Rayleigh velocity

In general a sextic equation has six roots, but an allowable solution for the Rayleigh velocity, vR, must be both real and positive, and only one such solution exists (Auld 1973 Brekhovskikh and Godin 1990). 

The value of the Rayleigh velocity depends primarily on the shear wave velocity of the material, and rather less strongly on the ratio of the shear velocity to the longitudinal velocity. The ratio of the Rayleigh velocity to the shear velocity may be considered as a function of the Poisson ratio cr. From Table 6.1, the Poisson ratio may be expressed as... 

For the permissible range of a from 0 to 0.5, over which vs/v ranges from 1/ /2 to 0, the Rayleigh velocity varies from about 87 to 95 per cent of the shear velocity, although at the latter limit both the shear and the Rayleigh velocities vanish. An approximate solution to (6.57) is (Scruby etal. 1987)... 

The most prominent feature in most V(z) curves, after the central maximum at focus, is the series of oscillations at negative defocus associated with Rayleigh wave excitation. It is perhaps therefore not surprising that the most accurate information in the reconstructions of R 6) concerns the Rayleigh velocity. The period of the Rayleigh oscillations is... 

Hence, by measuring the period of the Rayleigh oscillations, the Rayleigh velocity may be deduced directly. Likewise, the exponential decay of the Rayleigh oscillations is often of the form (Kushibiki et al. 1982)... 

Hence the Rayleigh velocity and normalized attentuation can be calculated ... 

The formula for determining the Rayleigh velocity vr from the spatial period Az of the oscillations in the V(z) curve, with water velocity v0 and frequency /, is... 

Thus the dependence of the measured Rayleigh velocity on the three experimental quantities is... 

This table gives the accuracy required in each experimental parameter in order to measure a Rayleigh velocity in the vicinity of 3000 m s-1 from V(z) with water as the coupling fluid, assuming the other parameters are exact. If each parameter contributes equal error, then from (8.47) each tolerance must be reduced by 1/ /3. 

For a measurement of a Rayleigh velocity in the vicinity of 3000 m s 1, if the fractional errors in / and Az are Sf/f and SAz/Az and the error in the temperature measurement is ST (measured in degrees kelvin), then the overall fractional measurement error is... 

Fig. 9.13. Quantitative analysis of the contrast from a white spot lesion in human tooth enamel (i) micrograph, 370 MHz (ii) V(z) curves of selected points along the line in the micrograph (iii) Rayleigh velocity and attenuation calculated from V(z) measured at each of the points on the line in (i) (Peck et al. 1989).

Rayleigh angle, and hence the Rayleigh velocity, by the formulae given in eqns (7.7) and (7.8). 

Fig. 9.14. Buccal surface of human molar tooth. Rayleigh velocity and attenuation measured using a line-focus-beam lens at 225 MHz. The sketches indicate the approximate measurement angles at which the Rayleigh wave propagation was in the direction of the arrows... 

Fig. 9.16. Single crystals of apatite compounds (a) V(z) measured with a line-focus-beam lens at 225 MHz (b) measured and calculated Rayleigh velocity in the surface of a hydroxyapatite crystal (c) measured and calculated Rayleigh velocity in the surface of... 

These values give an indication of the fractional variation of the optical refractive index and the Rayleigh velocity as a function of the dopant concentration. However, the relationship may not be linear over the range indicated (especially Ge02) and the original data should be consulted for accurate purposes (Jen etal. 1989). 

The Rayleigh velocity of the fibres and the matrix were measured in this study, and had average values of 3074 ms"1 for the matrix (cf. the value for titanium in Table 6.3), 6610 m s"1 for the /3-SiC in the fibres (somewhat higher than the range of values for most other solids), and 2766 ms"1 in the carbon core. Such measurements played an important part in the interpretation of results in the very extensive study from which these pictures... 

Fig. 11.9. Angular dependence of the Rayleigh velocity in Cu-20%Nb with draw ratio (a) = 3.6 (b) rj = 5.4 (Thompson etal. 1990a).

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