Reflection, Transmission, and Mode Conversion of Elastic Plane Waves at Planar Boundaries between Homogeneous Nondissipative Materials

At first, we investigate reflection and-eventually-mode conversion of elastic plane waves for the planar boundary of a homogeneous isotropic nondissipative elastic half-space, assuming the “material” of the complementary half-space to be vacuum. Because vacuum with mass density zero does not support elastic waves, the complementary half-space is field-free so that the field in the elastic half-space must satisfy the (homogeneous170) boundary condition

T(R, t) · n = 0, R ∈ S, (9.1) according to Section 3.3: The boundary S with normal n is stress-free (traction-free). This standard problem of elastic wave propagation has often been treated in the literature (e.g., Achenbach 1973; Auld 1973; Ben-Menahem and Singh 1981; Harker 1988; Langenberg 1983), and even US-NDT is well aware if it (Krautkra¨mer and Krautkra¨mer 1986; Kutzner 1983; Schmerr 1998). Nevertheless, we discuss it here: First for the sake of completeness and second to appreciate, a (nearly) coordinate-free calculation171 that essentially shortens the procedure and makes it clearer. In addition, we want to depict the result with reflection and mode conversion of pulsed waves because US-NDT is in fact pulsed testing; if at all, we only find illustrations for time harmonic waves in the literature.