By Adrian Doicu

The discrete assets procedure is a good and strong device for fixing a wide type of boundary-value difficulties in scattering conception. quite a few numerical equipment for discrete assets now exist. during this ebook, the authors unify those formulations within the context of the so-called discrete resources procedure. Key gains* accomplished presentation of the discrete assets strategy* unique concept - an extension of the normal null-field approach utilizing discrete assets* functional examples that display the potency and suppleness of elaborated tools (scattering through debris with excessive element ratio, tough debris, nonaxisymmetric debris, a number of scattering)* record of discrete assets programmes on hand through the net

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**Extra resources for Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources**

**Example text**

If Im(Afc) = 0 and Im(A:) > 0 we obtain Wa = 0 in D^. Finally, if Im(Afc) = 0 and Im(A:) = 0 we get WxaR-^oo Js l^ol ^'^ "= ^' whence Wa = 0 in Dg follows. 50) finishes the proof of the theorem. 4. For instance, let Ua be the single-layer potential with density a € 1/^(5) satisfying Wa = 0 in A . )) +a = 0. 59) 2,S SH 32 CHAPTER II THE SCALAR HELMHOLTZ EQUATION follows. 6. 61) Wa = 0 in Dg. Then a^O on S. ))-«| = 0. 63) we find that - Im(Afe) / |a|^ d 5 = Im(A:) / (|A:|^ \wa\'^ -f \Vwaf) S dV. 64) Di Since Im(Afc) > 0 and Im(fc) > 0 we conclude that a ~ 0 on 5 .

Before we state our next results, let us prove some auxiliary lemmas. 1: Let u G 5ft(Di) he a solution to the Helmholtz equation in Di and let u^ be its Fourier coefficients with respect to (f. 1 Illustration of the support of discrete sources. exists and represents an analytic function of z. Here {p,(p,z) are the cylindrical coordinates ofx and t] = {p^ z) eH. - = -^P„„ I [«(y)^^(y) - ^(y)«i_(y)] d5(y). rl^l T^ (X \ . I I I I . 1 ~ COS^\ cF f |m| - n, |m| + n -f 1, \m\ -f 1, 1. 23) 1. M. |m|! "^ sentation of the spherical Bessel function we see that v^{z) is an analytic function of z.

In addition to the completeness, the system of functions should be linearly independent since only then can the matrices appearing in the numerical schemes be inverted. This chapter is devoted to the analysis of complete and linear independent systems of functions for the Helmholtz equation. As complete systems of functions we will discuss the systems of discrete sources. There is a close relation between the properties of the fields of discrete sources and the structure of their support. In particular if the supports are chosen as a point, a straight line, or a surface, then the corresponding systems of functions are the localized spherical wave functions, the distributed spherical wave functions and the distributed point sources, respectively.