Acoustic and electromagnetic scattering analysis using discrete sources III systems of functions in acoustic theory

Acoustic and electromagnetic scattering analysis using discrete sources III systems of functions in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources III systems of functions in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources III systems of functions in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources III systems of functions in acoustic theory Acoustic and electromagnetic scattering analysis using discrete sources III systems of functions in acoustic theory

Ill SYSTEMS OF FUNCTIONS IN ACOUSTIC THEORY For solving the acoustic scattering problems in the framework of the discrete sources method and the null-field method it is necessary to approximate the surface fields by a complete system of functions on the particle surface 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 We begin this chapter by presenting some basic results on the completeness of localized spherical wave functions In order to preserve the completeness at irregular frequencies linear combinations of regular functions and their normal derivatives on the particle surface will be used We then proceed to describe a general scheme for complete systems construc- 39 40 CHAPTER III SYSTEMS OF FUNCTIONS IN ACOUSTICS tion using radiating solutions to the Helmholtz equation In particular, we will discuss the completeness of distributed radiating spherical wave functions After that, we will provide a similar scheme using entire solutions to the Helmholtz equations The next section then concerns the completeness of point sources Here, we will discuss the systems of functions with singularities distributed on closed and open auxiliary surfaces In addition, we will analyze the completeness of distributed plane waves The last section of this chapter deals with the linear independence of these systems COMPLETE SYSTEMS OF FUNCTIONS The completeness properties of the sets of localized spherical wave functions and point sources have been studied exhaustively by means of different representations theorems In this chapter we will present these basic results but our main concern is to enlarge the class of complete systems 1.1 Localized spherical wave functions We begin our analysis by establishing the completeness of the spherical wave functions in L'^{S) These functions form a set of characteristic solutions to the scalar wave equation in spherical coordinates and are given by ul;,lM = zi^^{kr)PJr^{cose) e^'^^, n = 0,1, , m = - n , , n (3.1) Here, (r, 0, (p) are the spherical coordinates of x, z^^ designates the spherical Bessel functions jn» ^n stands for the spherical Hankel functions of the first kind hn , and Pn denotes the associated Legendre polynomials Note that ulnn is an entire solution to the Helmholtz equation and u^^ is a radiating solution to the Helmholtz equation in R^ — {0} The expansion of the Green function in terms of spherical wave functions will frequently used in the sequel It is ,A ^ JL f ^-mn(y)^mnW, IYI > |x| ^(x,y,A:) = ^ E E ^ - M n=Om=-n ' (^-^^ [ ul^^{y)ul,^{x), |y| < |x| where the normalization constant Vmn is given by _2n-f l(n-|m|)! ^"^""" (n-fH)!' ^^'^^ The main result of this section consists of the following theorem COMPLETE SYSTEMS OF FUNCTIONS 41 THEOREM 1,1: Let S be a closed surface of class C*^ and let n denote the unit outward normal to Then each of the systems (a) {ul,ri^ n = 0,1, , m = -n, , n} , |t/^n ~ A ^ ^ , n = 0, l, ,m = ~n, ,n/ Im(AA:) > o | , (b) {wj^ni n = 0,l, ,m = -n, ,n/fc^p(Di)}, I ^ I S ^ ' n = 0, l, ,m = - n , ,n/fc ^ r / ( A ) | , {< -h A - ^ , n =: 0,1, , m = -n, , n/ Im(Afc) > is complete in L^{S) Proof: For proving the first part of (a) it suffices to show the closeness of the system {^mn^ n = 0,l, ,m = ~n, ,n} in L^{S), Let a € L^(5) and assume J(^" (y) t^mn (y)clS(y) = 0, n = 0,1, ,m = - n , ,n (3.4) With Ua' (x) being the single-layer potential with density a' = a* we choose X D[, where £)[ is the interior of a spherical surface S^ enclosed in D^ For |y| > |x| we use the spherical waves expansion of the Green functions and deduce that Ua' vanishes in D[ The analyticity of Ua' gives Ua' = in Z?i, whence, by theorem 