Z E A Fellah
email: fellah@lyon.inserm.fr

C Depollier

M Fellah

W Lauriks

J.-Y Chapelon

Influence of dynamic tortuosity and compressibility on the propagation of transient waves in porous media

Keywords: Pulse propagation, Transient signals, Porous material

This paper provides an analytical solution in the time domain for the propagation of transient ultrasonic waves in a homogeneous, isotropic porous material with a rigid frame. The originality of this propagation equation is the use of the Pride et al. and Lafarge models, which give a better description of acoustic wave attenuation. Two parameters, p and p ′ , are added to the description of losses in the porous material, given a modification of coefficients of the propagation equation in the time domain. The analytical solution to the wave equation has a different form as compared with that obtained from the Johnson-Allard model. This wave equation contains fractional derivative terms describing attenuation and dispersion in porous material. The Laplace transform method is used to solve the propagation equation. An experimental application to porous plastic foam is given to validate the mathematical solution to the propagation equation.

One important result of this work is that the introduction of the two parameters p and p ′ corrects the Johnson-Allard model by increasing attenuation with no change in dispersion. This phenomenon is much more significant for resistive porous materials.

Introduction

Porous media filled with air, such as fibrous mats, plastic foams and various felts, are used intensively in the automobile, aeronautical and building industries to attenuate sound waves. In the recent past, low frequency (i.e. in the range of 20-600 kHz) ultrasonic techniques have amply proven their worth as a powerful tool for probing the acoustic properties of these materials.

The geometry of the pores in ordinary porous media is not simple, and it is not easy to calculate the viscous and thermal interaction between fluid and structure directly. Useful information can be obtained from the simple case of porous materials with cylindrical pores. The Kirchhoff theory [START_REF] Krichhoff | Uber der Einfluss Wärmeleitung in einem Gase auf die Schallbewegung[END_REF] of sound propagation in cylindrical tubes provides a general description of viscous and thermal effects, but this description is unnecessarily complicated for many applications. Moreover, the fundamental acoustics equations used in the Kirchhoff theory can be very difficult to solve in the case of a non-circular cross-section. A simplified model in which thermal and viscous effects are treated separately has been developed by Zwikker and Kosten [START_REF] Zwikker | Sound Absorbing Materials[END_REF] for the case of circular cross-sections. The validity of this model was later justified [START_REF] Tijdeman | On the propagation of sound waves in cylindrical tubes[END_REF][START_REF] Stinson | The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape[END_REF], for a range of radii from 10 -3 cm to several centimeters at acoustic frequencies. This model was used to describe viscous effects in slits and cylindrical tubes with a circular cross-section. Thermal exchange effects were related to viscous effects using a model developed by Stinson [4]. For common porous materials, it is generally impossible to model the bulk modulus and effective mass forming the frame geometry. This explains why the models that describe sound propagation in these materials are mostly phenomenological. A review of the models developed before 1980 can be found in a work by Attenborough [START_REF] Attenborough | Acoustical characteristics of porous materials[END_REF]. More recent studies have been performed by Lambert [START_REF] Lambert | Propagation of sound in highly porous open cells elastic foams[END_REF] and Attenborough [START_REF] Attenborough | On the acoustic slow wave in air-filled granular media[END_REF].

In 1987, a substantial contribution was made by Johnson et al. [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF], who released a theory of dynamic fluid flow (i.e. as a function of frequency) in porous media, introducing the concept of dynamic tortuosity and permeability as well as viscous characteristics and length. Originally, the research mainly focussed on geophysical and petroleum industry applications. The Johnson et al. model was used for different kinds of porous materials [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF][START_REF] Champoux | Dynamic tortuosity and bulk modulus in air-saturated porous media[END_REF][START_REF] Fellah | Transient acoustic wave propagation in rigid porous media: a time-domain approach[END_REF][START_REF] Fellah | Direct and inverse scattering of transient acoustic waves by a slab of rigid porous material[END_REF][START_REF] Fellah | Solution in time domain of ultrasonic propagation equation in a porous material[END_REF]. For gas-saturated porous media, Allard [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] has produced an analogous theory for thermal effects, by introducing the concept of thermal characteristic length. Lafarge et al. [START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF][START_REF] Lafarge | Sound propagation in porous materials having a rigid frame saturated by gas[END_REF] have extended this theory by adding the concept of thermal permeability which plays an important part at low frequency approximation.

To cope efficiently with the specific problems appearing in transient acoustic field propagation, new approaches are required [START_REF] Szabo | Time domain wave equations for lossy media obeying a frequency power law[END_REF]. At present, most analyses of signal propagation are carried out in the frequency domain using the Fourier transform, and the results are translated to the time domain, and vice-versa. However this has several limitations. The first one is that the transformation is difficult to compute numerically with sufficient accuracy for non-analytical functions. For example, using the Fourier transform to obtain time domain results for a lossy material is a more complicated approach than using a true time domain analysis, and the numerical results are less accurate. The second disadvantage is that by working in the frequency domain, some numerical information is lost or hard to recover. For example, in the case of noisy data it may be difficult to reconstruct the chronological events of a signal by phase unwrapping. Consequently, it is difficult to obtain a deep understanding of transient signal propagation using the frequency domain method.

In our previous paper [START_REF] Fellah | Solution in time domain of ultrasonic propagation equation in a porous material[END_REF] we analytically calculated the solution in the time domain for the propagation of transient ultrasonic waves in a homogeneous isotropic porous material with a rigid frame, using the Johnson et al. [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF] and Allard [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] model. However Pride et al. [START_REF] Pride | Drag forces of a porous-medium acoustics[END_REF] and Lafarge [START_REF] Lafarge | Sound propagation in porous materials having a rigid frame saturated by gas[END_REF] show that the Johnson et al. [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF] and Allard [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] model underestimates the imaginary part of the relaxation functions of the porous medium. This underestimation affects attenuation of the ultrasonic pulse and then the physical parameters estimated by the inverse problem. In this work, a general analytical solution in the time domain is proposed, in which the Pride et al. [START_REF] Pride | Drag forces of a porous-medium acoustics[END_REF] and Lafarge [START_REF] Lafarge | Sound propagation in porous materials having a rigid frame saturated by gas[END_REF] models are taken into account. This solution has been validated by experimental data.

Model

In the acoustics of porous materials, a distinction can be made between two situations depending on whether the frame is deformable or not. In the first alternative, the dynamics of the waves due to coupling between the solid frame and the fluid are clearly described by the Biot theory [START_REF] Biot | The theory of propagation of elastic waves in fluid-saturated porous solid. I. Low frequency range[END_REF][START_REF] Biot | The theory of propagation of elastic waves in fluid-saturated porous solid. I. Higher frequency range[END_REF]. In air-saturated porous media, the structure is generally motionless and the waves propagate only in the fluid. This case is described by the equivalent fluid model, which is a particular instance of the Biot model.

Let a homogeneous isotropic porous material with porosity φ be saturated with a compressible and viscous fluid of density ρ f and viscosity η. It is assumed that the frame of this porous solid is not deformable when subjected to an acoustic wave. This is the case, for example for a porous medium with high skeleton density or a very large elastic modulus or weak fluid-structure coupling. To apply the results of continuum mechanics the wave-length of sound must be much larger than the sizes of the pores or grains in the medium.

