A general formalism is presented for the study of the transient wave fields in the vicinity of caustics of arbitrary complexity. The caustics are classified using catastrophe theory. In the present study, a general introduction to the subject is presented and explicit results (numerical and analytical) for the cuspoid family of catastrophes are given. Numerical results are presented for the fold, cusp, and swallowtail catastrophes which show the smooth variation of the acoustic field as the control parameters are independently varied. These results vividly demonstrate the phenomenon of ray focusing, showing the number of rays which converge at each catastrophe, the resulting phase shifts they undergo, as well as their decay in shadow regions. Analytical results are given which describe the time-dependent acoustic field on and at large distances away from each of the cuspoid catastrophes. Four sets of numbers are introduced which describe the behavior of the wave field in the vicinity of each catastrophe. These are: (1) an exponent which describes the temporal decay of the acoustic field on the most singular point of each catastrophe; (2) a number which describes the asymmetry of the time-dependent acoustic field on the most singular point of each catastrophe; (3) an exponent which describes the time separation as a function of control parameter between individual ray arrivals as each of the control parameters are independently varied; and (4) an exponent which describes the asymptotic decay as a function of control parameter of each constituent ray arrival. Sets (1) and (3) are shown to be simply related to the singularity and fringe indices previously introduced by Arnold [V.I. Arnold, Usp. Mat. Nauk. 30(5), 1–75 (1975)] and Berry [M.V. Berry, J. Phys. A10, 2061–2081 (1977) ], respectively. It is argued that many features of the frequency domain representation of the acoustic field in the vicinity of the catastrophes are most easily understood by examining the time domain representation.
ASJC Scopus subject areas
- Acoustics and Ultrasonics
- Arts and Humanities (miscellaneous)