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Time-harmonic elastic scattering by unbounded deterministic and random rough surfaces in three dimensions

Guanghui Hu, Tianjiao Wang, Xiang Xu, Yue Zhao

Abstract

In this paper, we investigate well-posedness of time-harmonic scattering of elastic waves by unbounded rigid rough surfaces in three dimensions. The elastic scattering is caused by an $L^2$ function with a compact support in the $x_3$-direction, and both deterministic and random surfaces are investigated via the variational approach. The rough surface in a deterministic setting is assumed to be Lipschitz and lie within a finite distance of a flat plane, and the scattering is caused by an inhomogeneous term in the elastic wave equation whose support lies within some finite distance of the boundary. For the deterministic case, a stability estimate of elastic scattering by rough surface is shown at an arbitrary frequency. It is noticed that all constants in {\it a priori} bounds are bounded by explicit functions of the frequency and geometry of rough surfaces. Furthermore, based on this explicit dependence on the frequency together with the measurability and $\mathbb{P}$-essentially separability of the randomness, we obtain a similar bound for the solution of the scattering by random surfaces.

Time-harmonic elastic scattering by unbounded deterministic and random rough surfaces in three dimensions

Abstract

In this paper, we investigate well-posedness of time-harmonic scattering of elastic waves by unbounded rigid rough surfaces in three dimensions. The elastic scattering is caused by an function with a compact support in the -direction, and both deterministic and random surfaces are investigated via the variational approach. The rough surface in a deterministic setting is assumed to be Lipschitz and lie within a finite distance of a flat plane, and the scattering is caused by an inhomogeneous term in the elastic wave equation whose support lies within some finite distance of the boundary. For the deterministic case, a stability estimate of elastic scattering by rough surface is shown at an arbitrary frequency. It is noticed that all constants in {\it a priori} bounds are bounded by explicit functions of the frequency and geometry of rough surfaces. Furthermore, based on this explicit dependence on the frequency together with the measurability and -essentially separability of the randomness, we obtain a similar bound for the solution of the scattering by random surfaces.
Paper Structure (6 sections, 12 theorems, 165 equations)

This paper contains 6 sections, 12 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Analysis of the variational problem for small frequency
  4. An a priori bound for smooth rough surfaces
  5. Well-posedness for random rough surfaces
  6. Conclusion

Key Result

Theorem 2.1

For any $\omega>0$, the Variational Problem I (3.16) is uniquely solvable in $V_h$. Moreover, there exists a constant $C$ independent of $\omega, h$ and the Lipschitz constant $L$ of $f$ such that the solution satisfies the estimate where and Here

Theorems & Definitions (20)

  • Theorem 2.1
  • Lemma 3.1
  • Remark 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 10 more