SHOCK WAVE FORMATION AT A CAUSTIC

The behavior of a weak acoustic compression near a caustic surface has been determined for a special class of specified incoming signals. This problem arises in various contexts, including the propagation of sonic booms generated by supersonic aircraft. The solution derives from a mapping of hodograph-like solutions to the physical plane. Themaximum amplification for this class of signals of fixed amplitude depends on the width of the incoming signal. The solutions contain reflected shock waves that satisfy the appropriate shock jump conditions, provided the width of the incoming signal is greater than a certain critical value. 1. Introduction. There are many sources of weak shock waves. These include the commonly experienced phenomena of thunder and the sonic boom of supersonic aircraft. At supersonic speeds, aircraft generate a nearly conical wave pattern. These wave fronts propagate along their normals at the local sound speed, i.e., along acoustic rays. Variations in the sound speed (due to temperature changes), aircraft maneuvers, and winds can lead to ray crossing and a focusing of the wavefront. In the case of the sonic boom this results in a so-called superb.oom if the rays form an envelope, or a super-superboom if they meet at a point. In the case of a sonic boom, the pressure signature is, nominally, an N-shaped wave. More specifically, a weak shock wave provides an essentially instantaneous rise in pressure; subsequently the pressure falls linearly (in space or time) and is returned to nearly the ambient pressure through a second weak shock wave. The focusing of this weak shock wave is examined here. For simplicity, we phrase the problem in the context of the behavior of a weak shock wave generated by an aircraft in slightly supersonic flight in a flow with a Mach number gradient. Such gradients occur naturally in the troposphere where the ambient temperature decreases nearly linearly with altitude. Sonic booms as well as weak shock waves from other sources are, under normal circumstances, adequately described by a nonlinear adaptation of geometric acoustics (1), (2), (3). In geometric acoustics, as in geometric optics, a ray is the path of a signal point on a wave. Neighboring rays form an infinitesimal ray tube. The area of a ray tube will, in some situations, vanish at a point in space-time, whichwe shall call a focal point. Such focal points may form a hyper-surface in space and time that, in most cases, is an envelope of the rays in physical space. Such a hypersurface is an acoustic caustic, and will be referred to hereafter as a caustic or caustic surface.