In such porous material, acoustic waves propagate only in the fluid, so it can be seen as an equivalent fluid, the density and bulk modulus of which are "renormalized" by fluid-structure interactions. A prediction of the acoustic behavior of the porous material requires the dynamic tortuosity α(ω) and the dynamic compressibility β(ω) to be determined. These functions depend on the physical characteristics of the fluid in the pore space of the medium and are independent of the dynamic characteristics of the structure. The basic equations of the equivalent fluid model are given in frequency domain [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] by

ρ f α(ω)iωv i =∇ i p, β(ω) K a iω =∇•v. (1)
In these relations, v and p are the particle velocity and acoustic pressure, K a = γP 0 is the compressibility modulus of the fluid. The first equation is the Euler equation, and the second one is a constitutive equation. α(ω) and β(ω) are the dynamic tortuosity of the medium and the dynamic compressibility of the air in the porous material. These two factors are complex functions which depend heavily on the frequency f = ω/2π. Their theoretical expressions are given by Johnson et al. [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF], Allard [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] and Lafarge [START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF]

α(ω) = α ∞   1 - ηφ iωα ∞ ρ f k 0 1 -i 4α 2 ∞ k 2 0 ρ f ω ηΛ 2 φ 2   , (2)
β(ω) = γ -(γ -1)   1 - ηφ iωρ f k ′ 0 Pr 1 -i 4k ′2 0 ρ f ωPr ηφ 2 Λ ′2   -1 , (3)
where i 2 =-1,γ represents the adiabatic constant, Pr the Prandtl number, α ∞ the tortuosity, η is fluid viscosity, k 0 viscous permeability, k ′ 0 thermal permeability [START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF], Λ and Λ ′ viscous and thermal characteristic lengths [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF][START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF]. This model was initially developed by Johnson [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF], and completed by Allard [START_REF] Allard | Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials[END_REF] by adding the description of thermal effects. Later on, Lafarge [START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF] introduced the parameter k ′ 0 which describes the additional damping of sound waves due to thermal exchanges between fluid and structure at the surface of the pores.

The functions α(ω) and β(ω) express the viscous and thermal exchanges between the air and the structure which are responsible for sound damping in acoustic materials. These exchanges are due, on the one hand, to relative fluid-structure motion and, on the other hand, to air compression-dilatations produced by the wave motion. The parts of the fluid affected by these exchanges can be estimated by the ratio of a microscopic characteristic length of the media, for example pore size, to viscous and thermal skin depth thickness δ = (2η/ωρ 0 ) 1/2 and δ ′ = (2η/ωρ 0 Pr) 1/2 . For viscous effects, this domain corresponds to the region of the fluid in which the velocity distribution is disturbed by frictional forces at the interface between the viscous fluid and the motionless structure. For thermal effects, it corresponds to the fluid volume affected by heat exchange between the two phases of the porous medium. In this model, sound propagation is completely determined by the following six parameters: φ, α ∞ ,σ = η/k 0 (flow resistivity), k ′ 0 ,Λand Λ ′ . An underestimation of the imaginary part of the relaxation functions α(ω) and β(ω) was corrected by Pride et al. [START_REF] Pride | Drag forces of a porous-medium acoustics[END_REF] and Lafarge [START_REF] Lafarge | Sound propagation in porous materials having a rigid frame saturated by gas[END_REF] by adding two geometrical parameters p and p ′ . The authors [START_REF] Lafarge | Sound propagation in porous materials having a rigid frame saturated by gas[END_REF][START_REF] Pride | Drag forces of a porous-medium acoustics[END_REF] suggest the following expressions of dynamic tortuosity and compressibility

α(ω) = α ∞   1 - ηφ iωα ∞ ρ f k 0   1 -p + p 1 -i 2α 2 ∞ k 2 0 ρ f ω ηΛ 2 φ 2 p 2     , (4)
β(ω) = γ -(γ -1)   1 - ηφ iωρ f k ′ 0 Pr   1 -p ′ + p ′ 1 -i 2k ′2 0 ρ f ωPr ηφ 2 Λ ′2 p ′2     -1
.

(

) 5
It can be seen that for p = p ′ = 1, the Pride-Lafarge model reduces to the Johnson-Allard model. Some values of parameters p and p ′ have been calculated by Lafarge for particular pore geometry [START_REF] Lafarge | Sound propagation in porous materials having a rigid frame saturated by gas[END_REF].

Asymptotic domain

The asymptotic domain is defined when the viscous effects are concentrated within a small volume near the frame and the compression/dilation cycle is faster than the heat transfer between the air and the structure, and it is a good approximation to consider that compression is adiabatic.

This domain corresponds to the range of frequencies such that viscous skin thickness δ = (2η/ωρ 0 ) 1/2 is smaller than pore radius r, and corresponds to a high-frequency approximation of the response factors α(ω) and β(ω) when ω →∞, their expressions are given by

α(ω) = α ∞ 1 + 2 Λ η -iωρ f 1/2 - σφ(1 -p) iωρ f α ∞ , (6)
β(ω) = 1 + (γ -1) 2 Λ ′ η -iωPrρ f 1/2 - ηφ(1 -p ′ ) iωk ′ 0 Prρ f . (7)
In the time domain, α(ω) and β(ω) are expressed by the temporal operators α(t) and β(t)

α(t) = α ∞ δ(t) + 2 Λ η ρ f 1/2 ∂ -1/2 ∂t -1/2 + σφ(1 -p) ρ f α ∞ ∂ -1 ∂t -1 , (8)
β(t) = δ(t) + (γ -1) 2 Λ ′ η Prρ f 1/2 ∂ -1/2 ∂t -1/2 + ηφ(1 -p ′ ) k ′ 0 Prρ f ∂ -1 ∂t -1 . (9)
In these equations, δ(t) is the Dirac delta function, the temporal operator ∂ -1 /∂t -1 represents an Integral

∂ -1 x(t) ∂t -1 = t 0 x(t ′ ) dt ′ ,
and the semi integral operator ∂ -1/2 /∂t -1/2 is given by the following definition of the fractional integrals of order

ν given in Samko et al. [20] D ν [x(t)] = 1 Γ(-ν) t 0 (t -u) -ν-1 x(u) du, ( 10
)
where Γ(x) is the gamma function.

In this framework, the basic equation ( 1) of this model can be expressed as follows:

ρ f α(t) * ∂v i ∂t =-∇ i p and β(t) K a * ∂p ∂t =-∇•v, (11)
where * denotes the time convolution operation. For a wave propagating along the x-axis, using ( 8) and ( 9), one finds that Eq. ( 11) yields

ρ f α ∞ ∂v(x, t) ∂t + 2 ρ f α ∞ Λ η πρ f 1/2 t 0 ∂v(x, t)/∂t ′ √ t -t ′ dt ′ + σφ(1 -p) • v(x, t) =- ∂p(x, t) ∂x , (12) 1 K
a ∂p(x, t) ∂t + 2 γ -1 K a Λ ′ η πPrρ f 1/2 t 0 ∂p(x, t)/∂t ′ √ t -t ′ dt ′ + (γ -1)ηφ(1 -p ′ ) k ′ 0 Prρ f p(x, t) =- ∂v(x, t) ∂x . (13)
In these equations, the convolutions express the dispersive nature of the porous material. They take into account the memory effects, where the response of the medium to the wave excitation is not instantaneous but needs some time to take effect.

