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

dis-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 dent systems of functions for the Helmholtz equation As complete systems

indepen-of functions we will discuss the systems indepen-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 func-tions 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 pleteness of localized spherical wave functions In order to preserve the completeness at irregular frequencies linear combinations of regular func-tions and their normal derivatives on the particle surface will be used We then proceed to describe a general scheme for complete systems construc-

com-39

tion using radiating solutions to the Helmholtz equation In particular, we

will discuss the completeness of distributed radiating spherical wave

func-tions 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

singu-larities distributed on closed and open auxiliary surfaces In ađition, we

will analyze the completeness of distributed plane waves The last section

of this chapter deals with the linear independence of these systems

1 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

so-lutions to the scalar wave equation in spherical coordinates and are given

by

ul;,lM = zi^^{kr)PJr^{cose) ế^^, 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

func-tions will frequently used in the sequel It is

,Ạ ^ J L 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

_ 2 n - f l ( n - | m | ) !

^"^""" 4 ( n - f H ) ! ' ^^'^^ The main result of this section consists of the following theorem

THEOREM 1,1: Let S be a closed surface of class C*^ and let n denote

the unit outward normal to 5 Then each of the systems

With Ua' (x) being the single-layer potential with density a' = a* we choose

X 6 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' = 0 in

Z?i, whence, by theorem 2.2 of Chapter 2, a ~ 0 on 5 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

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»<5j), /c, j = l, ,mD, is

nonsingular Coming to the proof we observe that the closeness relations

Jg a'uln^ d S = 0, n = 0,1, , m = —n, , n, leads to the vanishing of the

single-layer potential Ua' in Dg- Then proceeding as in theorem 2.2 of the

precedent chapter we find that a' ^ UQ e C^'^'iS)^ where ( ^ J — /C') a^ =

0 Therefore, GQ = Yljl^i^jXj- Now, conditions / ^ a o X j d 5 = 0, j =

1, , m£), gives ag = 0 on 5, whence a ~ 0 on 5 follows On the other hand

if instead of the system {xj} ^^ we consider the set {6j}^J[ we observe

that from Jg a^Sj dS = 0, j = 1, , m o , we arrive at

when k e r]{Di) In this case a finite set of functions representing a basis for

iV ( ^ / -h /C) should be added to the original system Note that analogously

to the interior Dirichlet problem if {0j} J! is a basis for iV(^I-|-/C')

and Uj stands for the single-layer potential with density 0^, then 0^ =

—dujj^/dn on 5 , the functions ^^ = ""^^4- ^^ '^^J ~ l, ,mAr, form a

basis of N {\l 4- K) , and the matrix Tjv = \T^} , I^^ = (^it, 0^) , A;, j =

l, ,miv, is nonsingular

We will now formulate the null-field equations in terms of spherical

wave functions

T H E O R E M 1.2: Let the surface field hg satisfy either the set of

null-field equations for the Dirichlet problem

n hs{y)nl,{y) + My)^iy) d5(y) = 0, n = 0,1, , m = - n , , n,

Then kg solves either (2.83) or (2.87), and conversely

Proof: The proof proceed as in theorem 1.1 by making use on the

spherical waves expansion of the scalar field given by

«(x)=y"[- /».(y)5(x,y,fc) + u o ( y ) ^ ^ ^ ^

S

d5(y), X € A , (3.8)

for the Dirichlet problem and by

"(X) = / [fe»(y)^^^^'(y)^^ + ^{y)p(x,y,fc)] d5(y), X e A , (3.9)

for the Neumann problem

1.2 Distributed spherical wave functions

In this section we will analyze the completeness properties of the spherical

wave functions distributed on a segment of the j^-axis For the moment

we confine ourself to the system of functions consisting in radiating

solu-tion to the Helmholtz equasolu-tion A specific scheme for complete systems

construction can be formulated as follows:

