Matrix 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 *x*_{2} *>* 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

Generally, the vectors * a* and

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 when the surface is free (=__T____0__); - – the emission matrix
*E*gives, in the absence of incident waves (=__a____0__), the amplitudesof bulk waves created by a uniform distribution of forces__b__on the surface;__T__ - – the reception matrix
*R*is useful for calculating the mechanical displacementinduced, on the free surface (__u__=__T____0__), by bulk waves of amplitude;__a__ - – the admittance matrix
*Y*gives, in the harmonic case, the mechanical displacement created by a uniform distribution of forces exerted on the surface. The components of its Fourier transform are the Green’s functions of the half-space for mechanical surface phenomena.

Consider an incident, homogeneous plane wave of amplitude *a*, whose polarization is defined by the unit vector * p*. For transversely independent problems, the wave vector

Upon omitting the propagation factor, the stresses *σ*_{i2} are in the form:

By introducing the dimensionless parameters *m* = *k*_{2}*/k*_{1}, which defines the propagation direction in the *x*_{1}*x*_{2} plane, the mechanical traction involved in the boundary conditions on the surface *x*_{2} = 0 is given by:

The *mechanical traction* on the surface is the sum of those created by each incident wave (denoted by the index *I* = 1, 2, 3); by writing *t _{i}* =

where *A* = [*A _{iI}*] is the (3 × 3) matrix obtained by juxtaposing the column vectors

The total mechanical traction is given by the sum:

Since * t* =

The nine components of the diffraction matrix *D* are the *reflection coefficients*. For example, by attributing the index 1 to the quasi-longitudinal (QL) wave and the indices 2 and 3 to the quasi-transverse (QT) waves, the vector * a*= (

the denominator for all the reflection coefficients is the determinant of the matrix *B*.

In the absence of any incident wave (* a*=

An elastic wave with non-zero displacement * b* can propagate on the free surface of a solid. This wave was discovered in the case of an isotropic solid by Lord Rayleigh in 1885. The characteristic equation of a Rayleigh wave is therefore obtained by canceling the denominator of the reflection coefficients on the free surface of an isotropic solid. This observation can also be applied to waves that propagate along the interface between a fluid and a solid, or between two solids, whether isotropic or anisotropic.

In the absence of any incident wave (* a*=

With *Q* = [*q _{lJ}*] denoting the polarization matrix of the emitted (reflected) waves, the mechanical displacement on the surface:

can be expressed as a function of the stress vector * T* using the admittance matrix

By using *P* = [*p _{lI}*] to denote the polarization matrix of incident waves, the total mechanical displacement on the surface created by the incident waves of amplitude

Given that only two bulk waves can propagate in an isotropic solid and for the displacements * u* and mechanical tractions

where *s _{L}* = 1

with *C*_{22} = *C*_{11} and *C*_{21} = *C*_{11} − 2*C*_{66}, leads to:

For the *longitudinal wave*, the normalized components of the polarization vector are:

Using the relation the components of mechanical traction * T_{L}* are given by:

Similarly, for the *transverse wave*:

the components of the mechanical traction * T_{T}* are written as:

By introducing the Lamé constant *μ* = *C*_{66}, the stress vector * T* =

The inversion leads to the emission matrix:

where Δ is the Rayleigh determinant:

Considering relations [A2.19] and [A2.21], the polarization matrix of the reflected or emitted waves is:

and the admittance matrix *Y* can be deduced from relation [A2.14] *Y* = *QE*, with:

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