Semiclassical inverse spectral problem for elastic Rayleigh waves in isotropic media

The transmission problem in linear isotropic elasticity

We study the isotropic elastic wave equation in a bounded domain with boundary with coefficients having jumps at a nested set of interfaces satisfying the natural transmission conditions there. We analyze in detail the microlocal behavior of such solution like reflection, transmission and mode conversion of S and P waves, evanescent modes, Rayleigh and Stoneley waves. In particular, we recover Knott's equations in this setting. We show that knowledge of the Dirichlet-to-Neumann map determines uniquely the speed of the P and the S waves if there is a strictly convex foliation with respect to th…

Rayleigh-type Surface Quasimodes in General Linear Elasticity

Rayleigh-type surface waves correspond to the characteristic variety, in the elliptic boundary region, of the displacement-to-traction map. In this paper, surface quasimodes are constructed for the reduced elastic wave equation, anisotropic in general, with traction-free boundary. Assuming a global variant of a condition of Barnett and Lothe, the construction is reduced to an eigenvalue problem for a selfadjoint scalar first order pseudo-differential operator on the boundary. The principal and the subprincipal symbol of this operator are computed. The formula for the subprincipal symbol seems …

Geometric optics for surface waves in nonlinear elasticity

This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. We consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an \emph{approximate} Rayleigh wave solution to the underlying elasticity equations. This evolution equation,…

Propagation of Polarization in Transmission Problems

For geometric systems of real principal type, we define a subprincipal symbol and derive a transport equation for polarizations which, in the scalar case, is a well-known equation of Duistermaat and Hörmander. We apply the transport equation to propagation of polarization in transmission problems of elastodynamics, to interior bulk waves as well as to free (Rayleigh) surface waves. Using spectral factorizations of matrix polynomials having real spectrum, we establish reflection and refraction laws of polarizations at the boundary and at interior interfaces. The results are not limited to isotr…

Reconstruction of Lame moduli and density at the boundary enabling directional elastic wavefield decomposition

We consider the inverse boundary value problem for the system of equations describing elastic waves in isotropic media on a bounded domain in $\mathbb{R}^3$ via a finite-time Laplace transform. The data is the dynamical Dirichlet-to-Neumann map. More precisely, using the full symbol of the transformed Dirichlet-to-Neumann map viewed as a semiclassical pseudodifferential operator, we give an explicit reconstruction of both Lamé parameters and the density, as well as their derivatives, at the boundary. We also show how this boundary reconstruction leads to a decomposition of incoming and outgoin…

线性弹性各向同性Cosserat材料中Love波的传播

Propagation of Love waves in linear elastic isotropic Cosserat materials

本研究提出基于代数Riccati方程的方法分析Cosserat材料中Love波传播，保证亚声速波速存在唯一性，并确认无需人为表面层。

半经典格林函数与拉格朗日相交及其在非均匀介质中贝塞尔光束传播的应用

Semiclassical Green functions and Lagrangian intersection. Applications to the propagation of Bessel beams in non-homogeneous media

本文研究局域化右端项问题的半经典渐近，利用拉格朗日相交和Maslov双典范算子构造参数，并将其应用于非均匀介质中贝塞尔光束的传播分析。