Generalized equation for transient-wave propagation in continuous inhomogeneous rigid-frame porous materials at low frequencies

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Generalized equation for transient-wave propagation in continuous inhomogeneous rigid-frame porous materials

at low frequencies

Mohamed Fellah, Zine El Abiddine Fellah, Erick Ogam, F. G. Mitri, Claude Dépollier

To cite this version:

Mohamed Fellah, Zine El Abiddine Fellah, Erick Ogam, F. G. Mitri, Claude Dépollier. Generalized

equation for transient-wave propagation in continuous inhomogeneous rigid-frame porous materials at

low frequencies. Journal of the Acoustical Society of America, Acoustical Society of America, 2013,

134 (6), pp.4642. �10.1121/1.4824838�. �hal-00881886�

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ontinuous inhomogeneous rigid-frame porous materials

at low frequenies.

M. Fellah

Laboratoire de Physique Théorique,Faulté de Physique, USTHB, BP 32El Alia,Bab Ezzouar

16111, Algérie.

Z.E.AFellah,E. Ogam

LMA, CNRS,UPR7051, Aix-MarseilleUniv,Centrale Marseille, F-13402MarseilleCedex 20,

Frane.

F.G. Mitri

Los Alamos National Laboratory, MPA-11, Sensors Eletrohemial Devies, Aoustis Sensors

Tehnology Team,MS D429, Los Alamos,NM87545, USA.

C. Depollier

LUNAM Universite du Maine. UMR CNRS 6613 Laboratoire d'Aoustique de l'Universite du

Maine UFR STSAvenue O. Messiaen 72085 Le MansCEDEX 09 Frane.

Submitted topubliation intheJournal of Aoustial Soietyof Ameria.

1

running title : inhomogeneous porous material

1. SpeialissueonAoustisofPorousMedia

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This paper provides a temporal model for the propagation of transient aousti waves in

ontinuousinhomogeneousisotropiporousmaterialhavingarigidframeatlowfrequenyrange.

A temporal equivalent uid model in whih the aousti wave propagates only in the uid

saturating the material, is onsidered. In this model, the inertial eets are desribed by the

inhomogeneous inertial fator [A.N. Norris., J.Wave Mat. Interat. 1365 (1986)℄. The visous

and thermal losses of the medium are desribed by two inhomogeneous suseptibility kernels

whih depend onthe visous andthermal permeabilities .The mediumis one dimensional and

its physial parameters (porosity,inertial fator, visous andthermal permeabilities)aredepth

dependent. A generalized wave propagation equation inontinuous inhomogeneous material is

establishedand disussed.

(4)

The propagationofsoundinuid-saturatedporous mediawithrigid solidframesisof great

interestforawiderangeofindustrialappliations.Withairastheporeuid

1

appliationsanbe

found in noise ontrol, nondestrutive material haraterization, thermoaoustially ontrolled

heattransfer, et.

Theaoustipropagationinhomogeneousporousmaterialshavingrigidframehasbeenwell

studied,dierentmethodsandtehniquesweredevelopedinfrequeny

1 − 5

andtimedomains

6 − 12

for the aousti haraterization. All these tehniques are valid only for homogeneous porous

materials, inwhih,theirphysialparameters areonstant insidetheporousmedium.However,

inthegeneral ase, theporousmedia areinhomogeneous

13 − 15

and theirphysial propertiesare

loallyonstants,i.e.theyareonstantintheelementary volumeofhomogenization

13

,butthey

mayvaryfrompointtopointintheporousmedium.Forthisgeneralase,agoodunderstanding

of the aousti propagation is neessary for developing a new methods of haraterization. A

generalized hyperboli frational equation for transient wave propagation in inhomogeneous

rigid-frame porous materials has been established in the asymptoti domain (high frequeny

range)

15

,but not inthe visous domain(low frequenyrange), inwhih anothersetof physial

parameters (inertial fator, visous and thermal permeabilities) intervene in the propagation.