(a) Let { x n } ^ i C E represents the position vectors of the discrete

sources

(b) Let Ua be the single-layer potential with density a and let us consider

the set of conditions

(£iXa) (xn) = 0, n = l , 2 , , (3.10)

which provide that Ua = 0 in Df Here C is some operator whose

significance will be clarified latter

Let us define the scalar functions fn by setting

/n(y) = ( £ x 5 ) ( x n , y ) , n = l,2, (3.11) Since

(Cua) (xn) = J a{y)fn{y)dS{y) = {fn, a*>2,5 (3.12)

s

we see that the closure relations for the system { / n } ^ i are equivalent to

the vanishing conditions (3.10) Thus, the following result is valid

T H E O R E M 1.3: Let S be a closed surface of class C^ Then the system

{ / n } ^ i is complete in L'^{S)

Two parameters are essential for complete system construction: the

support H of discrete sources and the vanishing conditions for the

single-layer potential in Di Both parameters determine the type of discrete

sources In general we can use as support a point, a curve, a surface,

etc

Let E consists of the point O which coincides with the origin of a

Cartesian coordinate system Then, the corresponding complete system is

the system of radiating spherical wave functions w^„ This system was

analyzed in the previous section We intend now to derive necessary and

sufficient conditions leading to the vanishing of the single-layer potential

in A

In the sequel we will denote the half-plane (p = const by E We will

assume that O is the center of a sphere S^ enclosed in Di and will denote its

interior by D [ For x G P [ , the single-layer potential Ua can be expressed

as a Fourier series with respect to the azimuthal angle ip The Fourier

coefficients u^ = u^(r, 9) are given by

<{r,9)= Y 0nJnikr)Pir^icose), (3.13)

n=|mj

where

/?- = ^2)^„ I a(y)«3 ^„(y)d5(y) (3.14)

Let us formulate the vanishing conditions in terms of the Fourier coefficients

lim - 5 - i - l r i = 0 for all m G Z, / > \m\ (3.15)

^-*o {kry and any 6 G [0, TT] Note, that similar conditions can be provided by impos-

ing that Ua vanish at the point x = 0 together with all derivatives Next,

we will prove that the set of vanishing conditions (3.15) implies

/Sr" = 0 for all m G Z and / > |m| (3.16) Fix m and construct

^ = t ^n'i^Pl-^i^^osO) (3.17)

For / = |m| we pass to the limit when r —» 0 and use the asymptotic form

of the Bessel function:

jn{x) = j£^J^" [1 + 0{x')] as X - 0, (3.18)

to obtain

y\m\

/ ? M < | ( c o s 0 ) = O (3.19)

Since the last relation is valid for any 0 £ [0,7r], it follows that 0"^^ = 0

Taking / = |m| + 1 we arrive at /3|^j_j.i = 0 and the same technique can be

used to conclude Thus, condition (3.15) yields t i ^ = 0 in E fi Z)[ for all

m € Z; whence ita = 0 in Di follows The converse result is immediate

Evidently, arguing as in the precedent section the conclusion i^o = 0 in

Di follows directly from the closeness relations for the system of radiating

spherical wave functions However, we would like to draw attention to

the above set of implications since this strategy will be used in the sequel

Vanishing conditions similar to (3.15) will be derived for the system of

distributed spherical wave functions

Let the support of discrete sources be the segment F^ of the 2:-axis

Assume F^ C I>[ and choose a sequence of points {zn) C F^ The support

of discrete sources is depicted in Figure 3.1 Before we state our next

results, let us prove some auxiliary lemmas

LEMMA 1.1: 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 Then, the limit

limw^(r;)/(M''"^^^Z, (3.20)

4 2

FIGURE 3.1 Illustration of the support of discrete sources

exists and represents an analytic function of z Here {p,(p,z) are the

cylin-drical coordinates ofx and t] = {p^ z) eH

Proof: For x € D[, the Fourier coefficients are given by

Then, using the relation F(|m| — n, |m| -f- n -f 1, |m| +1,0) = 1, we evaluate

the limit when p —• 0 as

lim w-Cr/)/ (M'*"' = v^{z) = f ; a - ^ M , (3.23)

