingprocedure [43, Section 112]—even when refined by using Russakoff’s method of convolution with a test function [44].16In fact, while the microstructures in the homogeneous media considered by Lorentz are the molecules themselves, whichcan be coarse-grained by spatial averaging in quite small volumes – smaller thanwavelengths in general – herewe deal withstructured media defined by a pore geometry which we allow to be drawn at a much larger scale: we intend to let open thepossibility that Helmholtz-resonator’s structures are present inside the medium. The wavelengths will not necessarily haveto be large with respect to the microstructure, and the material will become homogeneous only by ensemble averaging, ingeneral.17

As usual in the ensemble-average conception, we imagine that we are given infinitely many samplesω of the medium,18taken from a probability space�, the ensemble of which defines the homogeneousmacroscopicmedium. In each realizationω, themedium is composed of two regions: the void (pore) regionVf (ω)which is a connected region permeated by the fluid,and the complementary solid-phase region Vs(ω). The pore–wall region or solid–fluid interface is denoted by ∂V(ω). Thecharacteristic function of the pore region is defined by

I(x;ω) =

1, x ∈ Vf (ω)0, x ∈ Vs(ω).

(50)

The ensemble-average operation at position x is denoted by ⟨ ⟩(x). It gives the expectation value of the considered fieldat the given position. As such, for instance, ⟨I⟩(x) is a constant named the porosity φ, giving the probability that the positionx lies in the fluid, over an infinite number of realizations.

In our acoustic wave context, as we assume the solid motionless and thermally inert19 (see Eqs. (58)), all microscopicfields a(t, x;ω) defined in Vf (ω) and specifying the fluid motion, can be by convention extended to be zero in the solidVs(ω). We then define the macroscopic mean A(t, x) of the field a(t, x;ω), by setting A(x) = ⟨a⟩(x). With this convention,the realizationsω such that x ∈ Vf (ω), bring null terms to themean, so thatwemay think of ⟨a⟩ as statistically representativeof the value the field a has at position x in the fluid-phase, multiplied by the porosity.

With the mean operation ⟨ ⟩ and fields a(t, x;ω) defined in this manner, it is easy to see that the averaging operator ⟨ ⟩

and the operator ∇ commute as follows

∇⟨a⟩ = ∇⟨Ia⟩ = ⟨I∇a⟩ + ⟨a∇I⟩ = ⟨∇a⟩ + ⟨a∇I⟩. (51)

The commutator∇⟨a⟩−⟨∇a⟩ is in general nonzero because of the last term, wherein∇I is a distribution−nδ(x) supportedon ∂V(ω), δ is the Dirac delta, and n is the outward unit normal to the fluid region.

To avoid handling a distribution and write the commutation relation (51) in a more usable form, the average ⟨ ⟩

can be redefined to include also an integration over an infinitesimal volume around x. This integration is best made inRussakoff’s manner, by convolution with a smooth test function such as fL(x) =

1N e

−x2/L2 , normalized over the whole space:dx′ fL(x′

− x) = 1; and whose characteristic width L is taken to be indefinitely small. Thus we redefine the averagingoperator ⟨ ⟩ as

A(t, x) = ⟨a⟩(t, x) ≡

limL→0

dx′ fL(x′

− x)I(x′;ω)a(t, x′

;ω)

. (52)

The additional term ⟨a∇I⟩ in (51) now can be written explicitly:

⟨a∇I⟩ =

limL→0

∂V(ω)

dx′ fL(x′− x)n(x′

;ω)a(t, x′;ω)

. (53)

16 An obviously better smoothing procedure than Lorentz’s abrupt integration over an averaging sphere—see Jackson [45, Section 6.6].17 As a further stress of the importance of the ‘Gibbsian’ ensemble-average conception in a macroscopic theory, recall that the Navier–Stokes–Fouriersingularities observed in the electromagnetic reformulation, manifest the inadequacy of the equations when the wavelengths become comparable andsmaller than microscopic distances such as mean free paths. Thus it is evident that they could be removed only by using ensemble-averaged quantitiesand equations, not requiring any coarse-grained spatial integration.18 No confusion arises with frequency, due to the bold font.19 The last hypothesis is not essential and could be easily removed.