Thestatithermalpermeability

16 k ₀ ^′

ôf^the ^porous^material îsâ^geometrial ^parameterêqual^to

theinverse trapping onstant of the solidframe

17

.In the desription of thethermal exhanges

between theframeand the saturating uid,the statithermal permeabilityplays arole similar

to the visous permeability in the desription of the visous fores. The inertial fator

18 α ₀

orrespondstothe lowfrequenyapproximationofthedynamitortuosity

1,6

givenbyNorris

18

;

α ₀ = ^< ^v ⁽ ^r ⁾

²

^>

< v ( r )>

²

,

^where

< v ( r ) >

îs^theâverage ^veloityôf^the^visous ûid^for ^diret ûrrent ôw

withinavolumeelement,smallompared totherelevantwavelength,butlarge omparedtothe

individualgrains/pores of the solid.

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is an advantage of the time-domain method

6 − 11,14,15,19 − 22

that the results are immediate and

diret.Theattrationofatime domainbasedapproahisthatanalysisisnaturally bounded by

the nite duration of aousti pressures and it is onsequently the most appropriate approah

fortransientsignals.However,for wave propagationgeneratedbytimeharmoniinidentwaves

and soures(monohromati waves), thefrequeny analysisismore appropriate

1 − 5

.

ThisworkfollowstheinvestigationpreviouslydoneinRefs.6and15,inwhihatime-domain

approahwasdevelopedandageneralizedhyperbolifrationalequationofpropagationhasbeen

establishedintheasymptoti domain(highfrequeny range).Here, ageneral expressionforthe

equationofwavepropagationinontinuousinhomogeneousporousmediumisderivedatvisous

domain (low frequenyrange) .

The outline of this paper is as follows. Setion II shows the equivalent uid model, the

relaxation funtionsdesribing theinertial, visous and thermal interations between uid and

struturearerealled.In thissetion, theonnetionbetween thetemporaloperatorsand wave

propagationinrigidhomogeneousporousmediainthelowfrequenyrangeisestablished.Finally,

in Setion III the analytial derivation of the general propagation equation is given in time

domain.The dierent termsof this equationaredisussed.

II. THE EQUIVALENT FLUID MODEL

Inairsaturatedporousmedia,thestrutureisassumedtobemotionless:theaoustiwaves

travel only in the uid lling the pores. The wave propagation is desribed by the equivalent

uid model whih is a partiular ase of the Biot's theory

23

. In this model, the interations

betweentheuidand thestruturearetaken intoaount intwofrequeny dependent response

fatorswhih arethe generalized suseptibilities :the dynamitortuosity ofthemedium

α(ω) ³

andthedynamiompressibilityoftheairinluded inthemedium

β(ω) ^1,16

^.^These^two^response

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fatorsareomplexfuntionswhihheavilydependonthefrequeny

f = ω/2π

^,

ω

^is^the^angular

frequeny. These funtions represent the deviation from the behavior of the uid in the free

spae asthefrequenyinreases. Theirtheoretial expressionsare given byJohnson

et al ³

^,^and

Allard

1

and Lafarge

et al ¹⁶

^:

α(ω) = α ∞ 1 + φ σ iωα ∞ ρ

s

1 + i 4α ² ∞ ηρω σ ² Λ ² φ ²

!

,

⁽¹⁾

β(ω) = γ − (γ − 1)



1 + ηφ iωρk ^′ ₀ P _r

s

1 + i 4k ₀ ^′ ² ρωP _r ηφ ² Λ ^′ ²





− 1

,

⁽²⁾

where

i ² = − 1

^,

γ

^represents ^the^adiabati^onstant,

P _r

^the^Prandtl ^number,

α ∞

^the^tortuosity^,

σ

^the ^ow resistivity,

Λ

^and

Λ ^′

^the ^visous ^and ^thermal harateristi lengths

1,3

,

η

^is ^the^uid

visosity,

φ

^is ^the ^porosity ^and

ρ

îs ^the ûid ^density^. ^This ^model ^was înitially ^developed ^by

Johnson

3

, and ompleted by Allard

1

by adding the desription of thermal eets. Later on,

Lafarge

16

introdued theparameter

k ^′ ₀

^whih ^desribes ^the^additional ^damping ^of ^sound^waves

dueto thethermalexhanges between uidandstruture atthesurfaeof thepores.Generally

theration between

Λ ^′

^and

Λ

^is^between ¹ ^and ^3.

