Asymptotic Evaluation of the Field At a Caustic

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Asymptotic Evaluation of the Field At a Caustic
Irvin W Kay
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+ D is a fvmction of the integration point on C, and can he considered as a function of arc length s along C. The points of stationsoy phase are determined "by . 2 'Then if — r- (\)/ + D) ^ at the stationary point, (l4) yields ds^ (16) E^(x. Y) - -^ ( ^ - ^) D-^/2 1 ^ (Vi + D) 1-^2 exp ik^ + D) - ^ 1 - sgn — 2 (\|/^ + VJ ds In (l6) all quantities on the right side are to "be evaluated at the point of stationary phase determined "by (15). If there is no station- ary point the right side is to "b...e replaced "by zeroj if there is more thsn one stationary point, the right side is to "be a sum with one term of the a"bove type for each such point.
It follows from (15) that at the stationary points either the law of reflection is satisfied or the reflected ray is the con- tinuation of the incident ray. In the latter case, evaluation of (16) shows that E ~' -«E. So that E^x-' 0. This occurs in the shadow, r 1 In the former case (l6) is found to coincide with (12), as is to "be e3qi>ected.


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