Axisymmetric Scattering of a Plane Longitudinal Wave by a Circular Crack in a Transversely Isotropic Solid

1991 ◽  
Vol 58 (3) ◽  
pp. 695-702 ◽  
Author(s):  
T. Kundu ◽  
A. Bostro¨m
Keyword(s):  
Integral Equation ◽  
Longitudinal Wave ◽  
Legendre Polynomials ◽  
Circular Crack ◽  
Transversely Isotropic ◽  
Edge Condition ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Isotropic Solid ◽  
Plane Longitudinal Wave

The scattering of elastic waves by a circular crack situated in a transversely isotropic solid is studied here. The axis of material symmetry and the axis of the crack coincides. The incident wave is taken as a plane longitudinal wave propagating perpendicular to the crack surface. A Hankel transform representation of the scattered field is used, and after some manipulations using the boundary conditions this leads to an integral equation over the crack for the displacement jump across the crack. This jump is expanded in a series of Legendre polynomials which fulfill the correct edge condition and the integral equation is projected on the same set of Legendre polynomials. The far field is computed by the stationary phase method. A few numerical computations are carried out for both isotropic and anisotropic solids. Results for the isotropic solid compare favorably with those available in the literature.

Determining the distribution of ion channels from experimental data

2018 ◽  
Vol 52 (5) ◽  
pp. 2083-2107
Author(s):  
Thibault Bourgeron ◽  
Carlos Conca ◽  
Rodrigo Lecaros
Keyword(s):  
Experimental Data ◽  
Inverse Problem ◽  
Integral Equation ◽  
Olfactory System ◽  
Numerical Scheme ◽  
Mellin Transform ◽  
Electrical Signal ◽  
Original Model ◽  
Phase Method ◽  
Stationary Phase Method

The authors study an integral inverse problem arising in the biology of the olfactory system. The transduction of an odor into an electrical signal is accomplished by a depolarising influx of ions through cyclic-nucleotide-gated (CNG for short) channels on the cilium membrane. The inverse problem studied in this paper consists in finding the spatial distribution of the CNG channels from the measured transduce electrical signals. The Mellin transform allows us to write an explicit formula for its solution. Proving observability and continuity inequalities is then a question of estimating the Mellin transform of the kernel of this integral equation on vertical lines. New estimates using arguments in the spirit of the stationary phase method are proven and a numerical scheme is proposed to reconstruct the density of CNG channels from modeled current representing experimental data, for an approximated model. For the original model an identifiability and a non observability (in some weighted L2spaces) results are proven.

A Method to Evaluate Three-Dimensional Time-Harmonic Elastodynamic Green’s Functions in Transversely Isotropic Media

1992 ◽  
Vol 59 (2S) ◽  
pp. S96-S101 ◽  
Author(s):  
H. Zhu
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Isotropic Case ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Transversely Isotropic Media ◽  
Time Harmonic ◽  
Isotropic Media

The three-dimensional time-harmonic elastodynamic Green’s functions in infinite transversely isotropic media have been derived explicitly. The Green’s functions consist of the corresponding static Green’s functions and double integral representations over a finite domain with the integrands being continuous. The Green’s functions will reduce to those for the isotropic case when the isotropic elastic constants are substituted. The singular parts of the Green’s functions have been shown to be the same as those of the static ones. The far-field approximations have been obtained by using the stationary phase method. In addition, a simpler method to construct wave front curves has been presented.

scholarly journals Dispersive Estimates for Full Dispersion KP Equations

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Didier Pilod ◽  
Jean-Claude Saut ◽  
Sigmund Selberg ◽  
Achenef Tesfahun
Keyword(s):  
Stationary Phase ◽  
Initial Value Problem ◽  
Bessel Functions ◽  
Linear Part ◽  
Independent Interest ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Initial Value ◽  
Kp Equations ◽  
Well Posed

AbstractWe prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev–Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev–Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of independent interest. As a consequence, we prove that the initial value problem associated to the Full Dispersion Kadomtsev–Petviashvili is locally well-posed in $$H^s(\mathbb R^2)$$ H s ( R 2 ) , for $$s>\frac{7}{4}$$ s > 7 4 , in the capillary-gravity setting.

Splitting in lateral shift induced by strong coupling in Kretschmann configuration involving molecular J-aggregates

2019 ◽  
Vol 33 (30) ◽  
pp. 1950370
Author(s):  
Kunwei Pang ◽  
Haihong Li ◽  
Gang Song ◽  
Pengfei Zhang
Keyword(s):  
Strong Coupling ◽  
Metal Film ◽  
Previous Experiment ◽  
Lateral Shift ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Incident Wavelength ◽  
Rabi Splitting ◽  
Kretschmann Configuration ◽  
J Aggregates

Molecular J-aggregates are widely used as emitters to achieve the quantum effects, such as the strong coupling phenomenon. We investigate the lateral shift splitting/Goos–Hänchen (GH) shift splitting induced by strong coupling in Kretschmann configuration involving molecular J-aggregates by using classical methods. The optical response of molecular J-aggregates is modeled by a single Lorentzian oscillator, and Fresnel equations and the stationary phase method are employed to solve our proposed structure. Our results show that the lateral shift versus the incident wavelength shows Rabi splitting-like line shape and the reflection spectrum exhibits the strong coupling phenomenon. Based on the results of the previous experiment work, we well explain the relation between Rabi splitting and the thickness of the metal film and provide a new method to choose the parameters of the structure for experiment.

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