It can be underlined that the above ensemble-average conception, well-adapted to the precise introduction of statisticalnotions, performs the macroscopic averages at one single point of space: as such it does not entail any separation conditionon the range of wavelengths that can be treated in the macroscopic ensemble-average theory. We note also that, themacroscopic properties represented in the theory are a priori not properties of one realization, but statistical propertiesof the ensemble of realizations. Thus for example, when discussing the propagation constants of the medium, these willrefer to the propagation constant of ‘coherent waves’ in the sense this wording has in multiple-scattering theory. However,we are not always given an ensemble ofmedia and have often to dealwith a single sample of themedium. The possible use ofmacroscopic concepts and theories must then be carefully reconsidered. A special case is that of stationary random media.In that case, one single copy ω of the medium exhibits some properties of the ensemble, via ergodicity: for an arbitrarystochastic variable w ergodicity enables us to equate ensemble averages with volume averages

⟨w⟩ = limL→∞

dx′ fL(x′

− x)w(t, x′;ω) (54)

where in the left ⟨ ⟩ is the ensemble average, and in the right, the limL→∞ corresponds to integrating over the whole spacein a given realization.

Lorentz’s or Russakoff’s conception of the average then comes from the fact that, in practice, the sample of the mediumwe deal with, is homogenized to some extent in the above sense, as soon as the dimension L reaches some threshold orcoarse graining value Lc . Then ignoring the label ω, the average of a wavefield variable a(t, x) in the fluid, can be definedregarding (54) with L set to the coarse graining value Lc

A(t, x) = ⟨a⟩(t, x) =

dx′I(x′)a(t, x′)fLc (x − x′). (55)

In that case,macroscopic homogeneity is never achieved exactly. Themacroscopic theory only specifies the behavior of somevolume-averaged fields subject to the condition that the wavelengths remain sufficiently large compared to the averagingscale Lc . Notice that, using (55), the commutation relation writes by direct integration by parts as

⟨∇a⟩ = ∇⟨a⟩ +

∂V

dx′a(t, x′)n(x′)fLc (x − x′). (56)

Finally, the case of periodicmedia is of special importance.We consider for simplicity that the propagation occurs along asymmetry axis. Themacroscopic theory in its ensemble-average formulation, will describe the propagation in the ensemble� of media obtained by random translation (without rotation) of a given reference sampleω0 of the periodic medium. Someof the ensemble-average ⟨ ⟩ properties of the space� – e.g. the propagation constants and characteristic impedances – willthen be exactly computable by making cell volume-averages in the single realization ω0. We will return to this in Section 5.

Nowwith these averaging concepts in mind, and given a small amplitude wave perturbation to the ambient equilibriumin the fluid, we proceed to establish the macroscopic equations governing wave propagation of macroscopic averagedwavefield quantities.

3.2. Pore-level equations

At the pore-level, the linearized fluid-mechanics equations describing the fluid motion are written as

ρ0∂v∂t

= −∇p − η∇ × (∇ × v) +

4η3

+ ζ

∇(∇ · v) (57a)

∂b∂t

+ ∇ · v = 0 (57b)

γχ0p = b + β0τ (57c)

ρ0cp∂τ

∂t= β0T0

∂p∂t

+ κ∇2τ (57d)

in Vf (ω), and

v = 0, τ = 0 (58)on ∂V(ω). Again, we may view Eq. (57b) as a definition, which becomes the equation of continuity on account of the otherEqs. (57).

Comparing to the preceding longitudinal equations in the free fluid (37a)–(37d), the rotational viscous term is nowrequired to be added in (57a). Let us make very clear the reason of this addition. In reality the medium is bounded andthe origin of the wave field is a source placed in the external free fluid, sending longitudinal acoustic waves,20 which satisfy

20 Consistent with the introductory remarks in Section 2 and while devoid of practical significance, the incidence of entropic waves could be consideredas well; the incidence of viscous shear waves on the other hand, is excluded to avoid singularities in the forthcoming electromagnetic recasting: the theoryto be developed will not tell us what happens in this case—also devoid, however, of practical significance.


∇ × (∇ × v) = 0. Now, as soon as solid inclusions are present, non zero values of ∇ × (∇ × v) appear in the permeatingfluid, as a result of the conversion of longitudinal normal modes into shear normal modes at the pore walls, regarding theno-slip condition (58). Thus the rotational viscous term is required, and in the end, the effects it produces will be accountedfor in the macroscopic theory.

