| It is well-known that the representation of a high frequency field by geometrical optics fails in the vicinity of a caustic. Among several solutions to this problem (boundary layer expansions, integral expressions derived from an ansatz, etc.), a systematic approach due to Maslov consists essentially of operating in the phase-space M (position-wavevector) instead of the space X (position). A flow, which characterizes the system, is defined in M by Hamilton's equations. The flow lines, or phase-space rays, are projected on X along the rays of geometrical optics. If the field is given on some surface S in X by an expression of the form exp{ik(phi)(x)} (SIGMA) (ik)('-p) a(,p)(x), it can be continued in the vicinity of S by well-known methods of geometrical optics (GO). The phase (phi)(x) on S in conjunction with the eikonal equation can be used to construct the wavevector k at every point x on S. The locus of the pair (x,k) is a surface (GAMMA) in M which, carried by the flow, generates a subspace (LAMDA) called a Lagrangian manifold. The caustics are the projections on space X of the apparent contour (SIGMA) of (LAMDA). Rays that come together at a caustic point are projections of distinct phase-space rays on (LAMDA) which, in general, remain distinct when projected on the wavevector space K or on a mixed space Y defined by (n-j)-components of the position vector and the j-components of the wavevector corresponding to different directions. It has been shown that for any caustic point there is a space Y such that the phase-space projected on it do not coalesce.;A generalization of the procedure, the GO-AFT approach, is developed which circumvents the phase space description. First, v is replaced by an (m+1)-term asymptotic expansion v(,m) and then Fourier transformed: F('-1)v(,m). Furthermore, in regions where u and v can be represented by (m+1)-term expansions (u(,m),v(,m) (it is shown (GO-AFT theorem) that those are related through an asymptotic Fourier transformation (AFT) of range m; i.e., one resulting from an asymptotic evaluation carried out up to (m+1)-terms. Thus, v(,m) = F(,m)u(,m). Consequently, this expression may be used in lieu of the GO expansion and the generalized solution will be expressed by U = F('-1)F(,m)u(,m). If a caustic appears in Y, another mixed space, Y', can be found where one does not appear and a similar, related representation is obtained.;In contrast to F, the transformation F(,m) is local but otherwise it has properties similar to F. It effectively cancels the singularities that occur near a caustic. Boundary point contributions are considered in addition to those resulting from stationary points.;To illustrate the GO-AFT approach, it is applied to a general homogeneous medium problem (in particular, one involving a cusp caustic) and to a general linear-layer problem (in particular, a plane wave incident on a linear layer). Comparisons with other methods for computing the field near to a caustic are made. Problems concerning indirect and direct asymptotics and caustics in the presence of a diffracting half-plane are also considered.;To the field u(x) over X corresponds a field v(y) on Y related to it by a Fourier transformation F(Y/X) involving the variables that have been changed in going from X to Y. Thus, v = Fu. The partial differential equation for v is easily deduced from that for u through the Fourier transformation F. An important observation of Maslov is that when the geometric optic expansion for u fails, near to a caustic point, there is always a space Y such that geometrical optics applies to the corresponding wavefunction v. If v(,0) is the zero-th order GO expression for v, Maslov takes F('-1)v(,0) as the expression for u. This is an integral representation that may be evaluated numerically, analytically, or by applying known techniques to obtain a uniform asymptotic expression. |
