Rayleigh and Stoneley waves in linear elasticity

Abstract

We construct microlocal solutions of Rayleigh and Stoneley waves in isotropic linear elasticity with the density and the Lamé parameters smooth up to a curved boundary or interface. We compute the direction of the microlocal polarization and show a retrograde elliptical motion of these two type of waves.

Approximation of the elastic Dirichlet-to-Neumann map

“…2,∞ , j = 1, 2, are matrices defined by replacing in the definition of the matrix M 2,∞ above c p , c s by c p,j , c s,j . We now conjecture that, under the conditions (8), ( 9) and (10), the elastic transmission eigenvalues in {τ ∈ C : Re τ ≥ 1} are located in a parabolic region of the form {τ ∈ C : Re τ ≥ 1, |Im τ | ≤ C(Re τ ) α } with some 0 < α < 1, while those in {τ ∈ C : 0 < Re τ ≤ 1} are either finitely many or asymptotically close to the imaginary axis. Proving this rigorously, however, remains a difficult, open problem.…”

“…Note that microlocal parametrices have been recently constructed in [1], [4], [10] for the wave elastic equation in the case d = 3. All these parametrices, however, are very different from the parametrix we construct in the present paper.…”

“…In contrast, our parametrix remains valid even when the boundary data is microlocally supported near Σ s and Σ p , provided θ ≥ h 2/5− . In [10] the principal symbol of the DN map has been computed explicitly still in the context of the wave elastic equation and d = 3. Note that the formula in [10] agrees with that one we get in the present paper modulo a conjugation by a unitary matrix and after making a suitable change in the notations.…”

“…In [10] the principal symbol of the DN map has been computed explicitly still in the context of the wave elastic equation and d = 3. Note that the formula in [10] agrees with that one we get in the present paper modulo a conjugation by a unitary matrix and after making a suitable change in the notations. In [5] a full parametrix was constructed for the stationary elastic equation in the exterior of a strictly convex body when d = 3 and the Lamé parameters being constants.…”

“…It plays an important role because conjugation by U 0 (ξ) allows to diagonalize the principal symbol of the elastic operator −∆ λ,µ in the half-space {x 1 > 0}. Note that a similar matrix-valued function is also used in the analysis in [4] and [10]. We use it to build the amplitudes A s and A p in the parametrix construction in Section 5.…”