Direct problem

The direct scattering problem is that of determining the scattered field as well as the internal field, that arises when a known incident field impinges on the porous material with known physical properties. To compute the solution of the direct problem one need to know the Green's function of the modified wave equation in the porous medium. In that case, the internal field is given by the time convolution of the Green's function with the incident wave and the reflected and transmitted fields are deduced from the internal field and the boundary conditions.

The generalized lossy wave equation in the time domain is derived from the basic equations ( 12) and ( 13) by elementary calculation in the following form

∂ 2 p(x, t) ∂x 2 -A ∂ 2 p(x, t) ∂t 2 -B t 0 ∂ 2 p(x, t)/∂t ′2 √ t -t ′ dt ′ -C ∂p(x, t) ∂t = 0, (14)
where coefficients A, B and C are constants expressed by

A = 1 c 2 = ρ f α ∞ K a ,B = 2α ∞ K a ρ f η π 1 Λ + γ -1 √ PrΛ ′ , (15)
C = 4α ∞ (γ -1)η K a ΛΛ ′ √ Pr + σφ(1 -p) K a + α ∞ (γ -1)ηφ(1 -p ′ ) K a k ′ 0 Pr . ( 16
)
The first one is related to the velocity c = 1/ √ ρ f α ∞ /K a of the wave in the air contained within the porous material. The coefficient α ∞ is the refractive index of the medium that changes wave velocity from c 0 = √ K a /ρ f in free space to c = c 0 / √ α ∞ in the porous medium. The other coefficients are essentially dependent on the characteristic lengths Λ and Λ ′ , and the two new parameters p and p ′ introduced by the Pride et al. and Lafarge models, and express the viscous and thermal interactions between the fluid and the structure. Coefficients A and B are the same in both the Johnson-Allard model and Pride-Lafarge models, but parameter C has been modified by the introduction of parameters p and p ′ . We will see in the next section how modification of the coefficient C can considerably change the analytical solution in the time domain of this Eq. ( 14).

Reflection and transmission scattering operators

Consider a homogeneous porous medium which fills the region 0 ≤ x ≤ L. This medium is assumed to be isotropic and to have a rigid frame. A short sound pulse impinges normally on the medium from the left. It gives rise to an acoustic pressure field p(x, t) and an acoustic velocity field v(x, t) within the material, which satisfying the propagation Eq. ( 14) written also as:

∂ 2 p(x, t) ∂x 2 - 1 c 2 δ(t) + CH(t) + B √ t * ∂ 2 p(x, t) ∂t 2 = C ∂p ∂t (x, 0), (17)
where H(t) is the Heaviside function: H(t) = 0 for t ≤ 0 and H(t) = 1, for t>0.

To derive the reflection and transmission scattering operators, it is assumed that the pressure field and flow velocity are continuous at the material boundary.

p(0 + ,t) = p(0 -, t), p(L -,t) = p(L + , t), v(0 -,t) = φv(0 + , t), v(L + ,t) = φv(L -, t), ( 18
)
where φ is the porosity of the medium and ± superscript denotes the limit from left and right, respectively. Assumed initial conditions are:

p(x, t)| t=0 = 0 and ∂p ∂t t=0 = 0, [START_REF] Biot | The theory of propagation of elastic waves in fluid-saturated porous solid. I. Higher frequency range[END_REF] which means that the medium is idle for t = 0. If the incident sound wave is generated in region x ≤ 0, then the expression of the acoustic field in the region to the left of the material is the sum of the incident and reflected fields

p 1 (x, t) = p i t - x c 0 + p r t + x c 0 ,x < 0, (20)
here, p 1 (x, t) is the field in region x<0, p i and p r denotes the incident and reflected fields respectively. In addition, a transmitted field is produced in the region to the right of the material. This has the form

p 3 (x, t) = p t t - L c - (x -L) c 0 , x>L. (21)
(p 3 (x, t) is the field in region x>Land p t is the transmitted field.)

The incident and scattered fields are related by scattering operators (i.e. reflection and transmission operators) for the material. These are integral operators represented by

p r (x, t) = t 0 R(τ)p i t -τ + x c 0 dτ = R(t) * p i (t) * δ t + x c 0 . ( 22
)
p t (x, t) = t 0 T (τ)p i t -τ - L c - (x -L) c 0 dτ = T (t) * p i (t) * δ t - L c - (x -L) c 0 . (23)
In Eqs. ( 22) and ( 23) functions R and T are the reflection and transmission kernels, respectively, for incidence from the left. Note that the lower limit of integration in ( 22) and ( 23) is given as 0, which is equivalent to assuming that the incident wave front first impinges on the material at t = 0. The operators R and T are independent of the incident field used in the scattering experiment and depend only on the properties of the materials. Using boundary and initial conditions [START_REF] Biot | The theory of propagation of elastic waves in fluid-saturated porous solid. I. Low frequency range[END_REF] and [START_REF] Biot | The theory of propagation of elastic waves in fluid-saturated porous solid. I. Higher frequency range[END_REF], reflection and transmission scattering operators can be derived [START_REF] Fellah | Direct and inverse scattering of transient acoustic waves by a slab of rigid porous material[END_REF], taking into account the n-multiple reflections in the material:

R(t) = 1 -E 1 + E n≥0 1 -E 1 + E 2n G t, 2n L c -G t, (2n + 2) L c , (24)
T (t) = 4E (1 + E) 2 n≥0 1 -E 1 + E 2n G t + L c 0 ,(2n + 1) L c . (25)
In these equations, E = φ/ √ α ∞ and G is the medium's Green function. The solution p(x, t) of the propagation equation ( 14) is given by the convolution of the Green function G(x, t) with the input signal p(0,t),

p(x, t) = t 0 G(t -t ′ , x)p(0,t ′ ) dt ′ . ( 26
)
In most cases of air-saturated porous materials, multiple reflection effects may be negligible because of the high attenuation of sound waves in these media. So, by taking into account only the first reflections at interfaces x = 0 and x = L, the expressions of the reflection and transmission operators will be given by

R(t) = √ α ∞ -φ √ α ∞ + φ δ(t) - 4φ √ α ∞ ( √ α ∞ -φ) ( √ α ∞ + φ) 3 G t, 2L c , (27)
T (t) = 4φ √ α ∞ (φ + √ α ∞ ) 2 G t + L c , L c . (28)

General solution of the propagation equation

In order to obtain the Green function of the medium, we will solve Eq. ( 17) using the Laplace transform method [START_REF] Podlubny | Fractional Differential Equations[END_REF][START_REF]Applications of Fractional Calculus in Physics[END_REF], taking into account the conditions [START_REF] Biot | The theory of propagation of elastic waves in fluid-saturated porous solid. I. Higher frequency range[END_REF]. We note P(x, z) the Laplace transform of p(x, t) defined by:

P(x, z) = L[p(x, t)] = ∞ 0 exp(-zt)p(x, t) dt, (29)
and the inverse Laplace transform [START_REF] Podlubny | Fractional Differential Equations[END_REF] by

p(x, t) = L -1 [P(x, z)]. (30)
Using the following relations

L[δ(t)] = 1, L[H(t)] = 1 z and L 1 √ t = π z , (31)
the Laplace transform of the wave Eq. ( 17) yields

