TIME-EVOLUTION OF A CAUSTIC

The Lagrange manifold formalism is adapted to study the time-evolution of caustics associated with high frequency wave propagation in media with both spatial and temporal inhomogenities.

The standard aplxech for obtaining the Hamilton's equations =(r,)

=VH
where r is the ray-path Imrsmeter and a terized initial condition.But those space-time points where the ooordinate space mp beoomes singular i.e., (1.5) of (1.7) (1.8) (1.9) on the caustic curve where tt-GT : o with = (F,), the emetrical optics Inxxdue cannot he &pplied ctirectly.
Such difficulties at caustics can often be oircumvented by using the Lamange Msnifold formalism of Maslov [3] and Arnold [4], which has recently been extered to determine a class of asymptotic solutions [5] for phemmm modelled by Equation (1.1 }. Here we present a variation of this extension which enables a modellinE of the time evolution of the custic.This alEorithm s/so lesds to determination of the field on the csustic; but because so mch of this aspect of the procehe in [5] applies directly, for brevity we ize only those aspects pertinent to modellinE the evolution of the caustic.For clarity, we comsider the scalar wave equation given in Equation I. 1 ), al the analogous vector wave equation could also have been considered.An exle is included to illustrate the pzxure. (1.10) We assume that near caustics Equation (I. 1 has an astotic solution of the form (,t)-(A(,,t,k)exp{iA(--S(,t))]d : O(A-(R)). (2.1) The amplitude A(,,t,A) and its derivatives are assumed bounded and is regarded as a phase, i.e., @(,,t) .-S(,t).
In the LeMnifold formalism oaustio points are determined from the [5], or equivalently from S(,t).To obtain this S(,t), we first find trsOectories (through Hzilton's equations) (r,) ( ) , : then invert the wavevector and time transformations 5 5(, Substitution into the coordinate space map detezines the Lgrnge explicitly = ((5,t), Q(,t)) vps(5,t), where the time appears as a ter.P s(,t) I ' leads to the Iase (,5,t) -5 s(,t).(2.10) At a given time this condition leads to sets of triplets (), which upon substitution into Equation 2.5 determines the caustic in coordinate space.The time evolution of the caustic proceeds by considerinE Equation (2.I0) for several values of time.CorrespondinE to each such time is a set of triplets (), which upon substitution into the La&TanEe manifold yield the time evolution of the caustic in coordinate space.
The determination of the field on the caustic requires the develoient of a transport equation for the amplitudes.As this develolP_t so parallels that referred to above [5], we do not include it here.
(3.4) Next Hamilton's Equations (1.6) and (I.7), toEether with the in/tial cond/tions at r=0, are solved to obin the (3.7)While Hamilton' s Equations relate the canonical variables, the selection of the initial specific condition t=t o at y=0 introduces an additiorm/ couplinE between t o and g from the eikonal equation po 2 2 k 2 + at 0.
(3.8)It is this couplinE which allows the inclusion of time t as a parameter in the LaranEe Manifold.Specifically, elimination of t o between the time coordinate in Equation 3. ?and Equation 3.8), allow the arc I y to be parameterized in time t.Then, inversion of the map includes the time as a parameter, which is subsequently introduced into the LaranEe manifold.To illustrate, for clarity of exposition, we first use the simplicity of the example to eliminate e between (bPx_a) ; aQ_bPx 2_ a2_b 2 Q2+k2.px _py2_S t Finally, substituting into Equations (3.9) and (3.10), followed by with respect to Px and py respectively leads to the phase /(,,t) (We note only the minus sign in Eqution (3.11) leads to physically caustics.) As a specific example to illustrate the calculations/ aspects of the let us choose a -I, b= 6.655(= 45/7coe 15), k 2 = 23, e 15 , t o = 1, a lot 7 ai the initial amplitude at the emtter A o 1.