arXiv subjects

Ibrahim Baydoun

Publications and source records attributed to Ibrahim Baydoun.

4 recordsLinked to original sources

arXiv2018

Analytical formula for the roots of the general complex cubic polynomial

We present a new method to calculate analytically the roots of the general complex polynomial of degree three. Thismethod is based on the approach of appropriated changes of variable involving an arbitrary parameter. The advantageof this method is to calculate the roots of the cubic polynomial as uniform formula using the standard convention of thesquare and cubic roots. In contrast, the reference methods for this problem, as Cardan-Tartaglia and Lagrange, give theroots of the cubic polynomial as expressions with case distinctions which are incorrect using the standar convention.

math.GM↗

arXiv2016

Radiative transfer of acoustic waves in continuous complex media: Beyond the Helmholtz equation

Heterogeneity can be accounted for by a random potential in the wave equation. For acoustic waves in a fluid with fluctuations of both density and compressibility (as well as for electromagnetic waves in a medium with fluctuation of both permittivity and permeability) the random potential entails a scalar and an operator contribution. For simplicity, the latter is usually overlooked in multiple scattering theory: whatever the type of waves, this simplification amounts to considering the Helmholtz equation with a sound speed $c$ depending on position $\mathbf{r}$. In this work, a radiative transfer equation is derived from the wave equation, in order to study energy transport through a multiple scattering medium. In particular, the influence of the operator term on various transport parameters is studied, based on the diagrammatic approach of multiple scattering. Analytical results are obtained for fundamental quantities of transport theory such as the transport mean-free path $\ell^*$, scattering phase function $f$ and anisotropy factor $g$. Discarding the operator term in the wave equation is shown to have a significant impact on $f$ and $g$, yet limited to the low-frequency regime i.e., when the correlation length of the disorder $\ell_c$ is smaller than or comparable to the wavelength $λ$. More surprisingly, discarding the operator part has a significant impact on the transport mean-free path $\ell^*$ whatever the frequency regime. When the scalar and operator terms have identical amplitudes, the discrepancy on the transport mean-free path is around $300\,\%$ in the low-frequency regime, and still above $30\,\%$ for $\ell_c/λ=10^3$ no matter how weak fluctuations of the disorder are. Analytical results are supported by numerical simulations of the wave equation and Monte Carlo simulations.

physics.class-ph↗

arXiv2015

Scattering mean-free path in continuous complex media: beyond the Helmholtz equation

We present theoretical calculations of the ensemble-averaged (a.k.a. effective or coherent) wavefield propagating in a heterogeneous medium considered as one realization of a random process. In the literature, it is usually assumed that heterogeneity can be accounted for by a random scalar function of the space coordinates, termed the potential. Physically, this amounts to replacing the constant wavespeed in Helmholtz' equation by a space-dependent speed. In the case of acoustic waves, we show that this approach leads to incorrect results for the scattering mean-free path, no matter how weak fluctuations are. The detailed calculation of the coherent wavefield must take into account both a scalar and an operator part in the random potential. When both terms have identical amplitudes, the correct value for the scattering mean-free paths is shown to be more than four times smaller (13/3, precisely) in the low frequency limit, whatever the shape of the correlation function. Based on the diagrammatic approach of multiple scattering, theoretical results are obtained for the self-energy and mean-free path, within Bourret's and on-shell approximations. They are confirmed by numerical experiments.

physics.class-ph↗

arXiv2014

Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media

In this paper we develop a multiple scattering model for elastic waves in random anisotropic media. It relies on a kinetic approach of wave propagation phenomena pertaining to the situation whereby the wavelength is comparable to the correlation length of the weak random inhomogeneities--the so-called weak coupling limit. The waves are described in terms of their associated energy densities in the phase space position x wave vector. They satisfy radiative transfer equations in this scaling, characterized by collision operators depending on the correlation structure of the heterogeneities. The derivation is based on a multi-scale asymptotic analysis using spatio-temporal Wigner transforms and their interpretation in terms of semiclassical operators, along the same lines as Bal [Wave Motion 43, 132-157 (2005)]. The model accounts for all possible polarizations of waves in anisotropic elastic media and their interactions, as well as for the degeneracy directions of propagation when two phase speeds possibly coincide. Thus it embodies isotropic elasticity which was considered in several previous publications. Some particular anisotropic cases of engineering interest are derived in detail.

math-ph↗