The funtions

α(ω)

^and

β(ω)

êxpress ^the ^visous ând ^thermal êxhanges ^between ^the âir ând

thestruturewhihareresponsibleofthesounddampinginaoustimaterials.Theseexhanges

aredue ontheone handto the uid-struture relativemotion and ontheother handto theair

ompressions-dilatations produed bythe wave motion. The part of theuid aeted by these

exhangesanbeestimatedbytheratioofamirosopiharateristilengthofthemedia,asfor

example thesizesof thepores,to the visous andthermal skindepththikness

δ = (2η/ωρ) ^1/2

and

δ ^′ = (2η/ωρP _r ) ^1/2

^.^For^the^visous êets^this^domain ôrresponds^to^the^regionôf^theûid

inwhih the veloitydistribution is perturbed bythe fritionalfores at theinterfae between

the visous uid and the motionless struture. For the thermal eets, it is the uid volume

aetedbytheheatexhangesbetweenthe twophasesoftheporousmedium,thesolidskeleton

beingseen asa heatsink. At low frequenies(visous domain)

24

,the visousand thermalskin

thiknesses are muh larger than the radius of the pores

δ/r ≫ 1

^and

δ ^′ /r ≫ 1

^. ^The ^visous

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material is slow enough to favor the thermal interations between uid and struture. At the

same timethe temperature of the frame is pratially unhanged by the passage of the sound

wavebeauseof thehighvalueofits speiheat:theframeats asa thermostat.Inthisase,

theexpressions ofthe dynami tortuosityand ompressibilityaregiven bytherelations

24,25

:

α(ω) = α ₀

1 + ηφ

j ωα ₀ ρk ₀

,

⁽³⁾

β(ω) = γ − (γ − 1)ρk ^′ ₀ P _r

ηφ jω,

⁽⁴⁾

where

j ² = − 1

^,

γ

^represents ^the ^adiabati ^onstant,

P _r

^the ^Prandtl ^number,

k

^is ^the ^visous

permeabilityrelated to the ow resistivity

σ

^by^the^relation ^:

k = η/σ

^.

In thetime domain,the fators

α(ω)

^and

β(ω)

âre ôperatorsând ^their âsymptoti expressions

aregiven by

24,25

:

˜

α(t) = α ₀

δ(t) + ηφ α ₀ ρk ₀ ∂ _t ⁻ ¹

,

⁽⁵⁾

β(t) = ˜ γδ(t) − (γ − 1)ρk ^′ ₀ P _r

ηφ ∂ _t .

⁽⁶⁾

Inthese equations,

∂ _t ⁻ ¹

îs^theîntegralôperator

∂ _t ⁻ ¹ g(t) = R _t

0 g(t ^′ )dt ^′

^.Înêah ôf^theseêquations

thersttermintherighthandsideistheinstantaneousresponseofthemedium(

δ(t)

^is^the^Dira

funtion)whiletheseondtermisthe memoryfuntion.Ineletromagnetism,theinstantaneous

response is alled optial response. It desribes all the proesses whih annot be resolved by

thesignal.

Inthisframework,thebasiequations oftheaoustiwavespropagationalongthepositiveaxis

diretion are:

ρ α(t) ˜ ∗ ∂w

∂t = − φ ∂p

∂x ,

⁽⁷⁾

φ β(t) ˜ K a

∗ ∂p

∂t = − ∂w

∂x .

⁽⁸⁾

The rst equationis the Euler equation, the seondone istheonstitutive equation.

K _a

^is^the

bulkmodulusofair,

p

îsâousti^pressureând

w = φ v

where

v

isthepartileveloity,

∗

denotes

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(f ∗ g)(t) = Z _t

0 f (t − t ^′ )g(t ^′ )dt ^′ .

⁽⁹⁾

Thewave equation isdeduedfrom theseequations

24,25

:

∂ ² p(x, t)

∂x ² − 1 c ²

∂ ² p(x, t)

∂t ² − A ∂p(x, t)

∂t + B ∂ ³ p(x, t)

∂t ³ = 0,

⁽¹⁰⁾

where theoeients

c

^,

A

^and

B

^are^onstants respetively given by;

