A wave-fluid saturated poroelastic structure interaction model based on the modified Biot theory (MBT) and plane-wave decomposition using orthogonal cylindrical functions is developed. The model is employed to recover from real data acquired in an anechoic chamber, the poromechanical properties of a soft cellular melamine cylinder submitted to an audible acoustic radiation. The inverse problem of acoustic diffraction is solved by constructing the objective functional given by the total square of the difference between predictions from the MBT interaction model and diffracted field data from experiment. The faculty of retrieval of the intrinsic poromechanical parameters from the diffracted acoustic fields, indicate that a wave initially propagating in a light fluid (air) medium, is able to carry in the absence of mechanical excitation of the specimen, information on the macroscopic mechanical properties which depend on the microstructural and intrinsic properties of the solid phase.

## I. INTRODUCTION

The investigation focuses on the study of the interaction of acoustic waves with a soft, open-cell poroelastic cylinder immersed in a light fluid (air). The objective is to develop a method for the characterization of open-cell poroelastic materials such as foams in the audible frequency range where these materials are often designed to reduce noise and vibration in mechanical fields as the automobile, airplane and construction industries.

In previous studies,^{1,2} it was shown that the macroscopic structural, porosity and resistivity (related to permeability) of a cellular foam, could be recovered from the diffracted field at audible frequencies using the equivalent fluid model (EFM). In the EFM, the skeleton of the porous material is considered motionless (rigid-frame) in the absence of mechanical excitation and the waves inside the porous material to propagate only inside the fluid phase.^{3–6} However, it has been shown that it was possible to retrieve both the mechanical and the structural micro-geometric parameters of the soft, open-pore porous material, thanks to waves transmitted by a poroelastic foam panel at ultrasonic frequencies.^{7} In that study the waves in the soft cellular foam were considered to propagate, both in the fluid and the frame notwithstanding the supposed absence of mechanical excitation. The forward acoustic wave transmission problem was modeled using a modified Biot^{8,9} theory (biphasic theory) whereby some parameters have physical interpretation closely related to the pore structure at the microscale. The dynamic density function (that describes the viscous effects) is associated with the narrow sections of the pores and the dynamic bulk modulus (describing the thermal effects) is associated with the wider sections of the pores.^{10–13} The advantage of using the biphasic interaction model over the EFM is that it provides the possibility of retrieval of both the mechanical and the structural micro-geometric parameters of the open-pore porous material.

The aim of this study is to recover poromechanical parameters (density, Young's modulus and Poisson ratio) from the acoustic field, diffracted by a poroelastic specimen in form of a cylinder. We employ the biphasic interaction theory in the low frequency regime. The direct problem developed in this study is for a free-field configuration of the poroelastic cylinder compared to that of Hasheminejad *et al.*^{14} who have studied the diffraction of sound waves by an infinite cylinder near an impedance plane. Real data pertaining to the diffracted acoustic field is acquired in the smaller of the two anechoic chambers of the *Laboratoire de Mécanique et d’Acoustique* using the experimental method for acquiring the diffracted field described in reference 1. The diffracted field is obtained in two steps. First the incident field *p*_{inc} is acquired without the cylinder, then the total pressured field *p*_{tot} is acquired in the presence of the cylinder. The diffracted pressure field *p*_{sc} is then computed from the difference,

## II. THE FORWARD PROBLEM FOR THE DIFFRACTION OF AN ACOUSTIC WAVE BY A POROELASTIC CYLINDER

The forward diffraction problem (notably for simulating measured data) is formulated as follows. Given: (i) the geometry of the diffracting poroelastic cylinder body and its composition (material properties); (ii) the material properties of the host medium; (iii) the incident wavefield, determine: The pressure field measured at points circumscribing a circle around the object.

### A. Problem configuration

A monochromatic incident plane-wave (pressure) field with angular frequency ω, propagating in the host medium (air) Ω_{0}, impinges on a cylindrical, linear, macroscopically homogeneous, isotropic poroelastic target whose boundary is ∂Ω described by a circle of radius *r*_{c}. The pressure field is measured at a point *M* circumscribing the circle Γ around the object. The problem configuration is depicted on Fig. (1).

