arXiv subjects

Armand Wirgin

Publications and source records attributed to Armand Wirgin.

At least 19 recordsLinked to original sources

arXiv2026

Generalization of Rayleigh's high-frequency theory for the 2D Helmholtz equation in a half-space subject to a radiation condition at infinity and a Dirichlet condition on a 1D periodically-uneven boundary

The 2D Helmholtz equation, radiation condition and Dirichlet boundary condition, are the translation, in mathematical terms, of (at least) three physical 2D problems for the prediction of the total scalar wavefield on one side of an impenetrable 1D periodically uneven boundary when: a) a plane TE electromagnetic wave propagating in the vacuum strikes the boundary the other side of which is occupied by a perfectly-conducting medium, b) a plane SH elastic elastic wave strikes a rigid boundary, c) a plane acoustic wave strikes a pressure-release boundary. The first attempt to solve such problems in a non-heuristic manner was made by Lord Rayleigh (in his book, 'The Theory of Sound' which appeared in 1896). My task will be to revisit Rayleigh's theory of diffraction by a sinusoidal-shaped, impenetrable boundary, and more specifically, his perturbation method for obtaining a mathematically-explicit solution to the diffraction problem in the high-frequency regime. In so doing, I shall correct and generalize Rayleigh's method to obtain solutions for arbitrary angles of incidence as well as for 1D periodic impenetrable boundaries of quite-general shape.

physics.class-ph↗

arXiv2025

Resonant Seismic Motion Within a Sedimentary Basin Revisited

The problem, of 2D canonical nature, examined herein in the space-frequency framework, concerns a SH-polarized plane body seismic wave propagating in a hard, non lossy half space (bedrock) containing a soft, lossy cylindrical basin of semi-circular shape. The displacement response, at points both within and outside the basin is represented alternatively in integral (for basins of general shape) and partial wave (for basins of semi-circular shape) forms, the integral representation leading to an expression of a sort of conservation of energy relation, and an explanation of how the energetic gain provided by the incident wave, is divided essentially into two sources of loss: radiation damping (taking place in the bedrock) and volumic material damping (taking place within the basin), whereas the partial wave representation enables a detailed theoretical demonstration of the resonant nature of the response, marked by amplification and concentration of the displacement field essentially within the basin at certain (eigen-)frequencies and very little, or no amplification and concentration, at other frequencies. These phenomena carry over to the volumic material damping and radiation damping loss functions (of frequency), with the equivalent of resonant displacement response residing in peaks of the volumic material damping loss function and troughs of the radiation damping loss function. Maps, for the semi-circular basin, are computed herein of the displacement field both at resonant and non-resonant frequencies which strongly evoke those (of numerical nature resulting from the boundary element scheme) appearing in two publications by other authors for basins of more realistic shape, so as to suggest that the results in the present study are applicable, for a large part, to basins of general shape.

physics.geo-ph↗

arXiv2020

The spurious resonance disease and how to cure it: application to the seismic response of a canyon

Three types of boundary integral equation (BIE) methods are employed to obtain closed-form solutions of a wave-scattering problem which are compared to the exact, closed-form (reference), solution deriving from the separation-of-variables technique. The problem involves either Dirichlet (D) or Neumann (N) boundary conditions (BC) for a scatterer that is a circular cylinder submitted to one or two incident waves. The three BIE methods lead to different expressions for the traction (for D-BC) or boundary displacement (for N-BC) by which numerous resonances are predicted whose frequency of occurrence differs from one method to another. This is interpreted as being the sign that the three methods are generally-defective and the resonances are 'spurious'. This 'disease' is cured by combining two BIE into one in such a way that the resulting BIE gives rise to a closed-form solution identical to the exact reference solution devoid of spurious resonances.

physics.geo-ph↗

arXiv2020

Parametric study of the seismic response of a hill or mountain

The problem of the response of a cylindrical protuberance of rectangular shape to a SH seismic plane wave is studied in parametric manner so as to provide answers to the questions: (i) where and how should one measure this response, (ii) is the normal-incidence response a valid indication of response at other incident angles of the seismic plane wave, iii) is it possible to predict the maximal response without a detailed knowledge of the subsurface composition, iv) how does the aspect ratio of the protuberance affect the resonant response, and v) is the resonant wavefield uniformly distributed in the interior of the protuberance?