1 c ² = ρ

K a

α ₀ γ − (γ − 1)P _r k ₀ ^′ k 0

, A = ηφγ K a k 0

= φσγ K a

, B = ρ ² (γ − 1)k ₀ ^′ P _r α ₀

K a ηφ ,

⁽¹¹⁾

therstoneisrelatedtothewavefront veloity

c = 1/

r ρ

α ₀ γ − ^(γ ⁻ ^1)P

^r

^k

⁰^′

k

0

/K _a

^of^the^waveⁱⁿ

the air inluded in the porous material. The term

α 0 γ − ^(γ ⁻ ^1)P

^r

^k

^′⁰

k

⁰

appears as therefrative

index of the medium whih hanges the wave veloity from

c ₀ = p

K _a /ρ

ⁱⁿ ^free ^spae ^to

c = c ₀ / r

α ₀ γ − ^(γ ⁻ ^1)P

^r

^k

⁰^′

k

⁰

inthe porousmedium. The originalityof this wavefront veloityis

its dependeneon the inertial, visousand thermal eetsintheporousmaterial, ompared to

thewavefrontveloityinhighfrequenyrange

6,10,11,15

whihdependsonlyoninertialinterations

via the tortuosity

α ∞

^. ^The ^oeient

A

^is responsible of theattenuation of thewave without

dispersiondueto thevisouslossesvia thevisouspermeability

k 0

^.^The^onstant

B

^governs^the

spreading of the signal, and desribes the dispersion due to the thermal interations between

uid and struture via the thermal permeability

k ₀ ^′

^. ^T^o ^note ^that ⁱⁿ ^this ^regime ^of ^frequeny^,

thedispersionphenomena desribing bythe term

B ∂ ³ p(x, t)/∂t ³

îs^not âs împortant âsⁱⁿ^the

highfrequeny range, inwhihthefrational derivatives

6,10,11,15

areneededto desribeintime

domain the highdispersioninthe porous material. This propagationequation has been solved

analytially in Ref. 24. The diret

24

and inverse

25 − 27

sattering problem for a slab of porous

material has been studied given a good estimation of the physial parameters (visous and

thermalpermeabilities, andinertial fator).

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MOGENEOUS POROUS MATERIALS

Consider the propagation of transient aousti waves in ontinuous inhomogeneous porous

material having rigid frame. In this material, the aoustial parameters (inertial fator, poro-

sity,visous and thermal permeability) depend on thethikness, and areontinuous funtions.

For a wave propagating along the

x −

axis, the uid-struture interations are desribed bythe inhomogeneous relaxation operators

α(x, t)

^and

β(x, t)

^given ^by

˜

α(x, t) = α ₀ (x)

δ(t) + ηφ(x) α 0 (x)ρk(x) ∂ ⁻ _t ¹

,

⁽¹²⁾

β(x, t) = ˜ γδ(t) − (γ − 1)ρk ^′ (x)P _r

ηφ(x) ∂ _t .

⁽¹³⁾

In these equations, the porosity

φ(x)

^, ^the ^tortuosity

α 0 (x)

^, ^visous ^and ^thermal permeability

k(x)

^and

k ^′ (x)

^depend ^on^the^thikness^of^the^porous^material ^for^desribing^theinhomogeneous losses inthematerial.

Inthis framework, the basiequations

13 − 15

for our model anbewritten as

ρα(x, t) ∗ ∂w(x, t)

∂t = − φ(x) ∂p(x, t)

∂x ,

⁽¹⁴⁾

φ(x)

K _a β (x, t) ∗ ∂p(x, t)

∂t = − ∂w(x, t)

∂x .

⁽¹⁵⁾

In thenext setion, thegeneralized propagationequation inontinuous inhomogeneous porous

material having an aoustial parameters varyingwith depth is derived. The derivation of the

generalized wave equation in an inhomogeneous porous material is important for omputing

thepropagation of an aousti pulse inside themedium, and for solving the diretand inverse

sattering problems.

LetusonsidertheEulerequation(14)andtheonstitutiveone(15)inaninniteinhomogeneous

porousmaterial. Byputting

a(x) = ηφ(x)

α ₀ (x)ρk(x)

^and

b(x) = (γ − 1)ρk ^′ (x)P r

ηφ(x) ,

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ρα ₀ (x)δ(t) ∗ ∂w(x, t)

∂t + a(x)w(x, t) = − φ(x) ∂p(x, t)

∂x ,

⁽¹⁶⁾

φ(x)

K _a γδ(t) ∗ ∂p(x, t)

∂t − b(x) ∂ ² p(x, t)

∂t ² = − ∂w(x, t)

∂x .

⁽¹⁷⁾

We note

P (x, z)

^the ^Laplae ^transform ^of

p(x, t)

^dened ^by

P (x, z) = L [p(x, t)] =

Z ∞ 0

exp( − zt)p(x, t)dt.