2.2 of Chapter 2, a ~ on follows Analogously, theorems 2.4 and 2.6 of the precedent chapter may be used to conclude the proof of (a) The proof of the second part of the theorem proceed in the same manner For k G p{Di) the set of regular spherical wave functions {wmn» ^ = 0^ 1»-M m = - n , , n} is not complete in L^(5) The completeness can be preserved if a finite set of functions representing a basis of iV (^/ — /C') is added to the original system Before we prove this assertion let us recall some basic results 42 CHAPTER III SYSTEMS OF FUNCTIONS IN ACOUSTICS The null-space of the operator ^ J —/C' corresponds to solutions to the homogeneous interior Dirichlet problem, that means iV (^ J — /C') = V, where V stands for the linear space V= i | ^ /ve^{Di),Av-^k^v = OmDuv = Oons\ In addition dim N (h: - K'\ = dim AT ( ^ i l - X:") = 0, if k is not an interior Dirichlet eigenvalue, and d i m A r Q l - r ^ =AimN(h:-}C\ =mD, if k is an eigenvalue If {Sj}^J[ is a basis for AT ( ^ I — /C) and Vj stands for the double-layer potential with density 6j, then 6j = Vj^ on S and the functions Xj = dv*^/dn on 5, j = 1, ^TTIDI form a basis of N ( ^ J — /C') Furthermore, the matrix T ^ = M^^L Tj^j = (Xfc» | m | Thus, u^ = in E fl -D[ for all m G Z; whence by the analyticity of u the conclusion readily follows We pay now attention to the system of distributed spherical wave functions which form a set of radiating solutions to the Helmholtz equation They are defined by d n W = 0, fc ^ p{Di)] is complete in L^{S) We will now consider formulations of null-field equations in terms of the radiating functions ^ ~ T H E O R E M 1.14: Under the assumptions of theorem 1.12, replace in theorem 1.2 the radiating spherical wave functions n ^ „ , n = 0,1, ,m = —n, ,n, by the functions ^ ^ , n = 1,2, , with Im(fcA) > Let hs solve the resulting null-field equations (3.6) and (3.7) Then hg solves the general null-field equations (2.83) and (2.87), and conversely Proof: This is proved in the same manner as part (a) of theorem 1.12 Instead of the single-layer potential Ua' we consider the scalar fields u given by (3.8) and (3.9) Next, we will consider complete systems of functions with singularities distributed on open surfaces Before we state our results, let us note two theorems which are due to Mtiller [114] T H E O R E M 1.15: Consider Di a bounded domain of class C^ with boundary S and unit outward normal n Let u € 3?(Z)i) be a solution to the Helmholtz equation in Di If on a surface element SQ of S du ^ u = — = 0, ,^ ^^ (3.55) then u vanishes identically Proof: Green's formula and conditions (3.55) gives "w=-/hw^->»( ^' d S ( y ) , x € A , (3.56) S-So and / h^)- ,ag(x,y,fc) 5n(y) du d^^y)9^''^y''') dS{y), X e Ds, (3.57) S-So ' where as usually Ds = R^—Dj Consider an interior point xi in the vicinity of So and choose a sphere S'' of radius r around x i , such that S'" does not intersects the surface S — SQ The radius of the sphere is taken sufficiently small such that £>"", representing the interior of S^, is divided by 5o into 56 CHAPTER III SYSTEMS OF FUNCTIONS IN ACOUSTICS s-s FIGURE 3.4 The auxiliary surface S** exactly two parts of which the one part DJ Hes in the interior of and the other D2 exterior to SQ The position of the auxiUary surface is shown in Figure 3.4 Clearly, the integrals taken over S — SQ are analytic inside D^, The representation formula (3.57) shows that the integral vanishes in Djj whence, by the analyticity of the integral in D^, it follows that u vanishes identically in D^ and therefore in Di THEOREM 1.16: Let Di be a bounded domain of class C^ with boundary S and exterior D3 Denote by n the unit normal vector to S directed into Dg Let u € ^{Dg) be a radiating solution to the Helmholtz equation in Ds If on a surface element So of S du (3.58) then u vanishes identically Now we will derive some completeness results for systems of functions with singularities distributed on open surfaces THEOREM 1.17: Consider Di a bounded domain of class C^ with boundary S and exterior Dg- Let the set { x ~ } ^ j be dense on an open surface S" contained in Di and let the set {^n}^=i ^^ dense on an open surface SQ contained Dg, Then each of the systems (a) { , UL^ (b) - A ^ ^ , n = 0,1, , m = - n , , n/ Im(Afe) > o | , [yLn^ n = 0,1, , m = - n , , n / k ^ p{Di)} , (du^ | - ^ , n = 0,l, ,m = - n , , n / ^ fc^ry(A)|, | < n + A ^ ^ , n = 0,1, , m = - n , , n/ /m(Afc) > o | 15 linearly independent in L^{S) Proof: We prove (a) Before we present the proof, we recall that a countably infinite family of functions is linearly independent if every finite subset is linearly independent Let Y, S "mnuL(y) = 0, y € 5, (3.75) n = l Tn='-n for some constants ocmnt n = 1,2, ^Timax) ^ = ""^> ,n Then, the field «=5Z S ""»"««„ (3.76) n = l m=—n is a radiating solution to the Helmholtz equation which vanishes on S Therefore, u vanishes in Dg and in particular on any spherical surface S^ LINEAR INDEPENDENT SYSTEMS OF FUNCTIONS 61 enclosing The orthogonality of spherical wave functions on S^ finishes the proof of the first part of (a) To prove the linear independence of the second system we define u in a similar manner In this case u satisfies the homogeneous exterior Neumann problem, whence u = in D^ follows Conversely, for the third system we see that the linear independence relation implies that u satisfies the homogeneous exterior impedance boundary-value problem in Ds and therefore u vanishes in Ds The proof of the second part of the theorem proceeds analogously Next, we show the linear independence of the system of distributed spherical wave functions THEOREM 2.22: Consider the bounded sequence {zn) C F^, where Tz is a segment of the z-axis Assume S is a surface of class C^ enclosing Tg Replace in theorem 2.21 the localized spherical wave functions w^}j, n = 0,1, , m = —n, , n, by the distributed spherical wave functions ^^ni ^ =" 1,2, , m € Z Then the resulting systems are linearly independent in L^{S) Proof: Let us prove the linear independence of the system {4„, n = l,2, , mGZ} Suppose *^max E '^max E"'""^m„(y) = 0, y e , (3.77) m=—nimax n = l for some constants amn, fn = —mmax? •••?^max» ^ = 1,2, ,ninax' Then, the field ^'^max ''^max w= in=—nimax E E"'""^^" n=:l (^•'^^^ vanishes in D^ Since u is an analytic function we see that u vanishes in R^ — {'2^ne3}n=r- "^^^ Orthogonality of the exp{jm(p) yields '•'fnax E "-"'^M(^'-n)^!™! (cos0n) = (3.79) n=l for all m = —mmax? •••> ^max and any ?/ € E — {^inGa}^^!'' For each n we multiply (3.79) by rn '"'"^ Letting rn —• and accounting for the asymptotic form of the Hankel functions in the vicinity of 0, i.e (3.80) 62 CHAPTER III SYSTEMS OF FUNCTIONS IN ACOUSTICS we see that the conclusion readily follows The proof of the rest of part (a) proceed analogously For proving the linear independence of the system {(rj„„,n = l,2, , m € Z / f c ^ p ( A ) } we debut with mm ax max Y, n '•'max E "-"^mnCy) = 0, y € (3.81) m=—mmax n = l Let '^max "= E 'T'max E"-""^-"- (3.82) m=—rumax n=l Prom (3.81) and the assumption fc ^ p{Di) it follows that n = in Di Since u is an analytic function we deduce that u = in any bounded domain of R^ Choose now a spherical surface S^ enclosing S Clearly, w = on ^ and we may use the addition theorem for regular spherical wave function ^J„„(x) = u i | ^ | ( x - ^ „ e ) = E ^l'^'(-^n)«J„„-(x) (3.83) n'>\m\ to write this condition as mmax E /y^max E \ E"'"" |m| (3.85) n=l Multiplying the above equation by u^^, (x), where x G -Df and D f is the exterior of 5^, summing over m and n' and accounting for the addition theorem for radiating spherical wave functions we arrive at (•max E "-max E "'n"4n(x) = 0, X € Df (3.86) m=—mmax n = l The proof can now be completed as in the first part of the theorem In an analogous manner we can prove the rest of the theorem We will now investigate the linear independence of the systems of distributed point sources LINEAR INDEPENDENT SYSTEMS OF FUNCTIONS 63 THEOREM 2.23: Consider Di a bounded domain of class C^ Let the set { x ~ } ^ i be dense on a surface S~ enclosed in Di and let the set {^n } ^ i ^^ dense on a surface 5"^ enclosing Di Replace in theorem 2.21 the radiating spherical wave functions u ^ ^ , n = 0,1, , m = —n, ,n, by the functions (p~, n = 1,2, , and the regular spherical wave functions w^^, n = 0,1, , m = —n, ,n, by the functions ^^, n = 1,2, Then, the resulting systems of functions are linearly independent in 1/^(5) Proof: We prove only the linear independence of the system {