Radiation conditions for elastic waves

In the conclusion of this section we demonstrate that the Sommerfeld radiation conditions can be extended to the case of an elastic wavefield U (Kupradze, 1933, 1934, 

1963 Zhdanov et al., 1988 Aki and Richards, 2002). Let us consider an elastic wavefield U(r, t) characterized by the Lam6 equation in an unlimited domain  

Since this field may be represented as a superposition of two types of waves, compressional and shear waves, we are faced with the problem of formulating a certain analytical criterion (similar to the Sommerfeld radiation conditions) that provides for the exclusion from the solution of the elastic field equations of compressional and shear waves that are convergent at infinity. It should also be pointed out that the radiation conditions are not included as some kind of heuristic principle in the initial mathematical formulation of the problem. 

In the scalar case we obtained these conditions by analyzing (for the sake of simplicity) the asymptotic behavior of spherical waves. At the same time, as it has been demonstrated above, the same result could be obtained by analyzing the conditions required to ensure that the corresponding Kirchhoff integral goes to zero over a large sphere expanding to infinity. 

Following Kupradze (1963), we will demonstrate that the radiation conditions for the potential (compressional wave) and solenoidal (shear wave) components of the elastic field can be formulated as follows  

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We can formulate the radiation conditions for elastic waves as certain requirements for the field behavior at large distances from the source region CV. These requirements should ensure that the Kirchhoff integral over the sphere Or in formula (13.207) goes to zero as r —> cx). 

Krenk, S. Kirkegaard, P.H. 2001. Local tensor radiation conditions for elastic waves. Journal of Sound and Vibration, 247, 875—89. 

Other, related coherent Raman effects are also represented in Figure 5, such as the case (C) where the signal beam is detected at the Stokes frequency. The Raman-induced Kerr effect (B) may be interpreted as the quadratic influence of an electric field of frequency CO2 on the elastic scattering of radiation at a frequency or vice versa. In this case the phasematching (or wave-vector-matching) condition is fulfilled for any angle between beams 1 and 2, while in cases (A) and (C) it may only be met for certain angles of the beams with respect to each other. 

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