### B. The interaction model

#### 1. The acoustic wave propagation equations in the host medium (Ω_{0})

The field equations of state for the hydrostatic stress in the fluid can be expressed more conveniently in terms of a scalar velocity potential which is related to the acoustic particle displacement (*d*) and pressure (*p*) as follows

where

_{0}the fluid density,

*c*

_{0}is the wave velocity in the fluid.

Consider the poroelastic cylinder of radius *r*_{c} and its cylindrical coordinate system in Fig. (1). The plane wave solution of the incident wave propagating in the host medium (Ω_{0}) is first transformed into a superposition of cylindrical waves satisfying the Helmholtz equation in cylindrical coordinates using the integral representation for the Bessel function and employing orthogonality relations^{15,16} (assuming harmonic time *t* dependency, the factor exp ( − *i*ω*t*) with the angular frequency ω, is implicit),

in which *J*_{m} is the cylindrical Bessel function of the first kind^{15} and

The field scattered by the poroelastic cylinder,

where *A*_{m} are unknown scattering coefficients and *H*_{m}() is the *m*-th order cylindrical Hankel function which indicates that the scattered field satisfies the Sommerfeld radiation condition at infinity. This condition, which manifests the characteristic of the wave problem (the decay of waves) in far field, is commonly imposed to ensure the uniqueness of the solution in exterior acoustic scattering problems

The total acoustic velocity potential^{17} is expressed as

#### 2. The wave propagation equations in the poroelastic media

The equations of motion for a fluid-saturated poroelastic media were formulated by Biot.^{8,9} This motion is described by the macroscopic displacement of the solid and fluid phases represented by the vectors

where *P*, *Q*, *R* are generalized elastic constants which are related, via gedanken experiments, to other measurable quantities, namely ϕ, *K*_{f} (bulk modulus of the fluid), *K*_{s} (bulk modulus of the elastic solid), *K*_{b} (bulk modulus of the porous skeletal frame) and *N* (the shear modulus of the frame in vacuum).

The equations which explicitly relate *P*, *Q* and *R* to ϕ, *K*_{f}, *K*_{s}, *K*_{b} and *N* are given by^{3}

in which the bulk modulus of the frame can be evaluated from the relation,^{3} *K*_{b} = 2*N*(ν_{b} + 1)/[3(1 − 2ν_{b})] (ν_{b} is the Poisson ratio of the frame).

Expressions for the complex density and the bulk modulus considering different frequency regimes (frequency-dependent viscous and thermal effects respectively) that constitutes the modification of the Biot model are given in Appendix A. Consequently the equivalent mass density coefficients with their corrections (in the frequency domain)

where ρ_{11} = ρ_{1} + ρ_{a}, ρ_{12} = −ρ_{a}, ρ_{22} = ϕρ_{0} + ρ_{a}, ρ_{1} = (1 − ϕ)ρ_{s} + ϕρ_{0}, ρ_{a} = ϕρ_{0}(α_{∞} − 1) and ρ_{s} the density of the porous skeletal frame.

The displacement fields can be resolved into a superposition of longitudinal scalar potential φ and transverse vector

^{18}

where the superscripts *s* and *f*, are, respectively, associated with the solid-borne and fluid-borne waves. The solid-borne wave propagates in the two phases with similar amplitude, while the fluid-borne wave propagates mainly in the fluid phase, and is usually strongly damped. The above resolutions are substituted into Biot's field equations of motion. Then employing the vector identities, ∇(∇ · φ^{s}) = ∇^{2}φ^{s}, ∇ × (**∇**φ^{s}) = 0 and