physics.geo-ph↗

arXiv2020

Resonant amplified seismic response within a hill or mountain

We show theoretically what is meant by the term '(surface shape) resonance' in connection with the seismic response of a protuberance (emerging from flat ground) such as a hill or mountain of arbitrary shape. We address the specific problem of cylindrical protuberances of rectangular shape submitted to a SH plane wave. We find that the principal (i.e., qualitative) characteristics of the seismic response of a mountain are quite similar to those of a hill, and that the occurrence of significative amplification of the displacement field within these structures is due to the coupling of the incident wave to (surface shape) resonances.

physics.geo-ph↗

arXiv2020

A conservation law for testing methods of prediction of the seismic wave response of a protuberance emerging from flat ground

We establish the equations which translate a conservation law for the problem of the seismic response of an above-ground structure (e.g., building, hill or mountain) of arbitrary shape and inquire whether both the implicit (formal) and explicit (numerical) solutions for the response obey this law for the case of a cylindrical, rectangular protuberance. Both the low-order (poor approximations of the response) as well as higher-order (supposedly better approximations) turn out to satisfy the conservation of flux relation, which means that the satisfaction of this relation is a necessary, but not sufficient, means for determining whether a solution to the scattering problem is valid.

physics.geo-ph↗

arXiv2019

A Conservation law for a sedimentary basin submitted to a seismic wave

We establish the abstract and semi-abstract equations that translate the conservation law for a totally-filled, or totally-empty basin of arbitrary shape submitted to a SH plane seismic body wave. This law states that the input flux is equal to the sum of the scattered and absorbed fluxes, the latter being equal to zero when the filler medium is non-lossy. We show that the well-known, exact, solutions for the case of a semi-circular basin indeed satisfy this law. The latter can, and should, be employed to test the validity of any theoretical or numerical solution one might propose for a basin of arbitrary shape.

physics.geo-ph↗

arXiv2019

Forward and inverse acoustic scattering problems involving the mass density

This investigation is concerned with the 2D acoustic scattering problem of a plane wave propagating in a non-lossy fluid host and soliciting a linear, isotropic, macroscopically-homogeneous, lossy, flat-plane layer in which the mass density and wavespeed are different from those of the host. The focus is on the inverse problem of the retrieval of either the layer mass density or the real part of the layer wavespeed. The data is the transmitted pressure field, obtained by simulation (resolution of the forward problem) in exact, explicit form via separation of variables. Another form of this solution, which is exact and more explicit in terms of the mass-density contrast (between the host and layer), is obtained by a domain-integral method. A perturbation technique enables this solution to be cast as a series of powers of the mass density contrast, the first three terms of which are employed as the trial models in the treatment of the inverse problem. The aptitude of these models to retrieve the mass density contrast and real part of the layer wavespeed is demonstrated both theoretically and numerically. Keywords: 2D acoustics, forward scattering, inverse scattering, domain integral equation, constant mass density assumption, small density contrast, retrieval accuracy

physics.app-ph↗

arXiv2019

On the constant constitutive parameter (e.g., mass density) assumption in integral equation approaches to (acoustic) wave scattering

In 2D acoustic and elastodynamic problems the spatial variability of a constitutive parameter such as the mass density makes it difficult to employ boundary integral and domain integral techniques to solve the forward and inverse wave scattering problems. The oft-employed method for avoiding this problem is to assume this constitutive parameter (which is chosen herein to be the mass density) to be spatially-invariant throughout all space. The reliability of this assumption is evaluated both theoretically and numerically and it is shown, in the example of a canonical-shaped scattering obstacle, that the scattered field can be obtained in the form of a series of powers of the mass density contrast (the latter vanishing for constant mass density). The first term of this series is the solution for the scattered field corresponding to the constant density assumption and it is shown that taking into account only two more terms in the series enables to correct for practically all the errors incurred by the constant mass density assumption for a wide range of the other constitutive parameters and frequencies. It is shown how to apply this result for obstacles of non-canonical shape.

physics.geo-ph↗

arXiv2018

Incorporation of macroscopic heterogeneity within a porous layer to enhance its acoustic absorptance

We seek the response, in particular the spectral absorptance, of a rigidly-backed periodically-(in one horizontal~~ direction) ~inhomogeneous ~layer ~composed ~of ~alternating rigid and macroscopically-homogeneous porous portions, submitted to an airborne acoustic plane body wave. The rigorous theory of this problem is given and the means by which the latter can be numerically solved are outlined. At low frequencies, a suitable approximation derives from one linear equation in one unknown. This approximate solution is shown to be equivalent to that of the problem of the same wave incident on a homogeneous, isotropic layer. The thickness $h$ of this layer is identical to that of the inhomogeneous layer, the effective complex body wave velocity therein is identical to that of the porous portion of the inhomogeneous layer, but the complex effective mass density, whose expression is given in explicit algebraic form, is that of the reference homogeneous macroscopically-porous layer divided by the filling factor (fraction of porous material to the total material in one grating period). This difference of density is the reason why it is possible for the lowest-frequency absorptance peak to be higher than that of a reference layer. Also, it is shown how to augment the height of this peak so that it attains unity (i.e., total absorption) and how to shift it to lower frequencies, as is required in certain applications.

