12 INTRODUCTION If R is taken large enough we get c v_ v.w (P) s XR where 2 ^ is the sphere of radius R. Now Jkr \ Jkr if P G D, 0 if P g D. 9)-9[*-fr and * dr = R2du on 2^, where doj is the element of solid angle on HR. Thus the second integral becomes *2-//«M[^-*»H„/»- Thus the integral over the sphere will go to zero as R » oo if ff \u\ dec = o(\) and ff | ^ - ifcu| ^ = o(^ - 1 ). where the integrals are evaluated for r = R. These conditions are known as the Sommerfeld radiation conditions. Their significance is that they represent the condition that u consists of expanding waves radiating outward and no incoming waves1. Let us assume that this condition is satisfied. Then the value of u outside some surface S is given by «*-«F//[£•*"» •'(£)]• (H) 1 For a precise mathematical explanation of the Sommerfeld radiation conditions see the book by Lax and Phillips [4, pp. 120-128.] The gist of what they prove is the following: Let/ = [fxJi) be Cauchy data for the wave equation we thus seek a solution of the wave equation d2w 3r2 ' Aw = 0 with H(JC,0) = fx and (3W/3/)(JC,0) = / 2 . We say/is eventually outgoing if there is some constant c such that w = 0 for \x\ / - c. If we seek a solution of fixed frequency, then the appropriate Cauchy data are {H, ikw). Suppose that w is a solution of the reduced wave equation outside some bounded domain. Then {w, ikw} is eventually outgoing if and only if the Sommerfeld radiation conditions are satisfied.
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