Eqns. (10) and (11) can be rewritten in matrix form as follows

where [φ] = [φ^{s}, φ^{f}]^{T}, [ρ] and [M] are respectively

By imposing that the vector potential

The Helmholtz type of equations are obtained after solving the corresponding eigenvalue problems for the longitudinal and shear waves,

where ∇^{2} is the Laplacian and the propagation constants *k*_{1} and *k*_{2}, which characterize two compressional waves, are found from

with

Each wave propagates in both phases with different velocities and amplitudes. The ratio between displacement of fluid and solid phases, respectively denoted μ_{i} is given by

Likewise, a shear wave propagates in the porous media and is characterized by its wave number

and the ratio μ_{3} of displacement amplitude of the fluid and of the frame is (using Eqn. (15))

The fundamental field equations in cylindrical coordinate for the solid and fluid displacements in *r* and ψ direction in terms of displacement potentials in the poroelastic cylinder are written^{18}

and

where

where

The general expressions for the relevant stress components and the pore fluid pressure, are derived by substituting Eqns. (24) and (25) into Eqn. (23),

where *a*_{1, 2} = 2*N* − *P* − *Q*μ_{1, 2} and *b*_{1, 2} = *Q* + *R*μ_{1, 2}. Finally, the orthogonal field expansions for the fast and slow dilatational and shear waves propagating in the poroelastic cylinder may be expressed in the general form

where *B*_{m}(ω) , *C*_{m}(ω) and *D*_{m}(ω) are unknown coefficients to be determined using the boundary conditions.

### C. Use of wave transformations and boundary conditions to obtain *A*_{m}, *B*_{m}, *C*_{m} and *D*_{m}

The total acoustic field potential (for

Identically, the corresponding expression for the fluid particle normal displacement *d*_{n} is found as

The boundary conditions that have to be satisfied at the surface of the poroelastic cylinder to yield a unique solution for the scattering problem are

the continuity of pressure, i.e.,

$p_{f_p}=p$$pfp=p$.the compatibility of total stresses, i.e.,

$\sigma _{rr}^{s}=-(1-\phi )p$$\sigma rrs=\u2212(1\u2212\phi )p$ and$\sigma _{r\psi }^{s}=0$$\sigma r\psi s=0$.the conservation of fluid volume, i.e.,

$\displaystyle (1-\phi )\dot{u}_{r}+\phi \dot{U}_{r}=\dot{d}_{n}$$(1\u2212\phi )u\u0307r+\phi U\u0307r=d\u0307n$.

Incorporation of Eqns. (21)–(29), in the above boundary conditions yields the linear systems of equations

The resulting matrix equation can be written as

The elements of the matrix equation that enable to determine the unknown coefficient vectors

^{19}and the matrix equation solved using the

*LinearSolve function*. The scattered field measured in the host medium around the cylinder by the microphone is determined from the value of

*A*

_{m}, consequently this is the only coefficient extracted from the matrix equation.

The total pressure amplitude is finally given by,

## III. THE INVERSE PROBLEM FOR THE RECOVERY OF MECHANICAL PARAMETERS

The inverse acoustic diffraction problem is formulated as follows. Given: (i) the diffracted wavefield, (ii) the material properties of the saturating fluid and host medium, reconstruct the poroelastic parameters of the diffracting cylinder. The radius of the cylinder is determined by measurement using vernier calipers.

Precisely, the inverse problem herein involves the recovery of the mechanical parameters of the skeleton (density ρ_{b}, Young's modulus *E*_{b}, and Poisson ratio ν_{b}) of the poroelastic cylinder from the measured diffracted field and the MBT wave/poroelastic cylinder interaction model. The other parameters of the model are assumed to be known.

A function, constituting the measure of the discrepancy between the interaction model diffracted field and the experimental diffracted field is computed for each set of trial values of the parameters. We first divide [0, 2π] into *N*_{s} sectors so as to discretize ψ as: ψ_{n} = (*n* − 1)δ, *n* = 1…*N*_{s} where

*f*

_{dc}given by

where *p*_{sc} is the scattered pressure field.

The cost function

The model parameters are listed in Table (I).