∂ 2 P(x, z) ∂x 2 -z 2 1 c 2 + C z + B π z P(x, z) =- 1 c 2 + B π z zp(x, 0) + ∂p(x, 0) ∂t -Cp(x, 0). ( 32
)
Taking into account the initial conditions [START_REF] Biot | The theory of propagation of elastic waves in fluid-saturated porous solid. I. Higher frequency range[END_REF], Eq. ( 32) is simplified to

∂ 2 P(x, z) ∂x 2 -z 2 1 c 2 + C z + B π z P(x, z) = 0, (33)
which is a second-order differential equation with real constant coefficients. The characteristic equation of (33) is

r 2 -z 2 1 c 2 + C z + B π z = 0. ( 34
)
The general solution of ( 23) has the form

P(x, z) = e -x/c √ f(z) ϕ(z) + e x/c √ f(z) ψ(z), (35)
where ϕ(z) and ψ(z) are two functions independent of x and

f(z) = z 2 1 + Cc 2 z + Bc 2 π z = z 2 + Bc 2 √ πz √ z + Czc 2 = z(z + b ′ √ z + c ′ ). (36)
In Eq. ( 26), constants b ′ and c ′ are positive and are given by

b ′ = Bc 2 √ π and c ′ = Cc 2 . ( 37
)
By taking the finite solution at infinity, which corresponds to the physical solution of our problem

P(x, z) = e -x/c √ f(z) ϕ(z). (38)
The solution of Eq. ( 17) is the inverse Laplace transform of P(x, z). Thereby giving

p(x, t) = L -1 (e -x/c √ f(z) ϕ(z)) = L -1 (e -x/c √ f(z) ) * L -1 (ϕ(z)). (39)
In the next section, we will calculate the term L -1 (e -x/c √ f(z) ) which is the core of the problem.

Calculation of L

-1 (e -x/c √ f(z) )
By putting 2 = b ′2 -4c ′ , it is easy to verify that 2 is always positive when p = p ′ = 1, which is equivalent to the case of the Johnson-Allard model [START_REF] Fellah | Solution in time domain of ultrasonic propagation equation in a porous material[END_REF].

When = 0, the solution of the propagation Eq. ( 14) is given in Ref. [START_REF] Fellah | Solution in time domain of ultrasonic propagation equation in a porous material[END_REF] by

p(x, t) =        0i f 0 ≤ t ≤ x c , 1 4 √ π b ′ x c t x/c 1 (τ -x/c) 3/2 exp - b ′2 x 2 16c 2 (τ -x/c) p(0,t -τ) dτ if t> x c , (40)
where p(0,t) = L -1 (ϕ(z)). In this case, the Green function is given by

G(x, t) =        0i f 0 ≤ t ≤ x c , 1 4 √ π b ′ x c 1 (t -x/c) 3/2 exp - b ′2 x 2 16c 2 (t -x/c) if t> x c (41)
When 2 > 0, the general solution of the propagation Eq. ( 14) is given in Ref. [START_REF] Fellah | Solution in time domain of ultrasonic propagation equation in a porous material[END_REF] by

p(x, t) =      0 if 0 ≤ t ≤ x c , x c t x/c b ′ 4 √ π 1 (τ -x/c) 3/2 exp - b ′2 x 2 16c 2 (τ -x/c) + τ-x/c 0 h ′ (ξ) dξ p(0,t -τ) dτ if t> x c . (42)
where

h ′ (ξ) =- 1 4π 3/2 1 (τ -ξ) 2 -x 2 /c 2 1 ξ 3/2 1 -1 exp - (µ (τ -ξ) 2 -x 2 /c 2 + b ′ (τ -ξ)) 2 16ξ × (µ (τ -ξ) 2 -x 2 /c 2 + b ′ (τ -ξ)) 2 8ξ -1 µ dµ 1 -µ 2 . ( 43
)
In this case, the Green function is given by

G(x, t) =        0i f 0 ≤ t ≤ x c , x c b ′ 4 √ π 1 (t -x/c) 3/2 exp - b ′2 x 2 16c 2 (t -x/c) + t-x/c 0 h ′ (ξ) dξ if t> x c . (44)
For the Pride-Lafarge model, 2 can be negative. This is because the value of the coefficient C in the propagation Eq. ( 14) is modified by introducing the parameters p and p ′ . We will now consider the general solution of the propagation Eq. ( 14) by considering the Pride-Lafarge model and thus by considering the case where 2 is negative.

Case when

2 < 0 Let be ′2 =-2 ⇒ ′2 = 4c ′ -b ′2 . ( 45
)
In this case, f(z) can be put in the following form

f(z) = z + b ′ 2 √ z 2 + ′ √ z 2 2 . ( 46
)
Using Eqs. ( 45) and ( 46) and the analytical calculus given in Appendix A, we obtain the solution of the propagation Eq. ( 14) when 2 is negative

p(x, t) =      0i f 0 ≤ t ≤ x c , t x/c F 1 (τ) + τ-x/c 0 H(ξ, τ) dξ + i τ-x/c 0 Ξ(ξ, τ) dξ p(0,t -τ) dτ if t> x c . (47)
with

F 1 (τ) = 1 4 √ π b ′ x c 1 (τ -x/c) 3/2 exp - b ′2 x 2 16(τ -x/c) .
The functions H(ξ, τ) and Ξ(ξ, τ) are given by the following relations in which, t is replaced by τ:

H(ξ, t) = ′ 4π √ π x c 1 (t -ξ) 2 -x 2 /c 2 1 ξ 3/2 1 -1 AB 4ξ cos AB 8ξ - B 2 -A 2 8ξ -1 sin AB 8ξ × exp - B 2 -A 2 16ξ µ dµ 1 -µ 2 (48)
and

Ξ(ξ, t) =- ′ 4π √ π x c 1 (t -ξ) 2 -x 2 /c 2 1 ξ 3/2 1 -1 B 2 -A 2 8ξ -1 cos AB 8ξ + AB 4ξ sin AB 8ξ × exp - B 2 -A 2 16ξ µ dµ 1 -µ 2 , (49)
where

A = ′ µ (t -ξ) 2 - x 2 c 2 and B = b ′ (t -ξ).
The Green function is given by

G(x, t) =      0i f 0 ≤ t ≤ x c , F 1 (t) + t-x/c 0 H(ξ, t) dξ + i t-x/c 0 Ξ(ξ, t) dξ if t> x c .
(50)

Numerical simulations

Take a sample of plastic foam M1, which is a porous material saturated with air, with the following parameters: thickness 0.8 cm, tortuosity α ∞ = 1.5, viscous characteristic length Λ = 25 m, thermal characteristic length Λ ′ = 75 m, flow resistivity σ = 2 00 000 N m -4 s, porosity φ = 0.82 and thermal permeability k ′ 0 = 2.77 × 10 -10 m 2 . Simulated transmitted signal is computed from Eq. [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF]. The input signal is given in Fig. 1(a) and its spectrum in Fig. 1(b).

Fig. 2(a) shows a comparison between two simulated signals, the first (solid line) corresponds to the real part of the solution (50) when p = p ′ = 0.7 and the second (dashed line) corresponds to the solution (42) calculated in Ref. [START_REF] Fellah | Solution in time domain of ultrasonic propagation equation in a porous material[END_REF] according to the Johnson-Allard model when p = p ′ = 1. We chose the same values for p and p ′ to simplify this study. This specified value of 0.7 is valid only for porous material with circular pores [START_REF] Lafarge | Sound propagation in porous materials having a rigid frame saturated by gas[END_REF].Inthe general case, the values of p and p ′ may be different from 0.7 (see [START_REF] Lafarge | Sound propagation in porous materials having a rigid frame saturated by gas[END_REF]). Note that it is possible to have a positive value of 2 for other values of p and p ′ .