3.3. Macroscopic equations

Given the above pore-level equations, we seek themacroscopic equations governing wave propagation of the ensemble-averaged quantities

V ≡ ⟨v⟩ , and B ≡ ⟨b⟩ . (59)

Since the velocity vanishes at the pore walls, the following direct commutation relation always holds true: ⟨∇ · v⟩ =

∇ · ⟨v⟩ = ∇ · V . As a result, Eq. (57b) is immediately translated at the macroscopic level

∂B∂t

+ ∇ · V = 0. (60)

The electromagnetic analogy then suggests that the system of macroscopic equations can be carried through byintroducing new Maxwellian density fields H and D, and also linear difference-kernels operators ρ and χ−1, such that asecond field equation holds true

∂D∂t

= −∇H (61)

with constitutive fully nonlocal linear relations

D = ρV , H = χ−1B. (62)

As we have seen in Section 2.8, such form of equations, with the scalar H and scalar ρ, is suitable to treat nonlocalpropagation of longitudinal waves in an isotropic medium. Here, we assume in fact, either isotropy of our medium and theabsence of macroscopic shear waves (see footnote 20), or else when the medium is not isotropic, macroscopic longitudinalwave propagation along a symmetry axis—say an axis x, set in a direction e.21 Accounting for the time homogeneity and thematerial macroscopic homogeneity, the nonlocal relations (62) are written, in the former isotropic case

D(t, x) =

t

−∞

dt ′

dx′ρ(t − t ′, x − x′)V (t ′, x′) (63a)

H(t, x) =

t

−∞

dt ′

dx′χ−1(t − t ′, x − x′)B(t ′, x′) (63b)

with scalar kernels ρ(t, x) and χ−1(t, x) whose Fourier amplitudes verify ρ(ω, k) = ρ(ω, k), and χ−1(ω, k) = χ−1(ω, k).In the latter 1D case of propagation along a symmetry axis e, we have D = De and V = Ve, then the constitutive relationsbecome

D(t, x) =

t

−∞

dt ′

dx′ρ(t − t ′, x − x′)V (t ′, x′) (64a)

H(t, x) =

t

−∞

dt ′

dx′χ−1(t − t ′, x − x′)B(t ′, x′) (64b)

with scalar kernels ρ(t, x) and χ−1(t, x) whose Fourier amplitudes are ρ(ω, k), and χ−1(ω, k). The integrations over timet ′ in one hand, and over space coordinate x′ or x′ in another hand, express the so-called temporal dispersion and spatialdispersion effects, respectively. We need now to remove the ambiguity of the description by identifying the macroscopicfield H .

3.4. Identification of the field H via the notion of ‘acoustic part of energy current density’

The proper identification results directly from writing the following macroscopic versions of the two expressions (41)and (20) used in the viscothermal fluid for ‘the acoustic part of energy current density’ S = HV (Heaviside–Poynting’s or

21 Without any essential change, the general case of propagation in anisotropic media would be accounted for by introducing a tensor Hij related to thescalar B via a symmetric operator χ−1

ij . The theory, as formulated here, allows a determination of the properties of Bloch modes for propagation in periodicgeometries along a symmetry axis—see Section 5. The more general formulation would allow a determination of the properties of modes for the case ofpropagation in arbitrary oblique directions.


Umov’s form) and S = ⟨pv⟩ (Schoch’s form). Equating the two expressions,H must be the field satisfying the thermodynamicrelation–definition

H⟨v⟩ ≡ ⟨pv⟩. (65)

In these equations, ⟨v⟩ is parallel to ⟨pv⟩, since the propagation is considered either in an isotropic medium, or along asymmetry axis.

This identification of an effective macroscopic pressure field H different from the usual fluid-volume-averaged meanpressure p = φ−1

⟨p⟩, appears very natural. Generalized to the case of a bounded open material, it yields a field H that iscontinuous at the macroscopic interface of the material.22 This follows from the continuity of the normal component ofthe field V , required by mass flow conservation, and the continuity of the normal component of ‘the acoustic part of theenergy current density’, that may be supposed to hold true provided no resistive surface layer exists at the boundary ofthe material. Thus, in the present electromagnetic–acoustic analogy, the continuity of the normal component of velocity Vreplaces the continuity of the tangential components of the field E , and the continuity of the scalarH replaces the continuityof the tangential components of H [18, p. 272]. Previously we mentioned that the customary electromagnetic replacementHij = µ−1

0 Bij would have disadvantages for the description of impedances factors if the magnetic effects were not small.Indeed, if we were to use this point of view here, we would set H = χ−1

0 B and this field would no longer be continuous ingeneral, because the intervening bulk modulus would not coincide with the adiabatic one. Most importantly we expect, forthe reasons explained in Section 2.4, that the definition (65) is so as to render possible the determination of the operators ρand χ−1, by means of the solution of two independent action–response problems.

