THE METHOD OF STATIONARY PHASE 13 In this way, the solution exterior to S is described in terms of "radiation emitted from S".2 It was Huygens who originally had the idea that propagated disturb- ances in the wave theory could be represented as the superposition of "secondary disturbances" along an intermediate surface such as S but he did not have an adequate explanation of why there was no "backward wave", i. e. why the propagation was only in the outward direction. The idea that the backward waves would cancel one another out because of phase differences was due to Fresnel. Fresnel believed that if all the sources were inside S, the "secondary radiation" (i.e. the integrand in Helmholtz's formula) from each separate surface element would produce a null effect at each interior point due to interference. The above argument, due essentially to Helmholtz, was the first rigorous mathematical treatment of the problem, and shows that the internal cancellation is due to the total effect of the boundary. Nevertheless, as we shall see below, an application of stationary phase shows that (under suitable hypotheses) Fresnel was right, up to terms of order \/k . We now apply stationary phase to Helmholtz's formula. The function w, which occurs on the right hand side of the formula will itself be oscillatory, and we must make some assumptions about its form before we can proceed. We shall assume that, near S, u = aelkq) where a and p are smooth, and that ||grad q\\ = 1. This would be the case, for example, if u represented radi- ation from a single point, (?, lying inside S, where p(y) = \\y - Q\\. Also, we shall see in the next chapter how to construct "approximate solutions" to the reduced wave equation which are of this form with a and p arbitrarily prescibed along S (with cp subject to the constraint that grad 9 not be tangent to S). These "approximate solutions" satisfy the wave equation up to an error of order k~N for any large N\ and we can apply our calculations to these approximate solutions. Indeed, we shall carry out the stationary phase calculations here only to order \/k. We shall assume that we are sufficiently far from S so that \/r2 is negligible in comparison with k, and that a and da are also negligible in comparison with k. Substituting into (H), we find that the top order term (relative to powers of k) is £ff(a/r)eik^+r\*dp-*dr). s Now the points of stationary phase are those points, y, on S where 2 So far, we have dealt with "monochromatic radiation", u, corresponding to the time dependent function v where u(x,y,z, t) = u(x,y,z)e~ikt. For a fixed point, P, let vP denote the function vP(x,y,z,t) = v(x,y,z,t - r), where r is the distance from P to (x,y,z). Then substitution into Helmholtz's formula shows that v(P,t) = ^ff(vP * d{\/r) - (l/r)(du/dtP) * dr - (1/r) * dvP). s This is Kirchhoffs formula. Since it is linear in v, and does not explicitly involve the frequency, it is true for any superposition of monochromatic waves of varying frequencies, and hence for an arbitrary solution of the wave equation. In this form, the relation with Huygens' principle is very apparent, cf. [5]-
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