It is well known that although the usual harmonic ansatz of geometrical optics fails near a caustic, uniform expansions can be found which remain valid in the neighborhood of the caustic, and reduce asymptotically to the usual geometric field far enough from it. Such expansions can be constructed by several methods which make essentially use of the symplectic structure of the phase space. In this paper we efficiently apply the Kravtsov-Ludwig method of relevant functions, in conjunction with Hamiltonian ray tracing to define the topology of the caustics and compute high-frequency scalar wave fields near smooth and cusp caustics. We use an adaptive Runge-Kutta method to successfully retrieve the complete ray field in the case of piecewise smooth refraction indices. We efficiently match the geometric and modified amplitudes of the multi-valued field to obtain numerically the correct asymptotic behavior of the solution. Comparisons of the numerical results with analytical calculations in model problems show excellent accuracy in calculating the modified amplitudes using the Kravtsov-Ludwig formulas.
|Original language||English (US)|
|Number of pages||30|
|Journal||Mathematical Models and Methods in Applied Sciences|
|State||Published - Mar 1 2001|
ASJC Scopus subject areas
- Modeling and Simulation
- Applied Mathematics