4. Clarification of the relationship between the constitutive operators and the microgeometry

We are now in a position to determine the right action–response problems and upscaling procedures, by generalizingthe two action–response procedures that have been established in Section 2.9 for the longitudinal motions in viscothermalfluids, to the present case of longitudinal macroscopic motions in the fluid-saturated material. Given the fixed direction eof the macroscopic spatial variations, these procedures lead us to suggest that the above Fourier coefficients ρ(ω, k) andχ−1(ω, k) are directly related to the macroscopic response of the permeating fluid subject to a harmonic fictitious pressuretermP (t, x) = P0ei(ke·x−ωt) added to the pressure, either in the Navier–Stokes equation (57a), or the Fourier equation (57d).

Thus to determine the kernel ρ(ω, k)we first consider solving the action–response problem defined by Eqs. (57) and (58)with external bulk force

f = −∇P = −ikeP0ei(ke·x−ωt) (66)

inserted in the momentum balance equation (57a). The unique solution to this system of equations for the fields v, b, p, τ ,take the form

v(t, x;ω) = v0(ω, k, x;ω)ei(ke·x−ωt), b(t, x;ω) = b0(ω, k, x;ω)ei(ke·x−ωt),

p(t, x;ω) = p0(ω, k, x;ω)ei(ke·x−ωt), τ (t, x;ω) = τ0(ω, k, x;ω)ei(ke·x−ωt). (67)

The response amplitudes v0, b0, p0, and τ0 are bounded functions of x, which are uniquely determined by themicrogeometry(the realization ω) and the imposed temporal and spatial frequencies ω and k.

The above problem, once solved, we can extract of the response pressure p(t, x;ω) = p0(ω, k, x;ω)ei(ke·x−ωt), itsmacroscopic part, denoted by P(t, x) = P0(ω, k)ei(ke·x−ωt) whose amplitude P0(ω, k) is determined through the equation

P0(ω, k) =⟨p0(ω, k, x;ω)v0(ω, k, x;ω)⟩ .e

⟨v0(ω, k, x;ω)⟩ .e. (68)

This expression comes from the identification P⟨v⟩ = ⟨pv⟩, which has been inspired by the fundamental thermodynamicrelation–definition (65). Then using the relation−iωρ(ω, k) ⟨v0⟩ = −ik(P0(ω, k)+P0)e, which follows from (48), gives thenonlocal equivalent-fluid density ρ(ω, k)

ρ(ω, k) =k(P0(ω, k) + P0)

ω ⟨v0(ω, k, x;ω)⟩ · e. (69)

At this point, we see that the amplitude fields p0(ω, k, x;ω) and v0(ω, k, x;ω) are needed to be known in the wholecollection� of realizations ω in order to determine frommicrogeometry the effective density of the fluid-saturated porous

22 Strictly speaking the macroscopic theory developed here does not apply to bounded materials; but the sketched principle of the continuity of the fieldH is expected to be the same, in the generalized macroscopic theory to be developed for bounded materials.


medium. These amplitudes are the solutions of the following set of equations

−iωρ0v0 =

4η3

+ ζ

(∇ + ike) (∇ · v0 + ike · v0) − η(∇ + ike) × (∇ + ike) × v0 − (∇ + ike)p0 − ikeP0 (70a)

−iωb0 = −∇ · v0 − ike · v0 (70b)γχ0p0 = b0 + β0τ0 (70c)

−ρ0cpiωτ0 = −β0T0iωp0 + κ(∇ + ike) · (∇ + ike)τ0 (70d)

in Vf (ω), and

v0 = 0, τ0 = 0, on ∂V(ω). (71)

Theprocedure to determine the kernelχ−1(ω, k) is quite similar.We consider again, initially, solving the action–responseproblem defined by Eqs. (57) and (58) with, this time, bulk rate of heat supply

Q = β0T0∂P

∂t= −β0T0iωP0ei(ke·x−ωt) (72)

inserted in the energy balance equation (57d). The solutions to the above problem take the same form as specified beforethrough Eqs. (67)—with now, like in Section 2, a prime on the condensation for later convenience. As previously done, weextract of the response pressure p(t, x;ω), its macroscopic part P(t, x) = P0(ω, k)ei(ke·x−ωt) whose amplitude P0(ω, k)is determined by the Eq. (68), expressing an identification P⟨v⟩ = ⟨pv⟩ inspired by the fundamental thermodynamicrelation–definition (65). Then generalizing the free fluid equation (49) to the present case, results in P0(ω, k) + P0 =

