Elastic Waves in Solids, Volume 2

Appendix 2Matrix Representation of a Dioptric System

The objective of this time-harmonic representation is to express the amplitudes of bulk waves that propagate in a solid occupying the half-space x₂ > 0. These amplitudes are expressed as a function of the mechanical quantities that are conserved at an interface, that is, the three components of mechanical displacement u and those of mechanical traction: for a surface of normal l parallel to the x₂ axis. The phenomena associated with the emission, detection and diffraction of elastic waves can be formulated using a mixed matrix M, whereby the vector b for the amplitudes of the reflected (emitted) bulk waves and the mechanical amplitude u on the surface is related to the vector a for the amplitudes of the incident bulk waves and to the mechanical traction T. This matrix M consists of four sub-matrices D, E, R and Y :

[A2.1]

Generally, the vectors a and b have three components (one for each bulk wave), and the vectors u and T also have three components:

[A2.2]

The sub-matrices are thus square (3 × 3) matrices:

• – the diffraction matrix D provides the amplitudes of reflected waves as functions of the amplitudes of the incident waves ...