Rayleigh wave exponential decay

Figure 6.2(a), which was calculated using the results to be derived here, shows the amplitudes of the longitudinal and shear components of a Rayleigh wave in fused silica, and their exponential decay below the surface. 

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... 

Tamm [3] was first to show that the special states of electrons exist near crystal surface (so-called surface states), which have the discrete energy spectrum and their wave functions decay exponentially on both sides of the surface. Similarly, the vibrations of crystal surface atoms can be considered, which also decay on both sides of the surface. In the long wave limit, one of such surface phonon branches transforms into well-known surface Rayleigh waves, while the others yield special optic branches [4, 5]. 

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