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By Becker G. F.

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Extra resources for A Possible Origin for Some Spiral Nebulae (1916)(en)(8s)

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This procedure will justify itself shortly. The phase of the integrand is ^0 y0 + ^n Yn (x x0r y0) 48 so the stationary phase conditions are ^0 + ^n @Yn0 = 0 @y ^n @Yn0 = 0 : @xr Throughout the following, these are to be regarded as determining y0 and x0r . From the rst condition ^n 6= 0, else ^ could not be a unit vector. So the factor of ^n can be dropped from the second condition. Understanding the rst condition requires computation of @Yn =@y0. From its de nition @ (x x) ; (x y0 Y (x x0 y0))] 0 = @y r r n r 0 !

These will not gure in the computation of the principal symbol, and in any case have the same importance as contributions already neglected in the approximation L La. Then: Remarks (4) and (5) above combine to yield our principal result: := L c]L c] = " # is a two-by-two matrix of pseudodi erential operators of order 2. 56 The principal symbols of , etc. are products of the geometrical factor @ r (x x x) ! 2 @x g(xs x ^) = jr (xs xr x)j det r0r (x sx rx) s r and terms from the integral kernel de ning La, above.

A robust approach to acoustic imaging must drop this assumption, which underlies almost all contemporary work. I review brie y some recent progress towards removing the simple geometry assumption at the end of the chapter. Since p=f G where G is the perturbation in the fundamental solution, it su ces to compute G, which is the solution of ! 1 @ 2 G ; r 1 r G = 2 c @ 2G ; 1 r rG c2 @t2 @c3 @t2 G 0 t<0: 33 The key to an e ective computation is the Green's formula ! Z ! g. if u v are smooth and the support of the product uv is bounded.

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