From Fig. 2(a), we can see that an important change occurs in wave amplitude. By increasing p and p ′ from 0.7 to 1, wave amplitude increases by 65% of its initial value. This result can be predicted by the fact that when p and p ′ increase, coefficient C decreases and thus wave attenuation decreases due to inertial, viscous and thermal interactions between fluid and structure. This phenomenon is much more significant for resistive porous materials. Fig. 2(b) shows the same comparison as Fig. 2(a) for another less resistive sample M2, with the following parameters: thickness 0.8 cm, tortuosity α ∞ = 1.05, viscous characteristic length Λ = 300 m, thermal characteristic length Λ ′ = 900 m, flow resistivity σ = 20 000 N m -4 s, porosity φ = 0.96 and thermal permeability k ′ 0 = 2.77 × 10 -9 m 2 .In Fig. 2(b), the influence of the parameters p and p ′ on the attenuation is lower than in Fig. 2(a). We can conclude that the parameters p and p ′ play an important role in the acoustic attenuation, especially for resistive porous materials.

It can be seen from Fig. 2 that when p and p ′ changed from 0.7 to 1, the waveform is changed only in attenuation (wave amplitude) and not in dispersion. Fig. 3 shows the spectra of the two simulated signals given in Fig. 2(a). From the spectra of the two simulated signals, it can be seen that they have the same bandwidths which means that there is no dispersion. This end result shows that p and p ′ play an important role in acoustic wave attenuation but not in dispersion, as is the case for viscous and thermal characteristic lengths [START_REF] Fellah | Transient acoustic wave propagation in rigid porous media: a time-domain approach[END_REF][START_REF] Fellah | Direct and inverse scattering of transient acoustic waves by a slab of rigid porous material[END_REF]. 50)) (solid line) and spectrum of simulated transmitted signal for p = p ′ = 1(Eq. ( 42)) (dashed line). Fig. 4 shows the imaginary part of the solution (50) for p = p ′ = 0.7 (sample M1). The amplitude of the imaginary part of the solution is very small compared to the real part (Fig. 2(a)). This is the reason why only the real part of the solution corresponding to the physical solution is taken into account when a comparison with experiments is made. It is possible to write the complex Green function given in Eq. ( 50) as

G = G 1 + iG 2 =|G| e iθ , tan θ = G 2 /G 1 ≪ 1.
This leads to the conclusion that all the signal components get the same (very small) phase θ which is from this fact an inessential physical factor. This means that though the Pride-Lafarge model extends the range of validity of the fluid equivalent model, the Pride's parameters need to be interpreted physically.

In the next section, experimental validation of the solution (50) of the propagation equation for another plastic foam is given.

Ultrasonic measurements

As an application of this model, certain numerical simulations are compared to experimental results. Experiments are performed in air using two broadband Panametrics V389 piezoelectric transducers with 250 kHz central frequency in air and a bandwidth at 6 dB OF 60-420 kHz. 900 V pulses are provided by a 5058PR Panametrics pulser/receiver. The signals received are amplified to 90 dB and filtered above 1 MHz to avoid high frequency noise. (The energy is totally filtered by the sample in this upper frequency domain.) Electronic interference is removed by 1000 acquisition averages. The experimental setup is shown in Fig. 5.

Take a sample of air-saturated plastic foam M3 with the following parameters determined by conventional methods [START_REF] Fellah | Transient acoustic wave propagation in rigid porous media: a time-domain approach[END_REF][START_REF] Fellah | Direct and inverse scattering of transient acoustic waves by a slab of rigid porous material[END_REF]: thickness 0.8 cm, tortuosity α ∞ = 1.5, viscous characteristic length Λ = 35 m, thermal characteristic length Λ ′ = 105 m, flow resistivity σ = 125 000 N m -4 s, porosity φ = 0.9 and thermal permeability k ′ 0 = 4.44 × 10 -9 m 2 . Fig. 6 shows the experimental incident signal generated by the transducer. In Fig. 7, we compare the experimental transmitted signal (solid line), the simulated transmitted signal given by the real part of the Eq. ( 50) when p = p ′ = 0.7 (dashed line) and the simulated transmitted signal given by the solution (42) when p = p ′ = 1 (dashdot line). This comparison shows that the theoretical data obtained by Pride-Lafarge model leads to better results than those given by Johnson-Allard model.

Conclusion

In this paper, an analytical solution in the time domain for the propagation of ultrasonic waves in porous media with a rigid frame is established using the Pride-Lafarge model. This model is more general than the Johnson-Allard one which leads to underestimation of the imaginary part of the viscous and thermal relaxation functions. The solution of the propagation equation is solved using the Laplace transform and an experimental validation of the solution of the propagation equation is given. The direct problem is solved by taking into account the two new parameters p and p ′ introduced by Pride et al. and Lafarge. One important result of this work is that the introduction of the two parameters p and p ′ corrects the Johnson-Allard model by increasing attenuation with no change in dispersion. This phenomenon is much more significant for resistive porous materials. This analytical solution will help us to improve the characterization of porous material by solving the inverse problem in the time domain. To progress in this area, coming works will must improve the physical interpretation of the Pride-Lafarge parameters.

By choosing in Eq. (A.4) the expressions y

= x/c, a = z + b ′ /2 √ z and b = ′ √ z/2, we find that exp - x c f(z) = exp -z + b ′ 2 √ z x c - ′ √ z 2 x c +∞ x/c exp -z + b ′ 2 √ z ζ × J 1 ′ √ z/2 ζ 2 -x 2 /c 2 ζ 2 -x 2 /c 2 dζ. (A.5)
The first term in the right-hand side of Eq. (A.5) was calculated in Ref. [START_REF] Fellah | Solution in time domain of ultrasonic propagation equation in a porous material[END_REF] L -1 exp - We then deduce the following relation

x c z + b ′ 2 √ z = 1 4 √ π b ′ x c 1 (t -x/c)
L -1 exp - b ′ 2 ζz J 1 (℘z) =- i π 1 ℘ 2 t -ζb ′ /2 1 + (t -ηb ′ /2) 2 /℘ 2 for -i℘ ≤ t - b ′ 2 ζ ≤ i℘, (A.12)
which gives (Appendix B) We then choose ℘ such that

L -1 √ z exp - b ′ 2 ζ √ z J 1 (℘ √ z) = -i 2π √ π 1 ℘ 2 1 t 3/2 b ′ /2ζ+i℘ b ′ /2ζ-i℘ exp - u 2 4t u 2 2t -1 u -ζb ′ /2 1 + (u -ζb ′ /2)
℘ = ℘(ζ) = ′ 2 ζ 2 - x 2 c 2 . (A.16)
Consequently, Eq. (A.15) becomes

L -1 √ z exp -ζ z + √ zb ′ 2 J 1 [ ′ /2 √ z ζ 2 -x 2 /c 2 ] ζ 2 -x 2 /c 2 = i 2π √ π 1 ζ 2 -x 2 /c 2 1 (t -ζ) 3/2 1 -1 exp - (i ′ µ ζ 2 -x 2 /c 2 + ζb ′ ) 2 16(t -ζ) × (i ′ µ ζ 2 -x 2 /c 2 + b ′ ζ) 2 8(t -ζ) -1 µ dµ 1 -µ 2 .
(A.17)

