Ray model of V z

The expression for the period of the oscillations in V(z) can also be derived by ray theory. The ray model is particularly powerful in the analysis of V(z) data. 

In the simple form given at the start of this chapter, the ray picture correctly predicts the period of the oscillations in V(z), but it does not enable any other aspect of V(z) to be calculated. To do that it is necessary to look more carefully at the form of the reflectance function (Bertoni 1984 Brekhovskikh and Godin 1990). Writing the expression for the reflectance function in (6.90) explicitly in terms of the wavenumber in the fluid k and the longitudinal and shear wavenumbers in the solid k and ks, and with the tangential component of the wavevector = kx, which, by Snell s law, is the same in both media, and with 

When kx k the last term in the numerator and the denominator are real, and the reflectance function is real, so that there is no variation of phase with incident angle. When kx = k all terms except the first term in the numerator and the denominator vanish, and R(kx) = 1. When k kx k, the form of (7.24) makes it explicit that the last term in the numerator and the denominator are pure imaginary. The term in square brackets depends only on kx and the properties of the solid. Provided that there is no dissipation in the solid, it vanishes when kx is equal to the Rayleigh wavenumber kR = co/vR, as can be seen by substituting X = (ks/kR)2, Y = (k /ks)2, and comparing with (6.55). 

This equation is, of course, simply a definition of Ro(kx). Its usefulness arises from the explicit separation of the geometrical and Rayleigh components of the reflectance. Moreover, the fraction (k — kfy/ik — k ) in (7.26) contributes significantly only when kx is close to kp. The Rayleigh wavenumber is always greater than the bulk shear wavevector ks by 5-10 per cent or so (Table 6.2), and since R(kx) = 1 for kx ks, again cf. Fig. 6.3(b)i, the approximation may be made that 

If the field incident on the surface of the specimen is described by pinc( ) and Pinc(kx) is its Fourier transform, then the reflected field at the surface can be found by taking the inverse Fourier transform of the product of PiBC kx) and R(kx). Lower case denotes real-space fields, upper case spatial frequency (or reciprocal-space) fields. A time dependence exp(iwt), corresponding to a frequency w/27r, is implicit throughout. If the reflected field is separated into the geometrical part Rq x) and the leaky Rayleigh wave part Pr(x), 

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