Wavefront Analysis in the Nonseparable Elastodynamic Quarter-Plane Problems, I—Part 1: The General Method

This work concerns the transient response in elastodynamic quarter-plane problems. In particular, the work focuses on the nonseparable problems, developing and using a new method to solve basic quarter-plane cases involving nonmixed edge conditions. Of initial interest is the classical case in which a uniform step pressure is applied to one edge, along with zero shear stress, while the other edge is traction free. The new method of solution is related to earlier work on nonseparable wavequide (semi-infinite layer) problems in which long time, low-frequency response was the objective. The basic idea in the earlier work was the exploitation of a solution boundedness condition in the form of an infinite set of integral equations. These equations were generated by eliminating the residues at an infinite set of poles stemming from the zeros of the Rayleigh-Lamb frequency equation. These poles were associated with exponentially unbounded contributions. The solution of the integral equations determined the unknown edge conditions in the problem, hence giving the long time solution. In the new method for the quarter-plane problems the analog of the Rayleigh-Lamb frequency equations is the Rayleigh function that yields just three zeros, i.e., say the squares of c_R1, c_R2, and c_R3 speeds for the Rayleigh surface waves. Of these six roots (and corresponding poles) only one is physical, i.e., c_R1. The other five are eliminated since they correspond to: (1) exponentially unbounded waves, or (2) waves with speeds c_R2 > c_d, c_R3 > c_d, all inadmissible. They analogously (to the layer case) give four integral equations for the edge unknowns, hence the formal solution for the problems. To date the method has been successful in obtaining all the wavefront expansions for the uniform step pressure case mentioned earlier. The dilatational and equivoluminal wavefronts were obtained for the interior and both edges and the Rayleigh wavefronts for both edges. This asymptotic solution has been verified and hence establishes the credibility of the new method.