χ−1(ω, k)b′

0(ω, k, x;ω)+ φγχ0P0

, where the factor of porosity is inserted in the last term to account for the fact that

⟨P0⟩ = φP0. As before, the interpretation of the splitting of the total amplitude B0 in two terms is thatb′

0(ω, k, x;ω)is

the nonisothermal response part, while φγχ0P0 is the isothermal part. That gives rise to nonlocal equivalent-fluid bulkmodulus χ−1(ω, k)

χ−1(ω, k) =P0(ω, k) + P0

b′

0(ω, k, x;ω)+ φγχ0P0

. (73)

Obviously, we need to know the amplitude fields p0(ω, k, x;ω), v0(ω, k, x;ω) and b′

0(ω, k, x;ω) in order to determinethe effective bulk modulus of the fluid-saturated porous medium. These amplitudes are the solutions of the following set ofequations

−ρ0iωv0 =

4η3

+ ζ

(∇ + ike) (∇ · v0 + ike · v0) − (∇ + ike)p0 − η(∇ + ike) × (∇ + ike) × v0 (74a)

−iωb′

0 = −∇ · v0 − ike · v0 (74b)

γχ0p0 = b′

0 + β0τ0 (74c)

−ρ0cpiωτ0 = −β0T0iωp0 + κ(∇ + ike) · (∇ + ike)τ0 − β0T0iωP0 (74d)

in Vf (ω), and

v0 = 0, τ0 = 0, on ∂V(ω). (75)

These two new upscaling procedures, allowing to determine nonlocal density and nonlocal bulk modulus Fouriercoefficients ρ(ω, k) and χ−1(ω, k) from microgeometry, combined with the Maxwellian acoustic equations (60)–(62),represent the essential results of this paper. They wholly express the proposed new nonlocal theory.

5. Characteristic wavenumbers and impedances, and special case of periodic media

The characteristic feature of the present nonlocal theory is that, without making any important simplification,23 itaccounts for both temporal and spatial dispersion.Within the classical local equivalent-fluid theory which only accounts fortemporal dispersion, for a given frequency ω there is only one single normal mode that can propagate in the given positivex direction, with fields varying as ei(qe·x−ωt): to this single mode is associated a wavenumber q(ω) verifying the relationq2 = ρ(ω)χ(ω)ω2, ℑ(q) > 0, and a characteristic impedance Z(ω) =

ρ(ω)/[ω2χ(ω)] relating the mean pressure P in

the fluid (P = ⟨p⟩/φ) and the macroscopic velocity, where ρ(ω) = ρ0α(ω)/φ and χ(ω) = χ0φβ(ω) are the local density

23 The essential simplification which traces back to the use of a singular Navier–Stokes–Fourier physics, is the disregard of motions that would originatefrom external shear actions.


and compressibility functions—see e.g. [12] for the definition of the functions α(ω) and β(ω). Here, since we fully take intoaccount the spatial dispersion, several normal mode solutions might exist, with q a solution of the dispersion equation

q2 = ρ(ω, q)χ(ω, q)ω2. (76)

We notice that a solution q at given realω is complex because of losses. In the procedures of Section 4 we considered a priorireal values of ω and k, but for the reasons discussed in Agranovich and Ginsburg [24, pp. 7–8], exactly the same proceduresdetermine the nonlocal functions ρ and χ at complex values of ω and/or k as well. With these functions hence defined forthe complex arguments, a simple Newton scheme is very powerful in general to determine the complex solutions q. If welabel by l = 1, 2, . . . , the different solutions ql(ω), ℑ(ql) > 0, to Eq. (76), the corresponding wave impedances Zl = H/Vfor propagation in the direction +x are Zl(ω) =

ρ(ω, ql(ω))/[ω2χ(ω, ql(ω))]. It is an interesting by-product of the

present nonlocal theory, that the characteristic impedances are unambiguously fixed through the unequivocal, fundamentalUmov–Schoch’s thermodynamic identification (65) of the field H which plays the role of an effective macroscopic pressurein the fluid.

