The transmission problem in linear isotropic elasticity

Rayleigh and Stoneley Waves in Linear Elasticity

“…In this section, we recall some notations and results in [21]. For a fixed point x 0 on the boundary, one can choose the semigeodesic coordinates x = (x , x 3 ) such that the boundary ∂Ω is locally given by x 3 = 0.…”

“…In this case, given the boundary data f = u| x 3 =0 , first we get w b ≡ w| x 3 =0 by considering the restriction operator U out of the ΨDO U to the boundary, which maps w| x 3 =0 to f . It is shown that U out is an elliptic one and therefore it is microlocally invertible in [21]. Then we seek the outgoing microlocal solution w to the two wave equations with the boundary data.…”

“…In [4], the authors develop a semiclassical analysis of elastic surface wave generated by interior (point) source and in [5] the inverse spectral problem of Rayleigh waves is studied. Most recently in [21], the authors describe the microlocal behavior of solutions to the transmission problems in isotropic elasticity with curved interfaces. Surface waves are briefly mentioned there as possible solutions of evanescent type which propagate on the boundary, see [21, §8.2].…”

“…Essentially, the existence of Rayleigh waves comes from the nonempty kernel of the principal symbol of the Dirichlet-to-Neumann map (DN map) Λ in the elliptic region. In Section 2, we briefly state some relevant results in [21] that we are going to use later. In Section 3, based on the analysis in [21], one can see the Rayleigh waves are corresponding to the solution to Λu = l on the boundary, where l is a source microlocally supported in the elliptic region.…”

“…In Section 2, we briefly state some relevant results in [21] that we are going to use later. In Section 3, based on the analysis in [21], one can see the Rayleigh waves are corresponding to the solution to Λu = l on the boundary, where l is a source microlocally supported in the elliptic region. Next, inspired by the diagonalization of the Neumann operator for the case of constant coefficients in [20], we diagonalize Λ microlocally up to smoothing operators by a symbol construction in [25].…”