We can now calculate the inverse Laplace transform of Eq. (A.5)

L -1 exp - x c f(z) =- i ′ 4π √ π x c t x/c 1 ζ 2 -x 2 /c 2 1 (t -ζ) 3/2 1 -1 exp - (i ′ µ ζ 2 -x 2 /c 2 + ζb ′ ) 2 16(t -ζ) × (i ′ µ ζ 2 -x 2 /c 2 + ζb ′ ) 2 8(t -ζ) -1 µ dµ 1 -µ 2 . (A.18)
Using the variable change tζ = ξ, we obtain

L -1 exp - x c f(z) = t-x/c 0 h(ξ, t) dξ, with h(ξ, τ) =- i ′ 4π √ π x c 1 (t -ξ) 2 -x 2 /c 2 1 ξ 3/2 1 -1 exp - (i ′ µ (t -ξ) 2 -x 2 /c 2 + b ′ (t -ξ)) 2 16ξ × (i ′ µ (t -ξ) 2 -x 2 /c 2 + b ′ (t -ξ)) 2 8ξ µ dµ 1 -µ 2
.

Fig. 1 .

 1 Fig. 1. (a) Input signal; (b) spectrum of the incident signal.

Fig. 2 .

 2 Fig. 2. (a) Comparison between simulated transmitted signal corresponding to the real part of the solution (50) for p = p ′ = 0.7 (solid line), and simulated signal corresponding to solution (42) for p = p ′ = 1 (dashed line), for the sample M1; (b) comparison between simulated transmitted signal corresponding to the real part of the solution (50) for p = p ′ = 0.7 (solid line), and simulated signal corresponding to solution (42) for p = p ′ = 1 (dashed line), for the sample M2.

Fig. 3 .

 3 Fig.3. Spectrum of simulated transmitted signal for p = p ′ = 0.7( Eq. (50)) (solid line) and spectrum of simulated transmitted signal for p = p ′ = 1(Eq. (42)) (dashed line).

Fig. 4 .

 4 Fig. 4. Simulated transmitted signal corresponding to the imaginary part of the solution (50).

Fig. 5 .

 5 Fig. 5. Experimental set-up of the ultrasonic measurements.

Fig. 6 .

 6 Fig. 6. Experimental incident signal generated by the transducer.

Fig. 7 .

 7 Fig. 7. Experimental transmitted signal (solid line), simulated transmitted signal given by the real part of the Eq. (50) when p = p ′ = 0.7 (dashed line) and the simulated transmitted signal given by the solution (42) when p = p ′ = 1 (dashdot line).

  3/2 exp -It should be remembered that an integral representation of the Bessel function J 1 (z) is[START_REF] Podlubny | Fractional Differential Equations[END_REF]

									b ′2 x 2 16c 2 (t -x/c)	.	(A.6)
	J 1 (z) =	z π	1 -1	exp(izt) 1 -t 2 dt,	(A.7)
	By substituting ℘z in place of z, we obtain
	J 1 (℘z) =	℘z π	1 -1	exp(i℘zt) 1 -t 2 dt,	(A.8)
					i℘ -i℘	exp(-zT) 1 +	T 2 ℘ 2 dT.	(A.9)
	we then have
								
	L -1 [J 1 (℘z)] =	d dt	 -	i π	1 +	t 2 ℘ 2	 for -i℘ ≤ t ≤ i℘,	(A.10)
	which means that
	L -1 [J 1 (℘z)] =-	i π	1 ℘ 2

which, with the variable change T =-i℘t, yields

J 1 (℘z) =-iz π t 1 + t 2 /℘ 2 for -i℘ ≤ t ≤ i℘. (A

.11)

Appendix A

We use the relation [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF] ∞ 0 exp -a x 2 + y 2

where J 0 (x) is the Bessel function, and a, b are constants. By changing the variables ζ = x 2 + y 2 ,weget

By differentiating the two sides of Eq. (A.2) with respect to y, and by using the results By separating the real and imaginary part of the function h(ξ, t), we then obtain h(ξ, t) = H(ξ, t) + iΞ(ξ, t).

Appendix B

If f(u) is the inverse Laplace transform of a function g(z)

The inverse Laplace transform of √ zg(z) is

In our case

which give us the relation (A.13).


Agnes Korn

Kumzari Ossetic

Taleshi Gilaki

Tati

AREAL FEATURES IN THE LANGUAGES OF SOUTH IRAN: FOCUS ON BALOCHI AND BASKHARDI

Dialects and isoglosses

State of the art

• two main dialect groups: Southern Bashkardi; North Bashkardi, with different identity (Molkigāl vs. Marzigāl, respectively) to Bam / Kerman

Bandar-e Abas

SW Iranian NW Iranian

⇒ VOSKANIAN & BOYAJIAN (2007: 122): "Though the extant materials on these dialects are still limited, they do not seem to be dialect variations of the same language, but, rather, build two distinct language groups, unified -regardless their genetic characteristics -under one conventional name (Baškardī) merely on the basis of territorial coverage."8 9)

- -(7a) habitual / iterative a- - future a-(3), (7d) bé- - conditional a- bé-(3) - subjunctive a-(5) bé- e- -(6f), (7a) imperative bé-(5), (6a, c) e- -(6b, e) progressive a-(4) be- (10)
• North Bsh. a-not only for present, habitual, iterative, future, but also in contexts where from a Persian perspective one would expect a SBJV (including conditional sentences) • North Bsh. be-not only for SBJV and IMP, but also for future • zero-prefixed PRS (not mentioned in the literature!) in all functions

• apparently same prefixes used for different functions?

• apparently same functions expressed by all possible means?

The progressive again

The Bashkardi subjunctive

• modal functions also found with prefix a-(5)

• be-SBJV vs. unprefixed forms: apparently same functions as present/future and subjunctive; variation in (6) suggests that unprefixed form is the older one ⇒ prefix be-borrowed from Persian (as seems likely for Balochi)?

13 SKJAERVØ (1989a:848f.; 1989b:367), who calls the progressive "continuous present tense".

A scenario for the development of the imperfective categories in contact

Stage 1: -PRS unmarked PRS1 (PRS/FUT) unspecified for mood (6b, f), (7a) • attested in Middle Iranian (which also preserves a SBJV that can also be used for FUT); various particles "in the air" for emphasising aspectual or modal nuances

Conclusion

• shared features of Bashkardi and Baloch mapped onto a (partial) tree: '[The boy sees:] Again that bird sits on the wheat, it is eating the wheat.'

ASPECTUAL prefix A 1 (IPFV) MODAL prefix A/E 3 (SBJV) N Bashkardi Balochi Persian S Bashkardi MODAL prefix BE 2 (SBJV) PROGRESSIVE: PRV-PST/INF=COP
• NB: Focal progressive: Look, that bird is eating the wheat right now.

5 be-yår-ie ke gwar=e hamie kabåb-ōn a-xwar-om IMP-bring.PRS-2PL SUB side=EZ DEM1 meat-PL IPFV-eat.PRS-1SG 'Bring [the bread] so that I might eat it with the meat!'

• NB: gwar 'side' = Balochi (Persian bar); used in a way parallel to Persian pahlū-ye 'id.'. 'I am collecting dung.'

