Synthesis of unilateral radiators

A radiator is typically a parabolic mirror illuminated by an electromagnetic source, or a cylindrical transducer of resonant vibrations. Both of these devices are designed to radiate either a beam of parallel rays or a (focused) beam that converges to a point or a line. Consequently, at the worst, the radiation pattern is largely restricted to a {\it half space}, and at the best, to a cone or cylinder-like subspace of this half space. Such devices can therefore be termed unilateral radiators. This study is devoted to the synthesis of the sources that can give rise to such radiation, the underlying motivation being the removal of the material presence of the mirror or transducer casing from which waves coming from other boundaries could reflect or diffract

Three methods for the description of the temporal response to a SH plane impulsive seismic wave in a soft elastic layer overlying a hard elastic substratum

We treat the case of a flat stress-free surface (i.e., the ground in seismological applications) separating air from a homogeneous, isotropic, solid substratum overlain by a homogeneous, isotropic, solid layer (in contact with the ground) solicited by a SH plane body wave incident in the substratum. The analysis is first carried out in the frequency domain and subsequently in the time domain. The frequency domain response is {\it normal} in that no resonances are excited (a resonance is here understood to be a situation in which the response is infinite in the absence of dissipation). The translation of this in the time domain is that the scattered pulse is of relatively-short duration. The duration of the pulse is shown to be largely governed by radiation damping which shows up in the imaginary parts of the complex eigenfrequencies of the configuration. Three methods are elaborated for the computation of the time history and give rise to the same numerical solutions for a large variety of configurations of interest in the geophysical setting under the hypothesis of non-dissipative, dispersionless media. The method appealing to the complex eigenfrequency representation is shown to be the simplest and most physically-explicit way of obtaining the time history (under the same hypothesis). Moreover, it is particularly suited for the case in which modes can be excited as occurs when the incident wave is not plane or the boundary condition is not of the stress-free variety for all transverse coordinates on the ground plane

The inverse crime

The inverse crime occurs when the same (or very nearly the same) theoretical ingredients are employed to synthesize as well as to invert data in an inverse problem. This act has been qualified as trivial and therefore to be avoided by Colton and Kress

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

Amplification and Increased Duration of Earthquake Motion on Uneven Stress-Free Ground

When a flat stress-free surface (i.e., the ground in seismological applications) separating air from a isotropic, homogeneous or horizontally-layered, solid substratum is solicited by a SH plane body wave incident in the substratum, the response in the substratum is a single specularly-reflected body wave. When the stress-free condition, equivalent to vanishing surface impedance, is relaxed by the introduction of a {spatially- constant, non- vanishing surface impedance}, the response in the substratum is again a single reflected body wave whose amplitude is less than the one in the situation of a stress-free ground. When the stress-free condition is relaxed by the introduction of a a {spatially-modulated surface impedance}, which simulates the action of an uneven (i.e., not entirely-flat) ground, the frequency-domain response takes the form of a spectrum of {plane body waves} and {surface waves} and {resonances} are produced at the frequencies of which one or several surface wave amplitudes can become large. It is shown, that at resonance, the amplitude of one, or of several, components of the motion on the surface can be amplified with respect to the situation in which the surface impedance is either constant or vanishes. Also, when the solicitation is pulse-like, the integrated time history of the square of surface displacement and of the square of velocity can be larger, and the duration of the signal can be considerably longer, for a spatially-modulated impedance surface than for a constant, or vanishing, impedance surface.Comment: Third International Symposium on the Effects of Surface Geology on Seismic Motion, Grenoble, 200

Simulation of seismic response in a city-like environment

We study the seismic response of idealized 2D cities, constituted by non equally-spaced, non equally-sized homogenized blocks anchored in a soft layer overlying a hard half space. The blocks and soft layer are occupied by dissipative media. To simulate such response, we use an approximation of the viscoelastic modulus by a low-order rational function of frequency and incorporate this approximation into a first-order-in-time scheme. Our results display spatially-variable, strong, long-duration responses inside the blocks and on the ground, which qualitatively match the responses observed in some earthquake-prone cities of Mexico, France, the USA, etc.Comment: 22 pages, 8 figures, submitted to SDE

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

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