physics.app-ph↗

arXiv2018

Effective medium description of the resonant elastic-wave response at the periodically-uneven boundary of a half-space

A periodically-uneven (in one horizontal direction) stress-free boundary covering a linear, isotropic, homogeneous, lossless solid half space is submitted to a vertically-propagating shear-horizontal plane, body wave. The rigorous theory of this elastodynamic scattering problem is given and the means by which it can be numerically solved are outlined. At quasi-static frequencies, the solution is obtained from one linear equation in one unknown. At higher, although still low, frequencies, a suitable approximation of the solution is obtained from a system of two linear equations in two unknowns. This solution is shown to be equivalent to that of the problem of a vertically-propagating shear-horizontal plane body wave traveling in the same solid medium as before, but with a linear, homogeneous, isotropic layer replacing the previous uneven boundary. The thickness of this layer is equal to the vertical distance between the extrema of the boundary uneveness and the effective body wave velocity therein is equal to that of the underlying solid, but the effective shear modulus of the layer, whose expression is given in explicit algebraic form, is different from that of the underlying solid, notably by the fact that it is dispersive and lossy. It is shown that this dispersive, lossy effective layer, overriding the nondispersive, lossless solid half space, gives rise to two distinctive features of low-frequency response: a Love mode resonance and a Fixed-base shear wall pseudo-resonance. This model of effective layer with dispersive, lossy properties, enables simple explanations of how the low-frequency resonance and pseudo-resonance vary with the geometric parameters (and over a wide range of the latter) of the uneven boundary.

physics.comp-ph↗

arXiv2018

Dynamical homogenization of a transmission grating

A periodic assembly of acoustically-rigid blocks (termed 'grating'), situated between two half spaces occupied by fluid-like media, lends itself to a rigorous theoretical analysis of its response to an acoustic homogeneous plane wave. This theory gives rise to two sets of linear equations, the first for the amplitudes of the waves in the space between successive blocks, and the second for the amplitudes of the waves in the two half spaces. The first set is solved numerically to furnish reference solutions. The second set is submitted to low-frequency approximation procedure whereby the pressure fields are found to be those for a configuration in which the grating becomes a homogeneous layer of the same thickness as the height of the blocks in the grating. A simple formula is derived for the constitutive properties of this layer in terms of those of the fluid-like medium in between the blocks. The homogeneous layer model scattering amplitude transfer functions and spectral reflectance, transmittance and absorptance reproduce quite well the corresponding rigorous numerical functions of the grating over a non-negligible range of low frequencies. Due to its simplicity, the homogeneous layer model enables theoretical predictions of many of the key features of the acoustic response of the grating.

physics.app-ph↗

arXiv2018

Computational parameter retrieval approach to the dynamic homogenization of a periodic array of rigid rectangular blocks

We propose to homogenize a periodic (along one direction) structure, first in order to verify the quasi-static prediction of its response to an acoustic wave arising from mixing theory, then to address the question of what becomes of this prediction at higher frequencies. This homogenization is treated as an inverse (parameter retrieval) problem, i.e., by which we: (1) generate far-field (i.e., specular reflection and transmission coefficients) response data for the given periodic structure, (2) replace (initially by thought) this (inhomgoeneous) structure by a homogeneous (surrogate) layer, (3) compute the response of the surrogate layer response for various trial constitutive properties, (4) search for the global minimum of the discrepancy between the response data of the given structure and the various trial parameter responses (5) attribute the homogenized properties of the surrogate layer for which the minimum of the discrepancy is attained. The result is that: (i) at low frequencies and/or large filling factors, the effective constitutive properties are close to their static equivalents, i.e., the effective mass density is the product of a factor related to the given structure filling factor with the mass density of a generic substructure of the given structure and the effective velocity is equal to the velocity in the said generic substructure, 2) at higher frequencies and/or smaller city filling factors, the effective constitutive properties are dispersive and do not take on a simple mathematical form, with this dispersion compensating for the discordance between the ways the inhomogeneous given structure and the homogeneous surrogate layer respond to the acoustic wave.