15 Duda is a Balochi name from the epic tradition; maybe the Kadkhoda was a Baloch. 16 The balōčkāra were a high-rank social group while the naxīb were servants to the upper classes (Gerardo Barbera, p.c.). 17 In this context, Ordu is more likely to be a place name than mean 'camp'.

work on Gershevitch's material in the Ancient India and Iran Trust, Cambridge

Stage 2 :

 2 a1-PRS a1-: aspectual (imperfective) prefix of unclear origin (3),(7b, d), (9) • innovation in South & North Bashkardi + Balochi, possibly areal phenomenon Stage 3: be1-INF/PST be1: preposition be 'to', forms PROG (locative pattern) (10) • innovation in South Bashkardi, where the PROG is common and also occurs negated Stage 4: a2-INF a2-: PROG in North Bashkardi, where it is rare and does not seem to occur negated (cf. Persian dāštan pattern, which is normally not negated 14 • innovation in North Bashkardi, a2-= preposition a 'to' (then a calque of the South Bsh. patter) or a generalisation of IPFV a1-(then a partial calque)? (4) Stage 5: a/e3-PRS a/e3-: modal prefix (5) • innovation in South & North Bashkardi + Balochi, possibly areal phenomenon Stage 6: bé2-PRS bé2-: modal prefix (3), (5), (6a, c), (10) • Balochi: borrowed from Persian (limited use in Eastern Balochi dialects) • North Bashkardi: inherited or borrowed from Persian? VAFAEIAN (2018: 79, 90), who has found isolated NEG examples, though (ibid. 102, 136-138). ASPECTUAL prefix A 1 -(IPFV) MODAL prefix 3 -(SBJV) PROGRESSIVE: PRV-PST/INF=COP PRS kan-'do' PRS nind-/ništ 'sit' COP.PST at

  6a

  10a [Kadxodā:] to če be-kert=o you.SG what IPFV-do.PST=COP2SG Kadkhoda: 'What are you doing?' 10b men samåt be-čint=īn I dung IPFV-collect.PST=COP1SG

  Problem: dialectal divergences of North and South Bashkardi 5 • South Bsh.: preservation of postsonantal voiceless stops otherwise typical for Balochi: "Von allen Dialekten hebt sich das Balūtschī durch grosse Ursprünglichkeit glänzend hervor. Es hat die alten Tenues in allen Stellungen (...) bewahrt"(GEIGER 1901: 417) Both is systematic (i.e. not limited to isolated items that could be explained as loans).

	2.2 Historical phonology:
	Development of postsonantal stops in Bashkardi, Persian and Balochi 6
	Middle Persian North Bš South Bš Balochi	Proto-Iranian
	ābus	aves	yōpes	āpus		*āpuθra-'pregnant'
	kird	kerd	Kahnuj kert	Rudbar kurt	*kṛta-'done'
	kadām (Parth.) karōn	katam	East Bal. kut h ān *katāma-'which'
	spēd	espīr	espīt		spēt		*spaita-'white'
	• North Bsh. shows lenition of postsonantal stops similar to Middle Persian;
	but postvocalic *t > *d (> *δ?) > r		Kerman	Iranshahr to
	to Tehran
							Sistan
							va
	Morphology:	Minab Berentin		Durkan Keshmiran	North Bashkardi	Baluchistan
	Pronominal clitics in Bashkardi, Middle Persian and Balochi 7	Rameshk
	Hormuz Middle Persian North Bashkardi South Bashkardi Balochi Old Iranian Beshnu Gaverx Darze Biverch Giran SG 1 =Vm Qeshm =(o/a)m (3) =om =un =mai, =mā
	Larak 2 =Vt / =Vd 3 =Vš	=(e)t =(e/a/i)h, =e, (3)	=t Angohran =ī	Dahwast	=it =iš, =ē,	=tai, =θwā *=šai, =hai,
	PL 1 =mān, =n Strait of Hormuz 2 =tān / =dān 3 =šān, =š	=i (3) =mōn =tōn =šån,=šōn (1) Shahbavek	Sardasht Jaghdan =an (7a, b, d) Kavaran South =ox Bashkardi =eš	=ī =in =ū =iš, =ēš *=šām =īm, etc. Balochi *=nah *=wah
						Hormozgan
							:
	to "Most conservative is Balōči" (WINDFUHR 1989: 257 about the pronominal clitics, also
	OMAN mentioning Bashkardi)					Chabahar
	⇒ Is South Bashkardi still "South-Western Iranian" or rather an offshoot of Balochi?
			Jask Western Iranian
			Western Iranian
							traditional family tree of
							Western Iranian
		SW Iranian		NW Iranian	(excerpt, simplified)
				???
	Middle Persian N Bashkardi Persian	S Bashkardi	Parthian Balochi
	Persian		Bashkardi	Balochi	Kurdish, Gilaki,
	Lori, Bakhtiari,					Zazaki, Taleshi,
		etc.					etc.

Garou

• affiliation as per available literature: Bashkardi is South-West Iranian, i.e. Persian subbranch of Western Iranian while Balochi is North-West-Iranian (Parthian sub-branch) 4

• North Bsh. shares the PL clitics in -ān (> -ōn / -ūn) derived from the SG as seen in Persian • South Bsh. shows the preservation of the inherited PL clitics also found in Balochi

  PRS stem nen-probably < nind-, i.e. the form seen in Balochi; 9 'do' PRS stem: traditionally held to be an important isogloss of SW Ir. (Old Persian kunau-) vs. NWIr. (Parthian, Zazaki /kar-/), 10 but Bashkardi & Balochi differ from both

	• Conversely, North and South Bashkardi do share non-trivial features. 3. A case study of convergence: The imperfective domain
	2.3 Features of North and South Bashkardi shared with Balochi 3.1 The Bashkardi formations used in the IPFV domain
	Some verbal stems (selected forms) i.e. present tense, progressive, subjunctive, etc. (non-past domain) Middle Persian North Bashkardi South Bashkardi Balochi Parthian COP.PST (būd) ar (2) at (8) at The imperfective domain according to the published literature (schematic) 13 (būd) 'sit down' PRS PST nišast, nišīn-nešt (4) nen-ništ nind-North Bashkardi South Bashkardi cf. Persian nišast present/future a-PRS-INFL mī-PRS-INFL nišīδ-'do' PRS kun-kan-(3), (9) subjunctive be-PRS-INFL be-/e-PRS-INFL be-PRS-INFL kar-progressive a-PST-INF=COP be-PST(-INF)=COP AUX-INFL + PRS-INFL
	• Non-trivial agreement of North and South Bsh. for some verbal stems with Balochi, Distribution of verbal prefixes as seen in the data against both Persian and Parthian • COP.PST: origin unclear (thus the commonality is particularly interesting); a-bé/e--
	North Bš South Bš 'sit down': ing' 11 North Bš South Bš North Bš South Bš present a-(7b), (
	• entirely parallel to Bashkardi: progressive (generalised as present tense) in Caucasian Tat
	(an Iranian language spoken in Azerbaijan): 12
	IPFV system of Caucasian Tat
		Jewish Tat	Muslim Tat
	"eventual"	mǝ-PRS-INFL
	subjunctive	(bi-)PRS-INFL (bı-)PRS-INFL
	present tense PST-INF=COP bä-/mi-PST-INF=COP
	⇒ Is Bashkardi a mini-Sprachbund (linguistic area), i.e. convergence of Iranian varieties of
	different origins?
	This could accommodate both differences and shared traits of North and South Bashkardi.