Parameter . | symbol . | melamine . |
---|---|---|

Density of the solid (kg/m^{3}) | ρ_{s} | ρ_{b}/(1 − ϕ) |

Density of porous skeletal frame (kg/m^{3}) | ρ_{b} | ? |

Young's modulus of the solid (Pa) | E_{s} | E_{b}/(1 − ϕ)^{2} |

Young's modulus of the porous skeletal frame (Pa) | E_{b} | ? |

Poisson ratio of the porous skeletal frame | ν_{b} | ? |

Porosity | ϕ | 0.99 |

Tortuosity | α_{∞} | 1.01 |

Viscous characteristic length (μm) | Λ | 100 |

Thermal characteristic length (μm) | Λ′ | 3 × Λ |

Airflow resistivity (Pa.s.m^{−2}) | σ | 12500 |

Density of the pore fluid (kg/m^{3}) | ρ_{0} | 1.297 |

Fluid bulk modulus (Pa) | K_{f} | 1.42 × 10^{5} |

Viscosity of pore fluid (Pa s) | η | 1.78 × 10^{−5} |

Parameter . | symbol . | melamine . |
---|---|---|

Density of the solid (kg/m^{3}) | ρ_{s} | ρ_{b}/(1 − ϕ) |

Density of porous skeletal frame (kg/m^{3}) | ρ_{b} | ? |

Young's modulus of the solid (Pa) | E_{s} | E_{b}/(1 − ϕ)^{2} |

Young's modulus of the porous skeletal frame (Pa) | E_{b} | ? |

Poisson ratio of the porous skeletal frame | ν_{b} | ? |

Porosity | ϕ | 0.99 |

Tortuosity | α_{∞} | 1.01 |

Viscous characteristic length (μm) | Λ | 100 |

Thermal characteristic length (μm) | Λ′ | 3 × Λ |

Airflow resistivity (Pa.s.m^{−2}) | σ | 12500 |

Density of the pore fluid (kg/m^{3}) | ρ_{0} | 1.297 |

Fluid bulk modulus (Pa) | K_{f} | 1.42 × 10^{5} |

Viscosity of pore fluid (Pa s) | η | 1.78 × 10^{−5} |

## IV. RESULTS

### A. Recovery of mechanical parameters from the measured, diffracted acoustic field by a melamine foam cylinder

The geometric parameters of the melamine foam characteristics were derived using ultrasonic wave propagation methods as reported in.^{7} The results of the inversions are presented by plotting the cost/objective functions for the three mechanical parameters using a 500 Hz probe (Fig. 2(a)–2(c)). The retrieved values (ρ_{b}, 8.30 kg/*m*^{3}, *E*_{b}, 187 ±5 kPa, and ν_{b}, 0.483), are solutions of the inverse problem. The recovered density was in excellent agreement with that obtained from the calculated volume and weight (7.95 kg/*m*^{3}) of the cylinder measured on an electronic balance (Denver Instruments SI-4200). The density given by BASF (Ludwigshafen, Germany) for Basotec foam is 8.35 kg/*m*^{3}. The value of the recovered *E*_{b} is in good agreement with that in reference 7 obtained using ultrasonic waves. Sharper minima for ν_{b} is obtained at 1500 Hz (Fig. 3). A ν_{b} of 0.48 ± 0.01 is recovered. This is also in good agreement with those obtained in reference 7. A sensitivity study of the mechanical parameters on the diffracted field has been undertaken after which it has been remarked that small variations in Poisson ratio provoke significant changes to the diffracted field.

### B. Comparison between the reconstructed and measured diffracted acoustic fields

The theoretically computed fields using the recovered values are finally compared against the measured total and diffracted fields. In Fig. 4(a) and 4(b) the measured total field and diffracted fields are compared with the theoretical ones. The diffracted field computed using the equivalent fluid model at this frequency is also shown in Fig. 4(c). The modified Biot method is shown to better reconstruct the back scattered field amplitude than the EFM at this frequency. The other comparisons with experimental data for 1500 Hz is depicted on Fig. (5). The measurement units are in Pascal. There is a good agreement as is with the EFM results in reference 2.

