Abstract
The structure of the wavefields in the vicinity of space–time surface gravity wave caustics is investigated within a linear theory framework. Owing to dispersion, transient surface gravity wave caustics are not frozen in space — as is the case in optics and acoustics when the wave source and environment are stationary — and the corresponding wavefields have rich spatio-temporal structure. Associated with these wavefields are extreme wave events — spatially and temporally localized occurrences of anomalously high wave crests and/or deep wave troughs. Our analysis exploits properties of both the Maslov and Fourier integral representations of the wavefield, and results from the study of singularities of mappings (popularly known as ‘catastrophe theory’). The latter dictates that structurally stable caustics take on only certain forms, each of which is describable by a canonical polynomial generating function whose form is known. Particular attention is paid to those caustics whose complete structure is observable in space–time without varying any external control parameters. These are the fold, the longitudinal cusp, the transverse cusp, the elliptic umbilic, the hyperbolic umbilic, the ‘lips’ and ‘beak-to-beak’ structures, and the swallowtail. Attention is focused on the surface gravity wave problem, but the results presented apply to a large class of linear dispersive waves in inhomogeneous moving media.
