14 INTRODUCTION grad p() + grad r(y) is normal to S. There are two possible situations: grad^OO, gradrO) grad p{y) = - grad r(y) * dp(y) = — * dr(y) P = y + r grad p(y) (a) case (a) grad (p(y) = 2(grad q(y),n)n - grad r(y) * dcp(y) = * dr{y) P = y - r(2(grad q(y),n)n - grad p(y)) grad vy) gradrO) case (b) Let us suppose for the moment that y is a non-degenerate critical point of type (b). The top order term in the stationary phase formula will vanish, and the total contribution coming from^ in Helmholtz's formula will be of order \/k. (Notice that if S were convex and grad (p pointed outward, then for any P inside S, all the critical points would be of type (b). This, in a sense, justifies Fresnel's view that there is "local cancellation" of the backward wave.) For nondegenerate critical points of type (a) since * dy(y) = — * dr(y), we may, in computing the highest order contribution to the stationary phase formula, replace the above integral by ik fJlialrV^d*, This shows, that (up to order \/k) the induced "secondary radiation" along S behaves as if it

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