⁽¹⁸⁾

TheLaplae transform ofEqs.16, 17 yields

ρα 0 (x)

1 + a(x) z

zW (x, z) = − φ(x) ∂P (x, z)

∂x ,

⁽¹⁹⁾

φ(x)

K _a [γ − b(x)z] zP (x, z) = − ∂W

∂x (x, z),

⁽²⁰⁾

where

W (x, z)

^is^the ^Laplae ^transform ^of

w(x, t)

^.

Using Eqs. 19 and 20 and the alulus developed in Appendix. A, we obtain the following

equation

∂ ² P(x, z)

∂x ² = ∂

∂x ln α 0 (x)

φ(x) + ∂a(x)

∂x

1 z + a(x)

∂P (x, z)

∂x + ρα ₀ (x)

K _a

− b(x)z ³ + (γ − a(x)b(x)) z ² + γa(x)z

P(x, z).

⁽²¹⁾

UsingtheinverseLaplaetransformofEq.21andtheinitialonditions

10

;

∂p

∂t (x, 0) = p(x, 0) = 0

^,

we nd thegeneralized propagationequationin timedomain.

∂ ² p(x, t)

∂x ² − 1 c ² (x)

∂ ² p(x, t)

∂t ² − A ^′ (x) ∂p(x, t)

∂t + B ^′ (x) ∂ ³ p(x, t)

∂t ³

− ∂a(x)

∂x Z _t

0 exp ( − τ a(x)) ∂p(x, t − τ )

∂x dτ − ∂

∂x

ln α ₀ (x) φ(x)

∂p(x, t)

∂x ,

⁽²²⁾

where

1 c ² (x) = ρα ₀ (x) [γ − a(x)b(x)]

K _a , A ^′ (x) = ρα ₀ (x)γa(x)

K _a ,

^and

B ^′ (x) = ρα ₀ (x)b(x) K _a .

Eq.(22)isthegeneralized propagationequationforlossyinhomogeneousporousmaterial inlow

frequeny range. This equation is very important for treating thediret and inverse sattering

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ofhomogeneous porous medium,i.e.when

α 0 (x)

^,

φ(x)

^,

k(x)

^and

k ^′ (x)

^beome^onstants^(inde-

pendent of

x

^), ^we ^nd

A ^′ (x) = A(x)

^,

B ^′ (x) = B

^,

∂a(x)/∂x = 0

^. ^In ^this ^ase, ^the generalized wavepropagation(Eq.22)isreduedtothe propagationequationinhomogeneousmaterial(Eq.

10).

Therstandseondterminthepropagationequation(22):

∂

²

p

∂x

²

(x, t) − ¹

c

²

(x)

∂

²

p

∂t

²

(x, t)

^desribe^the

propagation(timetranslation)viathefrontwaveveloity

c(x)

^.^The^term

r

α ₀ (x)γ − ^(γ ⁻ ^1)P

^r

^k

^′

^(x)

k(x)

appearsastherefrativeindexofthemediumwhihhangesthewaveveloityfrom

c 0 = p K a /ρ

in free spae to

c = c ₀ / r

α ₀ (x)γ − ^(γ ⁻ ^1)P

^r

^k

^′

^(x)

k(x)

in the porous medium. From this equation,

(asit hasbeen showninthe homogeneousase

24

),itan be seenthat theinertial, visous and

thermaleetsareallresponsibleofthehange inthewavefront veloityompared tothehigh

frequenyinhomogeneous ase

15

inwhihonlytheinertialeet modifythefront waveveloity.

The third term in the propagation equation (22) :

A ^′ (x) ^∂p(x,t) _∂t

îs ^the ^most împortant ône ^for

desribing theaousti attenuation inporous materials at low frequeny range

26

,it results on

the attenuation of the wave without dispersion. It depends on thevisous permeability, whih

is the most inuential parameter inthis domain of frequeny.This term desribesthe aousti

attenuation due to the visous and inertial interations between uid and struture. To note

thatthe thermal eetsdo not intervene inthis therm. Thisan be explained bythe fatthat

the visous eets are the most important at this range of frequeny. The ontribution of the

thermaleetsistakenintoaountonlyintheseondterminEq.4.Thistermisverysensitive

to the spatialvariation ofthevisous permeability

k(x)

^.