The case of periodic geometry is especially important and deserves a special discussion here. Let us sketch, how thepresent theory will predict the correct Bloch wave properties – propagation constants q and impedances Z – of modespropagating along a symmetry axis. Starting with a given periodic geometry ω0 possessing a symmetry axis x, along whichthe irreducible period is Lx, the procedure to apply the macroscopic theory developed here, is to construct an ensemble� including the realizations ω, each of which is obtained by a random translation of one reference sample ω0, in bothlongitudinal and transversal directions. Now, because of the periodicity, when solving the action–response problems inSection 4 and seeking the solution fields in the form (67), one should be careful that the unicity is lost: the requirementthat the amplitudes be bounded fields, is no longer sufficient to determine a unique solution. The requirement that they arebounded fields can be satisfied by amplitude fields which are periodic with period Lx, or 2Lx, and so on. Thus an arbitraryinteger n = 1, 2, . . . specifying the period nLx, becomes a parameter of the solution.

Choosing a particular n, a unique solution is again obtained. It can then be seen immediately that, taking the ensemble-average in the procedures of Section 4will nowmean taking a n-cell average in the reference copyω0 (a volume average overa cell of dimension nLx in direction x and dimension of irreducible period in the transversal directions). This means that, ina given periodic geometry, we can compute the properties of Bloch modes constructed with n periods Lx as follows: solvingin the given periodic geometry the action–response problems, selecting the amplitude solutions having the irreducibleperiod nLx, plug the solutions in the appropriate equations to get the functions ρn(ω, k) andχ−1

n (ω, k)—being aware that theaverages ⟨ ⟩ are now the volume n-cell averages, then solving the dispersion equation (76) to find the possiblewavenumbers(qn)l(ω), and finally compute the impedances (Zn)l(ω) =

ρn(ω, (qn)l(ω))/[ω2χn(ω, (qn)l(ω))].

The new theory clearly shows that, in general, the nonlocal functions ρ(ω, k) and χ(ω, k) do not systematically reduceto the local ones at longwavelengths. The asymptotic relations of the type ρ(ω) = limk→0 ρ(ω, k), χ(ω) = limk→0 χ(ω, k),which are in line with the perturbative philosophy of conventional homogenization theory and are also often assumedwithout proof for the electric permittivity of nonferromagnetic media [24, p. 5], will not hold true in the vicinity ofHelmholtz’s resonances.

6. Conclusion

Following a deep electromagnetic–acoustic analogy andmaking use of an overlooked thermodynamic concept of acousticpart of the energy current density, we have developed a new nonlocal Maxwellian linear macroscopic theory of soundpropagation in rigid-framed porous media. Taking into account for the first time spatial dispersion without assuming thatit is a small second order correction, this new theory remains valid in its ensemble-average version, for the full range ofgeometries and frequencies or wavelengths. It clarifies the micro–macro relationship for the propagation of longitudinalwaves, within Navier–Stokes–Fourier physics, superseding the previous perturbative descriptions given by the traditionaltwo-scale asymptotic homogenization process and its variants. It will allow consideration of resonant metamaterials. Inrandom media, it will describe the propagation of the so-called coherent waves; in periodic media it will provide thepropagation constants and define the impedances of the so-called Bloch modes.

The theory, best defined when using ensemble averaging, is wholly expressed by the macroscopic acoustic ‘Maxwell’field and constitutive equations (60)–(61) and (62) in Section 3, and the action–response procedures in Section 4 whichdetermine the Fourier amplitudes of the two constitutive difference-kernels operators. These procedures are the following.To compute the Fourier amplitude ρ(ω, k), we subject each realization of themedium to a time and space harmonic ei(kx−ωt)

longitudinal external bulk force. We solve the Eqs. (70)–(71) to find the field responses. Plugging the pressure and velocitysolutions in Eqs. (68) and (69) and taking the ensemble averages over realizations, we get the wanted nonlocal density. Tocompute the Fourier amplitude χ(ω, k), we subject each realization to a time and space harmonic ei(kx−ωt) external bulkrate of heat supply. We solve the Eqs. (74)–(75) to find the field responses. Plugging the pressure, velocity and condensationsolutions in Eqs. (68) and (73) and taking the ensemble averages, we get the wanted nonlocal compressibility.

The remarkable exactness of these procedures will be illustrated in forthcoming papers: they will be shown to yield theprecise properties of Kirchhoff’s modes in a circular tube filled with a viscothermal fluid [46], the correct wavenumbers inperiodic square arrays of solid cylinders embedded in the fluid [47], and the correct metamaterial behavior of daisy-chained


periodic arrays of Helmholtz’s resonators filled with the fluid [16]. They seem to be exact mathematical results which havebeen deduced here by physical reasoning.

