Rayleigh wave excitation

It will become very clear by the end of this book that in a great deal of acoustic microscopy of materials the contrast is dominated by Rayleigh waves excited in the surface of the specimen (Briggs 1985). A summary of the properties of Rayleigh waves will be given in Table 6.2, and their role in the contrast will be introduced in 7.2.1. What all that means in terms of acoustic pictures will be... 

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... 

This is the generalized three-dimensional scattering relationship for the response just above the surface at x to an oscillatory pressure just above the surface at x, due to Rayleigh wave excitation, in the case where the y component of the wavevector is constant. The three-dimensional scattering function can now be calculated. 

There are two particular discontinuities in R(t) that are of great importance for materials in which Rayleigh waves are excited. The first occurs at t0 = 1 /n, because beyond that value Q(f) changes discontinuously to zero. The second is at fR = cos r/tt, because around the Rayleigh angle 6r there is a phase change of 2n in R 6), cf. Fig. 6.3(b)i, and hence in Q(t). The Fourier relationship gives oscillations in V u) of periodicity... 

This equation has an extremely important interpretation. In its differential form it means that, in two dimensions (with the x-axis lying in the plane of the interface between the solid and the fluid and the z-axis lying normal to the plane), if a pressure pjnc(x ) with implicit frequency dependence exp(iwt) acts along a strip in the y-direction at x of width dx, then the Rayleigh wave that is excited will propagate and the response in the fluid immediately above the surface at x will be... 

The term in the curly brackets describes the response at x due to the excitation at x, transmitted by the Rayleigh wave mechanism it can therefore be thought of as a kind of Green function. The results described by eqns (7.33) and (7.34) are central to the theory of the contrast from cracks and interfaces that will be presented in Chapter 12. 

The exponential term can be thought of as a Green function, with the time dependence always implicit. Thus an excitation at x causes a response at x whose phase is delayed by the distance between them multiplied by the real part of kp (this corresponds approximately to 2 /Ar), and whose amplitude is decreased exponentially by the distance between them multiplied by the imaginary part of kp (this corresponds to the decay associated with the propagation of the leaky Rayleigh wave). The magnitude x — x is used because... 

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