physics.app-ph↗

arXiv2018

Seismic response in modern cities

The proposed homogeneous flat-faced layer-like model of a city (termed overlayer), covering what is generally considered to be a dangerous site (from the point of view of seismic hazard) lends itself to an explicit theoretical analysis of its response to a seismic body wave radiated by distant sources. This study is carried out for: ground response of the complete site/overlayer configuration which is compared to the response of the configuration in which the overlayer is absent, response at the top of the layer for various layer thicknesses, and determination, as a function of frequency, of the fraction of incident flux that is dissipated in the overlayer, the underlying layer and, by radiation damping, in the hard half space. It is shown that all of these entities are highly frequency-dependent and even large in certain frequency intervals, without any resonant (in the sense of mode excitation) phenomena coming into play. The results of this study also show that transfer functions do not necessarily reflect the global response in the built component of a city and that more-appropriate energy-related functions, termed spectral absorptance (in the blocks of the city or their layer-like surrogate at the characteristic frequency of the seismic pulse) and absorptance (integral over frequency of the spectral absorptance), can increase with increasing city density or increasing city height. In fact, it is shown that more than a third of the incident seismic energy can be sent into, and therefore cause serious damage to, the built component. On the basis of these findings, it appears that the probable evolution of the morphology and constitutive properties of cities with time will make the latter more vulnerable to damage and destruction when submitted to seismic waves.

physics.geo-ph↗

arXiv2017

On the velocity of sound in water: theoretical aspects of Colladon's nineteenth century experiments

In 1827, Colladon carried out a series of experiments in Lac Leman (Lake Geneva, Switzerland) to measure the speed of sound in water. The purpose of our contribution is to treat this measurement as an inverse problem, and show, by theory how to solve the latter. It is thus revealed under what circumstances it is legitimate to employ the time-of-flight scheme underlying the Colladon experiments and how to bypass this scheme in order to fully account for the temporal and geometric characteristics of the source (of sound), the temporal characteristics of the received signal and the error incurred by the finite distance between the source and receiver.

physics.hist-ph↗

arXiv2016

Earthquakes in cities revisited

During the last twenty years, a number of publications of theoretical-numerical nature have appeared which come to the apparently-reassuring conclusion that seismic motion on the ground in cities is smaller than what this motion would be in the absence of the buildings (but for the same underground and seismic load). Other than the fact that this finding tells nothing about the motion within the buildings, it must be confronted with the overwhelming empirical evidence (e.g, earthquakes in Sendai (2011), Kathmandu (2015), Tainan City (2016), etc.) that shaking within buildings of a city is often large enough to damage or even destroy these structures. I show, on several examples, that theory can be reconciled with empirical evidence, and suggest that the crucial subject of seismic response in cities is in need of more thorough research.

physics.geo-ph↗

arXiv2007

Acoustic response of a rigid frame porous medium slab with a periodic set of inclusions

The acoustic response of a rigid frame porous slab with a periodic set of inclusions is calculated by use of a multipole method. The acoustic properties, in particular the absorption, of such a structure are then derived and studied. Numerical results together with a modal analysis show that the addition of a periodic set of high-contrast inclusions leads to quasi-modes excitation of both the slab and the gratings, and to a large increase of the acoustic absorption of the initial slab, this being partly due to the quasi-modes excitation.

physics.class-ph↗

arXiv2006

Seismic motion in urban sites consisting of blocks in welded contact with a soft layer overlying a hard half space: II. large and infinite number of identical equispaced blocks

We address the problem of the response to a seismic wave of an urban site consisting of a large and infinite number ($N\_{b}$) of identical, equispaced blocks overlying a soft layer underlain by a hard substratum. The results of the theoretical analysis, appealing to a space-frequency mode-matching (MM) technique, are compared to those obtained by a space-time finite element (FE) method. The two methods are shown to give rise to much the same prediction of seismic response for $N\_{b}=10$. The MM technique is also applied to the case $N\_{b}=\infty$, notably to reveal the structure and natural frequencies of the vibration modes of the urban site. The mechanism of the interaction between blocks and the ground, as well as that of the collective effects of the blocks, are studied. It is shown that the presence of a large number of blocks modifies the seismic disturbance in a manner which evokes, and may partially account for, what was observed during many earthquakes in Mexico City. Disturbances at a much smaller level, induced by a small number of blocks are studied in the companion paper.

physics.geo-ph↗