In the presentation of the general principle of the nonlocal full solution we have assumed homogeneity, isotropy orpropagation along a principal axis, and in periodic media, propagation along a symmetry axis. A next step will be to relaxthese symmetry assumptions unessential for the general principle. This will allow solving in an exact manner the problemof reflection–transmission at a finite anisotropic slab. The generalization of the theory to the poroelastic case could also beconsidered, leading to a complete solution superseding that of the Biot theory.

Finally, we note that, while it has been possible to develop an incomplete but usable conception of Maxwell acousticdensity fields using Schoch’s thermodynamic identification (65) of ‘the acoustic part of the energy current density’,a corresponding notion is missing in electromagnetics. We believe that a clarification of the notions of energy andelectromagnetic momentum in matter (that would provide a satisfactory end to the old Abraham–Minkowski controversy),might await the introduction of corresponding, so far elusive, thermodynamic concepts and quantities in electromagnetics.

Acknowledgments

The authors are grateful to one anonymous referee for a remark on averaging, and to the Editor-in-Chief, Andrew Norris,for his help and useful suggestions.

References

[1] A. Bensoussan, J.L. Lions, G.C. Papanicolaou, Asymptotic Analysis for Periodic Structure, North-Holland, Amsterdam, 1978.[2] E. Sanchez Palencia, Nonhomogeneous Media and Vibration Theory, in: Lectures Notes in Physics, vol. 127, Springer, Berlin, 1980.[3] J.L. Auriault, C. Boutin, C. Geindreau, Homogenization of Coupled Phenomena in Heterogenous Media, ISTE and Wiley, 2009.[4] T. Levy, Propagation of waves in a fluid-saturated porous elastic solid, Int. J. Engng. Sci. 17 (1979) 1005–1014.[5] J.L. Auriault, Dynamic behavior of a porous medium saturated by a newtonian fluid, Int. J. Engng. Sci. 18 (1980) 775–785.[6] M.Y. Zhou, P. Sheng, First principles calculations of dynamic permeability in porous media, Phys. Rev. B 39 (1989) 12027–12039.[7] D.M.J. Smeulders, R.L.G.M. Eggels, M.E.H. van Dongen, Dynamic permeability: reformulation of theory and new experimental and numerical data,

J. Fluid Mech. 245 (1992) 211–227.[8] R. Burridge, J.B. Keller, Poroelasticity equations derived from microstructure, J. Acoust. Soc. Am. 70 (1981) 1140–1146.[9] A.N. Norris, On the viscodynamic operator in Biot’s equations of poroelasticity, J. Wave Mat. Interact. 1 (1986) 365–380.

[10] D.L. Johnson, J. Koplik, R. Dashen, Theory of dynamic permeability and tortuosity in fluid-saturated porous media, J. Fluid Mech. 176 (1987) 379–402.[11] D. Lafarge, P. Lemarinier, J.F. Allard, V. Tarnow, Dynamic compressibility of air in porous structures at audible frequencies, J. Acoust. Soc. Am. 102

(1997) 1995–2006.[12] J. Kergomard, D. Lafarge, J. Gilbert, Transients in porous media: exact and modelled time-domain Green’s functions, Acta Acust. Acust. (2013)

(in press).[13] M.A. Biot, The theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low-frequency range, II. Higher frequency range, J. Acoust.

Soc. Am. 28 (1956) 168–191.[14] J.F. Allard, N. Atalla, Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, second ed., John Wiley & Sons, 2009.[15] P. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, Strong spatial dispersion in wire media in the very large

wavelength limit, Phys. Rev. B 67 (2003) 113103–113107.[16] N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, X. Zhang, Ultrasonic metamaterials with negative modulus, Nat. Mater. 5 (2006) 452–456.[17] M. Avellaneda, S. Torquato, Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porousmedia, Phys.

