Contrast theory of surface cracks

The presence of a crack or other discontinuity presents a serious difficulty for the standard V z) theory, because the reflectance function is defined for infinite plane waves that are reflected into infinite plane waves, and this requires a reflecting surface that is uniform. If a surface contains a crack then this requirement is violated, and an incident plane wave may be scattered into a whole family of waves (Tew et al. 1988). This scattering can be described in k-space by a scattering function S(kx, k x), where the prime refers to incident waves and the unprime to scattered waves. The x-direction is taken as tangential to the surface, and at this stage the theory is confined to two dimensions in the plane normal to both the surface and the crack. The response of the microscope can then be written in terms of the scattering function by integrating over the incident and reflected waves separately 

The experimental observation that Rayleigh wave scattering plays a dominant role in the contrast suggests that the reflection coefficient should be separated into the geometrical and Rayleigh parts in the way described in 7.2.1  

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 

When a crack is present eqn (12.4) must be extended to include a total of four terms. Let the crack be situated at the origin, and let it be characterized by coefficients of reflection Rr and transmission TR for Rayleigh waves. Consider first the response at a point x 0 to the left of the crack. The contributions are  

These four terms give an expression for the complete response at x  

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