⇒ Is North Bashkardi still "South-West Iranian" or do both South and North Bashkardi belong with Balochi? 2.4 Features of North and South Bashkardi not shared with Balochi Structure of progressive constructions in Bashkardi, Balochi and Persian North Bashkardi a-kerd-en=om (4) a-PST-INF=COP South Bashkardi be-kert(-en)=īn (10a, b) be-PST(-INF)=COP Balochi kanag-ā =un INF-OBL = COP Persian dār-am mī-kon-am AUX-INFL ('have / hold') + PRS-INFL • similar progressive in North and South Bashkardi (INF or PST with prefix); Balochi (inflected infinitive) and Persian (auxiliary construction) pattern differently South Bashkardi be-kert(-en) =īn 'I am doing' North Bashkardi a-kerd-en =om Structure be/a-PST(-INF) COP.1SG

• same slots filled with different elements: cf. definition of linguistic area: "one grammar with different lexicons" (KOPITAR 1829, quoted by FRIEDMAN 2000:2 for the classic case of the Balkan Sprachbund): convergence of languages of different origins that develop shared traits due to language contact

• contact-induced progressive in the NWIran / Caucasus area: Iranian and Aramaic varieties showing a locative-type progressive 'be in the state of X-

  borrowing• very preliminary suggestion of a model accounting for the data discussed in this paper (partial tree, highly simplified): interplay of material inherited in each of the sub-branches under study with areal influences showing a convergence of North and South Bashkardi, and of both and Balochi They had two goats (lit. to them were to piece [of] goat).'• NB: Expression of 'have' by 'to me is X' (no verb 'to have' in Bashkardi or Balochi);• counting word dåna also for animals. The shepherd] said: "Now if I... cure her, what would / will they give me?"'• NB: Pronominal clitics (=ī, =e, =om) in object function in the PRS domain.

	Appendix: Examples
	North Bashkardi:
	1 bü=šōn		do	dåna boz
	become.PST=PC3PL two piece goat
	'2 go	to	hamå darbīš	ī
	say.PST you.SG DEM2 dervish COP2SG
	ke	šöu	lahar mon	ar-ī
	SUB	night	hut	I		COP.PST-2SG
	'[The shepherd] said: "You are that dervish who [one] night was in my hut."'
	• NB: ar-shows the typically North Bsh. change of Old Ir. postvocalic t to r;
	• Ezafe often not present when possessor = pronoun (KORN 2017: 88).
	3 go	hålå aga mõ ǰost…	ǰōn=ī	be-kan-om=e
	say.PST now if	I		(false start) healthy=PC3SG SBJV-do.PRS-1SG=PC3SG
	č=om	a-dah-an
	what=PC1SG IPFV-give.PRS-3PL
	'[4 dega hamå morg nešt=e	sar gandom, gandom a-xwar(d)-én=ī
	again DEM2 bird	sit.PST=COP3SG on wheat	wheat	IPFV-eat.PST-INF=COP3SG
						postvocalic ptk,
		stops ptk > -bdg, PL PCs -ān, causative -ān-,	PL PCs inherited, causative -ēn-, 1 SG -Vn
		1 SG -om
	Persian		(some of)	(some of)	Balochi	Zazaki
			N Bashkardi	S Bashkardi
	PRS kun-
	PRS nišīn-					PRS kan-'do'
	(COP.PST būd)		PROGRESSIVE:	PRS nind-/ništ 'sit'
				PRV-PST/INF=COP	COP.PST at

ASPECTUAL prefix A 1 (IPFV) MODAL prefix A/E 3 (SBJV) PRS kar-PRS nišīδ-(COP.PST būd)

  I take the leaves to the goat, take milk [from] the goat, ..."' IPFV-kill-PRS.3SG=PC1PL power=PC3SG COP.3SG he will kill us; that's his capacity.'• NB (b): Expression of 'want' with a non-canonical subject ('to us is (not) desirable');interesting position of negation (after prefix)8 balōčkåra-n ham ord at-en yok balōčkåra=e yok naxīb freeman-PL also there COP.PST-3PL one freeman=IND one servant 'The freemen were there, one freeman and one servant.' 16 IPFV-do.PRS-1SG IPFV-go.PRS-1SG PN 'From Ordu, 17 I set out[and] go to Ahven.'

	go: say.PST 'He (the jackal) said: "Konar-tree, give me a handful of leaves." konår to ya moč=ī (tree) you.SG one handful=IND (aga na xo a-g-en konår if not well IPFV-say.PRS-3PL (tree) If not, well, they say [in fact]: "Konar, Konar, give [me] leaves!"?' ... konår tåg leaf (tree) [discussion in the background] be hamå-tau bo-go to that-way IMP-say.PRS [Someone else:] 'Say it that way.' hamå-tau bo-g-om? that-way SBJV-say.PRS-1SG 'Should I say it in that way?' [confirmation from the others present] xob. gu: konår konår tåg da, well say.PST (tree) (tree) leaf give.PRS be-da SBJV-give.PRS tåg da? leaf give.PRS 'OK. He said: "Konar-tree, give [me] leaves, tåg barr-om boz-å, boz ger-om šīr-å, leaf carry.PRS-1SG goat-DIR goat seize.PRS-1SG milk-DIR 7a yå 6b 6c 6d 6e 6f moraxxas=an kī or permitted_to_leave=PC1PL do.PRS.3SG ke ra-m ba xåreǰa SUB go.PRS-1PL to abroad '... or he sets us free so that we (can) go abroad. 7b kasxodå=i dūdå=n a-m-vå, headman=EZ PN=PC1PL IPFV-NEG-want.PRS.3SG We do not want the headman Duda. 15 7c bīzår om, bīzår om, desperate COP.1PL desperate COP.1PL We are desperate; we are desperate; 7d a-yōz-et=an salåh=ah e 9 az ordū ǰeīn a-kan-īn a-ra-īn ahven [so that] South Bashkardi: from PN place

As pointed out byKORN (2016[START_REF] Korn | Notes on the speech of the Afro-Baloch of the southern coast of Iran[END_REF] and others, this model is problematic for various reasons. In what follows, a feature being shared by X and Y does not necessarily imply that X and Y are the only Iranian varieties that shows this feature.

See alsoKORN (2021: 379) for a short list of differences between North and South Bashkardi.

Items from SKJAERVØ 1989a, based on GERSHEVITCH's data.

For 1PL =n and 3PL =š see JÜGEL(2015: 222, 225f.); for more details on the pronominal clitics and their development seeKORN 2009. Old Iranian is cited in an underlying form, abstracting the specificities of Old Persian and Avestan.

GERSHEVITCH (1970: 163) defines North and South Bashkardi as "two distinct language groups".

TEDESCO (1921:223f., 247, 254), PAUL (2003: 63-67).

SeeKORN (2005: 96, 127) for suggestions on the derivation of these stems.

Cf. VAFAEIAN (2018: 170-174), see also Section 3.2 below.

Cf.[START_REF] Korn | Contact-Induced Progressives in Iranian Languages: A Typological Comparison of Caucasian Tat and Bashkardi[END_REF], SULEYMANOV (2019: 200f.).