Hervé Franklin

Scattering by a fluid cylinder in a porous medium: Application to trabecular bone

In a trabecular bone, considered as a nondissipative porous medium, the scattering of an incident wave by cylindrical pores larger than the wavelength is studied. The goal is to know if scattering alone may cause such a high attenuation as that observed in calcaneus. The porous medium is modelized via Biots theory and the scattering by a single pore is characterized from the definition of a scattering matrix. An approximation of weakly disordered medium is then discussed to estimate the effective attenuation and dispersion as a function of frequency. These effective properties are shown to be different of those measured on calcaneus, due to the neglect of wave conversions during the scattering process.

S-matrix theory applied to acoustic scattering by asymmetrically fluid-loaded elastic isotropic plates

Exact and approximate formalisms describing the interactions of acoustic plane waves with an elastic isotropic plate immersed between two different fluids (asymmetrically fluid-loaded plate) are presented. This constitutes an extension of the Fiorito, Madigosky, and Uberall (FMU) [R. Fiorito et al., J. Acoust. Soc. Am. 66, 181 (1979)] theory, refined later by Freedman [A. Freedman, J. Sound Vib. 82, 181 (1982); ibid. 82, 197 (1982)]. The method, based upon the multichannel resonant scattering theory derived from quantum physics [A. Bohm, Quantum Mechanics, Foundations, and Applications (Springer, New York, 1993)], consists of two parts. First, a 2×2 scattering S-matrix in which the diagonal elements are the two reflection coefficients and the off-diagonal elements are the two transmission coefficients, is built. Second, in order to compare our results with the FMU theory, resonant approximations are given for these coefficients, assuming light fluids when compared to the plate. The approximated coefficien...

N-shell cluster in water: Multiple scattering and splitting of resonances

Clusters of N thin parallel and identical shells (aligned or not) in water are considered. Assuming a harmonic plane wave is normally incident upon one cluster, the scattered-field classical expression is recalled, and then computed for different types of clusters, along with resonance spectra. The scattering S matrix is defined, and its unitarity property used to check the numerical results. All spectra are compared with that of a single shell, in the frequency range where resonances are due to an A-wave phase matching only. Whatever the cluster, each resonance of the single shell is seen to split into M different ones. The value of M depends on the number of shells, the distance between them, and the symmetries of the cluster. Apart from the very special case of aligned shells (M=2N), no simple law has been found to predict the value of M.

Normal modes of a poroelastic plate and their relation to the reflection and transmission coefficients.

A poroelastic plate that obeys the Biot theory is considered. Compact new forms of its reflection and transmission coefficients, similar to those of the resonance scattering theory for an elastic plate, are derived. A numerical comparison of the reflection coefficient modulus with the plate normal modes, at low frequency, shows that a study of the reflection or transmission coefficient does not provide the same kind of information on the poroelastic plate than an investigation of guided leaky waves propagation.

Multiple scattering in a trabecular bone: Influence of the marrow viscosity on the effective properties

The Foldy and the Waterman and Truell approximations are used to determine the effective properties of the coherent wave that emerges after multiple scattering of a plane longitudinal fast wave by the largest pores in a trabecular bone. The unit scattering cell considered is either a single pore or two close cylindrical pores (cluster), at a fixed overall bone porosity. In the cluster case, the effective attenuation is about twice that obtained with one single pore per scatterer. It is shown that taking into account the marrow viscosity leads only to minor differences on the effective dispersion and attenuation.

Transition term method for the analysis of the reflected and the transmitted acoustic signals from water-saturated porous plates.

Ultrasound reflection (R) and transmission (T) from an immersed porous plate of QF-20 (a registered trademark of Filtros, Ferro Corporation) are investigated. Assuming open pores boundary conditions for the theoretical computations, values of the physical constants given by Johnson et al. [D. L. Johnson, D. L. Hemmick, and H. Kojima J. Appl. Phys. 76(1), 115-125 (1994)] are used at first. Comparisons of R and T with experimental results show a discrepancy. It is then assumed that visco-elastic losses exist inside the solid part. When a small imaginary constant part is considered for the bulk moduli, a better fit is obtained between theoretical and experimental values of /R/ and /T/. However, the numerous and very close peaks prevent easy measures of the resonance amplitudes and widths. The transition terms built up from the linear combinations R-T and R+T allow these peaks to be separated. Comparisons between theory and experiments are made. This work validates, in the frequency range where the agreement is good, a method for the location of the symmetrical and antisymmetrical modes of the plate. A discrimination is also possible between modes resulting from the fluid phase motions and from the solid phase motions.

Estimating frame bulk and shear moduli of two double porosity layers by ultrasound transmission

The acoustic plane wave transmission by water saturated double porosity media is investigated. Two samples of double porosity media assumed to obey Berryman and Wang (BW) extension (Berryman and Wang, 1995, 2000) of Biots theory in the low frequency regime are under consideration: ROBU® (pure binder-free borosilicate glass 3.3 manufactured to form the individual grains) and Tobermorite 11Å (the individual porous cement grains show irregular shapes). The de facto gap existing between theoretical and experimental data can be minimized by modifying adequately two of the parameters estimated from triaxial tests: the frame bulk and shear moduli. The frequency dependent imaginary parts that follow necessary from the minimization are in relation with the energy losses due to contact relaxation and friction between grains.

Multiple scattering in porous media: comparison with water saturated double porosity media.

Multiple scattering in a poroelastic medium obeying Biots theory is studied; the scatterers are parallel identical cylindrical holes pierced at random in the medium. The paper focuses first on the influence, on the effective wavenumbers, of the mode conversions that occur at each scattering event. The effect of the holes on the dispersion curves is then examined for two different values of the ratio of their radius to the pores mean radius. Depending on the latter, the dispersion curves of the pierced material are compared, for the fast and shear waves, with those of either a more porous medium or a double porosity medium.

Two-channel resonant scattering theory: Application to an absorbing elastic cylinder

The scattering of a plane sound wave normally incident on an absorbing elastic cylinder immersed in water is investigated theoretically. Because of the loss of acoustic energy due to absorption, the unitarity condition of the usual S matrix arising from the classical resonant scattering theory fails. However, it is shown that this unitarity condition can still be met by constructing a new S matrix which is nondiagonal. This construction is analogous to the multichannel scattering theory of particle physics. Computations are performed considering an absorbing elastic cylinder in order to check the validity of the theory.

Band gap structures in underwater screens of periodically spaced porous plates.

Acoustic properties of different periodic structures composed of alternating fluid and fluid-saturated porous layers obeying Biots theory are investigated. At first, the network of modes and the transmission coefficients of finite structures of six plates are studied in the frequency-angle of incidence plane. It is shown that the network of modes concentrates in localized domains of the plane where the transmission coefficients will take the greatest values. With this minimum of six plates, the structures exhibit the main features as for structures containing more plates, especially those with an infinite number of plates. Then, considering infinite structures the band gap calculations are led using the Bloch-Floquet theorem. The evanescent and propagative zones in the frequency-angle of incidence plane are determined. What is proposed here is a class of underwater porous screens that exhibits band gaps extending over great angular domains and enlarging in the frequency domain when the pores at the interfaces of the porous plates are sealed. The effect of porosity on the band gaps is also investigated.