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Abmct-kymptotic transient field mlutions of the form A(r, t ) exp [is@, r)] , where S is a rapidly and A a slowly varying function of space and time, may be analyzed in terms of wave packets with c e n e frequency w = -as/at and central wavenumber f = VS. When the (dis-@e) medium is lodes, stationary, and homogeneous, wave packets with constant real w aud k move dong straightline trajectories called space-time rays. In the presence of dissipation and (or) when the input signal has an exponential amplitude dependence, S is complex. The corresponding wave packets with constant complex w and f move dong complex -time rays, ia., dong trajectories defined in a complex (r, r) coordinate space. The properties of complex s p a c e t h e rays and of the fields proprgrting along them, and their relation to physical fields observed on real (r, t ) coordinates, are illustrated for a plane pulse with Guusian envelope and frequency swept carrier, launched into a lossy environment. Tracking of spatial and temporal maxima is performed by ray techniques, and a paraxial my regime is defined that permits discuMion of a signal velocity. Specipr attention is given to my focusing and the associated phenomena of pulse compression. It is shown how a complex input frequency profile can be synthesized 80 as to achieve optimum c o m p r d o n at a real space-time observation point in a lossy medium. The general results are applied in detail to a cold dissipative pLQnr, and a representative set of numerical calculations is included. ray technique, it is appropriate to ask whether a similar pro-Durham under Contract DAHC 04-69-c-0079, and in part by the Joint and quantitative information on, a more g e n e d and prac-Services Electronics Program under Contract F 44620-69-0047. Thh tically important class of wave phenomena. As a first step

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Several diffraction problems whose solutions involve incomplete Airy functions are briefly described; general and asymptotic characteristics of these functions are summarized. A uniform asymptotic representation, in which the incomplete Airy functions serve as canonical functions, is given for the class of integrals characterized by two saddle points arbitrarily positioned relative to an end point. Proceeding from an appropriate boundary-layer expansion, the functions and their properties are applied in a detailed analysis of the fields near the point of confluence of the caustic of a converging wave and the shadow boundary resulting from the presence of an opaque obstacle.

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