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Elastic scattering by locally rough interfaces

Chengyu Wu, Yushan Xue, Jiaqing Yang

TL;DR

The paper addresses elastic scattering by locally rough interfaces in 2D and 3D and develops a full well-posedness theory. It introduces a stress-vector identity from a Helmholtz-like decomposition that yields key surface-integral limits and a frequency-independent uniqueness result, and it leverages this to decouple waves and derive a direct-problem framework. A two-layered elastic Green’s tensor for the a0=1 transmission problem is constructed with explicit 2D and 3D representations via Fourier analysis and steepest descent, and its radiation properties are verified. Using this Green’s tensor, the authors formulate a Fredholm variational problem with a Dirichlet-to-Neumann map and prove existence of solutions, establishing the first all-frequency well-posedness result for elastic scattering by locally rough interfaces.

Abstract

In this paper, we present the first well-posedness result for elastic scattering by locally rough interfaces in both two and three dimensions. Inspired by the Helmholtz decomposition, we first discover a fundamental identity for the stress vector, revealing an intrinsic relationship among the generalized stress vector, the Lame constants and certain tangential differential operators. This identity leads to two key limits for surface integrals involving scattered solutions, from which we deduce the first uniqueness result of direct problem for all frequencies. Through a detailed analysis, applying the steepest descent method, subsequently we derive the existence and uniqueness of the corresponding two-layered Green's tensor along with its explicit expression when the transmission coefficient equals 1. Finally, by leveraging properties of the Green's tensor, we establish the existence of solutions via the variational method and the boundary integral equation, thereby achieving the first well-posedness result for elastic scattering by rough interfaces.

Elastic scattering by locally rough interfaces

TL;DR

The paper addresses elastic scattering by locally rough interfaces in 2D and 3D and develops a full well-posedness theory. It introduces a stress-vector identity from a Helmholtz-like decomposition that yields key surface-integral limits and a frequency-independent uniqueness result, and it leverages this to decouple waves and derive a direct-problem framework. A two-layered elastic Green’s tensor for the a0=1 transmission problem is constructed with explicit 2D and 3D representations via Fourier analysis and steepest descent, and its radiation properties are verified. Using this Green’s tensor, the authors formulate a Fredholm variational problem with a Dirichlet-to-Neumann map and prove existence of solutions, establishing the first all-frequency well-posedness result for elastic scattering by locally rough interfaces.

Abstract

In this paper, we present the first well-posedness result for elastic scattering by locally rough interfaces in both two and three dimensions. Inspired by the Helmholtz decomposition, we first discover a fundamental identity for the stress vector, revealing an intrinsic relationship among the generalized stress vector, the Lame constants and certain tangential differential operators. This identity leads to two key limits for surface integrals involving scattered solutions, from which we deduce the first uniqueness result of direct problem for all frequencies. Through a detailed analysis, applying the steepest descent method, subsequently we derive the existence and uniqueness of the corresponding two-layered Green's tensor along with its explicit expression when the transmission coefficient equals 1. Finally, by leveraging properties of the Green's tensor, we establish the existence of solutions via the variational method and the boundary integral equation, thereby achieving the first well-posedness result for elastic scattering by rough interfaces.

Paper Structure

This paper contains 6 sections, 15 theorems, 151 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Uniqueness of solution
  3. Two-layered Green's tensor
  4. Two-dimensional case
  5. Three-dimensional case
  6. Existence of solution

Key Result

Lemma 2.1

\newlabellem2.1 The following identities hold where $\mathcal{M}_{\boldsymbol{\nu}}$ is the differential operator defined by for $\mathbf{u}=(u_1,u_2,u_3)^\top$ and $\boldsymbol{\nu}=(\nu_1,\nu_2,\nu_3)^\top$.

Figures (3)

  • Figure 3.1: Branch cuts and domain of definition for the function $\beta_{a,\pm}$.
  • Figure 3.2: Paths of descent for the integrals \ref{['3.7']} in the case that $\sin\theta_x>0$ and $|\cos\theta_x|<\frac{k_{p,-}}{k_{p,+}}<1$, $\frac{k_{s,-}}{k_{p,+}}<\left|\frac{1}{\cos\theta_x}\right|<\frac{k_{s,+}}{k_{p,+}}$.
  • Figure 3.3: Integration path $\mathcal{C}$ in the Hankel transform and the steepest descent path $\mathcal{D}$ utilized in the asymptotic approximation.

Theorems & Definitions (28)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • ...and 18 more