Fluids A 3 (1991) 2529–2540.[18] L.D. Landau, E. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1960.[19] A. Schoch, Remarks on the concept of acoustic energy, Acustica 3 (1953) 181–184.[20] R.V. Craster, J. Kaplunov, A.V. Pichugin, High-frequency homogenization for periodic media, Proc. R. Soc. A 466 (2010) 2341–2362.[21] R.V. Craster, J. Kaplunov, E. Nolde, S. Guenneau, Bloch dispersion and high frequency homogenization for separable doubly-periodic structures, Wave

Motion 49 (2012) 333–346.[22] C. Boutin, A. Rallu, S. Hans, Large scale modulation of high frequency acoustic waves in periodic porous media, J. Acoust. Soc. Am. 132 (2012)

3622–3636.[23] J.C. Maxwell, A Dynamical Theory of the Electromagnetic Field, Thomas F. Torrance (Ed.), Wipf and Stock Publishers, Eugene, Oregon, 1996.[24] V.M. Agranovich, V.L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons, Interscience Publishers, London, 1966.[25] P.S. Epstein, R.R. Carhart, The absorption of sound in suspensions and emulsions. I. Water fog in air, J. Acoust. Soc. Am. 25 (1952) 553–565.[26] J.C. Maxwell, On the dynamical theory of gases, Philos. Trans. R. Soc. Lond. 157 (1867) 49–88.[27] J.C. Maxwell, On stresses in rarefied gases resulting from inequalities of temperature, Philos. Trans. R. Soc. Lond. 170 (1879) 231–262.[28] I.M. de Schepper, E.G.D. Cohen, Very-short wavelength collective modes in fluids, J. Stat. Phys. 27 (1982) 223–281.[29] C. Truesdell, Precise theory of the absorption and dispersion of forced plane infinitesimal waves according to the Navier–Stokes equations, J. Ration.

Mech. Anal. 2 (1953) 643–741.[30] J.S. Rayleigh, The Theory of Sound, Vol. II, Macmillan, London, 1877. Reprinted: Cambridge University Press, 2011.[31] L.D. Landau, E. Lifshitz, Fluid Mechanics, Pergamon Press, 1987.[32] J.D. Foch, G.W. Ford, The dispersion of sound in monoatomic gases, in: J. de Boer, G.E. Uhlenbeck (Eds.), Studies in Statistical Mechanics V, North-

Holland, Amsterdam, 1970.[33] J.R. Willis, Effective constitutive relations for waves in composites and metamaterials, Proc. R. Soc. A 467 (2011) 1865–1879.[34] L.D. Landau, E. Lifshitz, Theory of Elasticity, Pergamon Press, London, 1959.[35] J. Frenkel, Kinetic Theory of Liquids, Dover Publications, 1955.[36] D. Bolmatov, V.V. Brazhkin, K. Trachenko, The phonon theory of liquid thermodynamics, Sci. Rep. 2 (2012) 421.[37] H.A. Lorentz, The fundamental equations for electromagnetic phenomena in ponderable bodies, deduced from theory of electrons, Proc. Roy. Acad.

Amsterdam 5 (1902) 254–266.[38] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, 1972.[39] G. Fournet, Electromagnétisme à partir des Equations Locales, Masson, Paris, 1985.[40] L.D. Landau, E. Lifshitz, Statistical Physics, Pergamon Press, London, 1959.[41] D.B. Melrose, R.C. McPhedran, Electromagnetic Processes in Dispersive Media—A Treatment Based on the Dielectric Tensor, Cambridge University

Press, New York, 1991.[42] P. Epstein, Textbook of Thermodynamics, John Wiley & Sons, Inc., New York, 1937.


[43] H.A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat, second ed., B.G. Teubner, Leipzig, 1916.[44] G. Russakoff, A derivation of the macroscopic Maxwell equations, Amer. J. Phys. 38 (1970) 1188–1195.[45] J.D. Jackson, Classical Electrodynamics, third ed., John Wiley & Sons, 1999.[46] G. Kirchhoff, Über des Einfluss der Wärmeleitung in einem Gase auf die Schallbewegung, Ann. Phys. Chem. (Leipzig) 134 (1868) 177–193; English

translation: On the influence of thermal conduction in a gas on sound propagation, in: R.B. Lindsay (Ed.), Physical Acoustics, in: Benchmark Papers inAcoustics, vol. 4, Dowden, Hutchinson and Ross, Stroudsburg, PA, 1974, pp. 7–19.

[47] A. Duclos, D. Lafarge, V. Pagneux, Transmission of acoustic waves through 2D phononic crystal: visco-thermal and multiple scattering effects, Eur.Phys. J. Appl. Phys. 45 (2009) 11302–11306.

nonlocal maxwellian theory of sound propagation in fluid-saturated rigid-framed porous media

Documents