The fourth term in the propagation equation (22) :

B ^′ (x) ^∂

³

^p(x,t) _∂t

3 ^governs ^the ^spreading ^of ^the

signal, and desribes the weak dispersion due to the thermal interations between uid and

struture via the spatial thermal permeability

k ^′ (x)

^. ^T^o ^note ^that ⁱⁿ^this ^regime ^of ^frequeny^,

thedispersionphenomena desribed bythis termarenot asimportant asinthehighfrequeny

range, in whih the frational derivatives are needed to desribe in the time domain the high

(12)

The nal term :

∂

∂x

h

ln ^α _φ(x)

⁰

^(x) i

∂p(x,t)

∂x

^desribes ^the attenuation aused by the spatial variation

of the tortuosity and the porosity. In ontrast to the other terms, this term does not ontains

temporal derivative of the pressure, it is independent of the relaxations times of the medium

and thus to the frequeny omponent of the aoustisignal.

Finallytheterm inthepropagationequation(22):

− ^∂a(x)

∂x

R t

0 exp ( − τ a(x)) ^∂p(x,t _∂x ⁻ ^τ) dτ

^desribes

thespatialvariation ofthe inhomogeneity oftheporousmediumdueto theinertialand visous

interations (there areno thermaleets) of the medium.

The generalized propagationequation derived at the visous domain (low frequeny range)

and given by Eq. 22 is very dierent from the generalized frational equation derived at the

asymptoti domain (high frequeny range)

15

.The physial parameters desribing the propaga-

tion are not the same in the two domains, for example, the visous and thermal interations

between uid and struture are desribed bythe visous and thermal harateristi lengths

Λ

and

Λ ^′

ât ^theâsymptoti ^domain,^howeverât ^the ^visous ^domain,^these ⁱⁿ^terations âre^desri-

bedbythevisousand thermal permeabilities

k ₀

^and

k ^′ ₀

^.^Theînertial êets âreâlso^desribed

bydierent parameters atthetwo regimesoffrequenies, thetortuosity

α ∞

^is^used^for ^the^high

frequeny range,while theinertial fator

α ₀

îsûsedât ^the^low^frequeny^range. Înâddition^the

relaxations times responsible of the dispersion phenomenon and memory eets of the aous-

tiwave areexpressedbydierent temporaloperators. Ithasbeenshown

6,15

intheasymptoti

domain(highfrequenyrange),thatthetortuosityandompressibilityoperatorsdependonfra-

tionaloperatorsfor desribing thevisous andthermal interations. Thesefrational operators

giveafrational derivative terminthegeneralized propagationequationfor theinhomogeneous

material.Thisfrationaltermisnotfoundinthederivedequation(equation22).Forthevisous

domain orresponding to the lowfrequeny range, thelossoperator have a simple expressions,

whiharefuntionsof simplederivatives,andthusthereisno frationalterminthegeneralized

propagation equation. However, we nd a term with third derivative responsible of thedisper-

(13)

desribedbythefrational derivative

10,11

ismore important intheasymptotidomain thanthe

dispersionphenomenon desribed by thethird derivative term

24,25

inthe visous domain. The

porosityisthe onlyparameter whih playsan importantrole inboth thehighandlowfrequen-

ies domains. The variation of the porositywith the depth

φ(x)

^is ^found ⁱⁿ ^the^two ^generalize

equations (asymptoti and visousdomains).

Generally it is interesting to work at the very low frequenies, espeially when we want to

obtain the visous permeability or the ow resistivity by solving the inverse problem diretly

in timedomain via transmitted or reeted aousti waves

25 − 27

. In this ase, theeet of the

inertialfator

α ₀

^beomes^negligibleⁱⁿ^the^expression^of^theinhomogeneousrelaxationoperator

˜

α(x, t)

^given ^byÊq. ^12,ⁱⁿ^this âse, ^we ôbtain^the ^following êxpression ^:

˜

α(x, t) = ηφ(x) ρk(x) ∂ _t ⁻ ¹ .

Thevisousinterationsarethemostimportantinthisase, theinertialexhangebetweenuid

and strutureare negligible.The thermal permeabilityisnot involved inthebasi equations of

Generalized equation for transient-wave propagation in continuous inhomogeneous rigid-frame porous materials at low frequencies

HAL Id: hal-00881886

https://hal.archives-ouvertes.fr/hal-00881886

Submitted on 14 May 2014

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