36 Integrals with Coalescing SaddlesApplications36.13 Kelvin’s Ship-Wave Pattern36.15 Methods of Computation

The physical manifestations of bifurcation sets are caustics. These are the structurally stable focal singularities (envelopes) of families of rays, on which the intensities of the geometrical (ray) theory diverge. Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns. See Berry (1969, 1976, 1980, 1981), Kravtsov (1964, 1988), and Ludwig (1966).

Diffraction catastrophes describe the connection between ray optics and wave optics. Applications include twinkling starlight, focusing of sunlight by rippling water (e.g., swimming-pool patterns), and water-droplet “lenses” (e.g., rainbows). See Adler et al. (1997), Berry and Upstill (1980), Marston (1992, 1999), Nye (1999), Walker (1983, 1988, 1989).

Diffraction catastrophes describe the “semiclassical” connections between classical orbits and quantum wavefunctions, for integrable (non-chaotic) systems. Applications include scattering of elementary particles, atoms and molecules from particles and surfaces, and chemical reactions. See Berry (1966, 1975), Connor (1974, 1976), Connor and Farrelly (1981), Trinkaus and Drepper (1977), and Uzer et al. (1983).