## V. DISCUSSION

The measured, total acoustic fields depicted on Figures 4(a) and 5(b), are slightly shifted as compared to the theoretically computed ones. The total scattered pressure field which is a direct measurement, is affected by the degree of anechoicity of the room, that is, stray acoustic fields diffracted by objects that are not necessarily the test target but are present in the room e.g the turn-table. The diffracted field, which is the difference between the incident and scattered fields agree well with the theory i.e is not shifted because the “noise” present in the two latter fields have been subtracted. The difficulty in centering of the suspended target with respect to the table's center of rotation and also on the perpendicularity of the vertical plane of the object and the horizontal plane of the floor on which the turn-table is resting, can also contribute to the shift.

Even though the back-scattered field (from object to source) is very small compared to the forward one (downstream after the object) due to absorption by the poroelastic cylinder, the diffracted fields (Fig. 4(b) and Fig. 5(b)) are still important. Consequently it is necessary for applications necessitating acoustic (or electromagnetic wave) discretion to control the scattered field by other means other than the passive application of absorbing liners or paints.

### A. Limitations of the study

The transition frequency which separates low-frequency viscosity dominated flow from the high-frequency inertia dominated one is given by

where *q*_{0} is the viscous permeability. The transition frequency for the melamine foam employed in this study having a flow resistivity (σ) of 12500 N

^{2}is

^{20}can be attained by employing the Lafarge model

^{12,21}. The only inconvenience is that the Lafarge model necessities knowledge of an extra parameter, the thermal permeability

^{22}This refinement will constitute the topic of future studies.

Most foams are anisotropic to some extent due to the foaming process^{23} but in this study we have supposed an isotropic model.

## VI. CONCLUSION

This study highlights that employing an incident acoustic wave initially propagating in the air medium is able to carry some information on the mechanical properties of the frame (Young's modulus, Poisson ratio and density) of a diffracting poroelastic cylinder. Usually the solid phase is considered as immobile when solicited from air generated waves and approximation of the rigid frame is often assumed, i.e. the wave inside the porous material propagates only inside the fluid phase. Implicitly, if the waves carry some information on the elastic properties of the solid phase, this means that the waves are also propagating in the solid part and that the approximation of the rigid frame is valid only as a first approximation. This is the first time the modified Biot model is shown to be able to recover the poroelastic material parameters of soft plastic open cell foams from measurements of diffracted acoustic waves in air, in the frequency domain where these materials are employed.

### Appendix A: Porous material parameters considering different frequency regimes

Johnson *et al.*^{24} developed an expression for the complex effective density matching the high and low frequency regimes written as follows

where ρ_{0} is the density of the saturating fluid, ϕ is the porosity, α_{∞} is the tortuosity of the porous material, *v* = η/ρ_{0} is kinematic viscosity and *q*_{0} = η/σ is the static viscous permeability (η is the shear viscosity of the saturating fluid and σ the static flow resistivity). The viscous permeability *q*_{0} is an intrinsic parameter depending only on the geometry of the pores. *G*_{j}(ω) is the viscodynamic function that provides the description of both the magnitude and phase of the exact dynamic tortuosity of large networks formed from a distribution of random radii and is given by

where Λ is the viscous characteristic length (related to the size of the inter-connection between two pores).

The thermal transfer due to the interaction between the fluid (in our case air) and the structure is modeled in the frequency domain through the complex bulk modulus of the porous material using the Champoux-Allard model^{3} and is given by

where *P*_{0} is the ambient pressure, γ is the specific heat ratio,

^{25}),

*P*

_{r}is the Prandt number and

*v*′ =

*v*/

*P*

_{r}

^{2}.

*G*′

_{j}(ω) is the thermal correction factor which is written as follows

### Appendix B: The elements of the matrix equation

The elements of the 4 × 4 matrix

*X*

_{mn}, denoting element of row

*m*and column

*n*are given by

The elements of the vector

Finally, the elements of the vector