Showing papers by "Michael E. Brown published in 1986"

The transient wave fields in the vicinity of the cuspoid caustics

Abstract: 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. ...

The transient wave fields in the vicinity of the elliptic, hyperbolic, and parabolic umbilic caustics

Abstract: The caustics of high‐frequency wave propagation may be classified using catastrophe theory. In this paper, we examine the transient wave fields in the vicinity of the elliptic, hyperbolic, and parabolic umbilic catastrophes. Analytical results are presented which describe the time‐dependent wave field due to an impulsive source on and at large distances in each control direction away from the most singular point of each of these three catastrophes. Numerical results are presented which show the smooth variation of these transient wave fields as each control parameter is independently varied. These results are compared to previously derived results on the transient wave fields in the vicinity of the cuspoid catastrophes (the cuspoid and umbilic catastrophes have coranks 1 and 2, respectively). It is found that the transient wave fields in the vicinity of the cuspoid and umbilic catastrophes differ with regard to the temporal structure of the wave field on the most singular point of each catastrophe, the manner in which these temporal structures unfold, and the phase shifts which individual rays undergo as a result of touching one of the catastrophes.