'' n = | m | ('^^)

where a!!* = 2"^^^^-^—, M /^r- Now, accounting of the series

repre-"" (n-~|m|)!|m|! "^

sentation of the spherical Bessel function we see that v^{z) is an analytic

function of z

LEMMA 1.2: Let u e 3?(Di) be a solution to the Helmholtz equation

in Di and let u^ be its Fourier coefficients with respect to ip Define v^ by

(3,23)y and assume v^{zn) = 0, n = 1,2, , where {zn) CTz is a bounded

sequence of points Then u — 0 in Dị

Proof: The boundedness of the sequence {zn) implies the existence of

a convergent subsequence {zn^) ^ fc = 1,2, Then, since v^{z) is analytic

and v^{znk) = 0, fc = 1,2, , we use the uniqueness theorem of analytic

function (cf Chilov [28]) to obtain v'^iz) = Oior zeOzD i ) [ , ịẹ

y â^^^^ = 0, for zeOzn D[ (3.24)

Using the technique previously described we arrive at â = 0 for all m € Z

and n > | m | Thus, u^ = 0 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

func-tions which form a set of radiating solufunc-tions to the Helmholtz equation

They are defined by

d n W = < | m | ( x ~ ^ n e 3 )

(3.25)

= h\'2i{krn)P):;:)(cosen)ế^'', n = 1,2, , m G Z,

where (r^, 6nr ^) are the spherical coordinates of x^ = x — ^n^a, ịẹ r^ =

p^-^iz- Znf, P,5'(cos^n) = (2|m| - l)!!sinl^' Qn and sin^n = pAn- The

geometrical significance of the above notations is shown in Figure 3.2 We

are now in position to give the main result of this section

T H E O R E M 1.4: Consider the bounded sequence (zn) C F^, where F^

is a segment of the z-axis Assume S is a surface of class C^ enclosing

Tz' Replace in theorem l.ly part (a), the localized spherical wave functions

w^^, n = 0,1, , m = —n, , n, by the distributed spherical wave functions

^^n^ ^ ~ 1,2, , m € Z Then^ the resulting systems of functions are

complete in L^{S)

Proof: We prove only the first part of the theorem Let us establish

the explicit form of the vanishing conditions

C ( ^ n ) = 0, n = 1,2, , m € Z, (3.26)

FIGURE 3.2 The vectors Xn and x in the half-plane E

where v^ is given by (3.23) with Ua standing for the single-layer potential

with density a To this end let us consider the ring currents Smi

Smiv, ^y) "^ 27r / ^(^' ^' ^^ ^^P [-J'^i^ - ^y)] d<^» (3-27)

0

where T) = {p,z) and T}y = {Py,Zy), It is readily seen that the Fourier

coefficients of the single-layer potential Ua with respect to <f are given by

<iv) = Ja{y)Sm{v.rjy)exp {-JTrnpy) dS{y) (3.28)

s

Consider now the representation (cf Eremin and Sveshnikov [55])

smirf,vy) = cf: J'y_^^!TL (fci^)-^^'^'-'^4;i,^,(fci?), ^ /!(/ + |m|)!22'+l'"l

(3.29)

where B/^ = p^ + p^ + (z — Zy)"^ and c is a constant Then,

Jim 5„(»7,r7y)/(M''"' = c„/ig,(A;r,)Pg(cosd,), (3.30)

where r^ = Py + {z - ^y)^, sin^^ = Py/r^j and Cm is a constant It follows

that

C ( ^ ) = lim < ( ^ ) / ( M ' ' " ' = c „ | a ( y ) u i ^ | „ , ( y - ^e3)d5(y) (3.31)

and we see that (3.26) represents the closure conditions for the system of

functions (3.25) The precedent lemma now conclude the proof of the first

part of the theorem

Next, we formulate the null-field equations in terms of distributed

spherical wave functions

T H E O R E M 1.5: Under the assumption of theorem 1.4 replace in

the-orem 1.2 the localized spherical wave functions u^^^, n = 0,1, ,m =

—n, , n, by the distributed spherical wave functions <;^„, n = 1,2, , m G

Z Let hs solve the resulting null-field equations (3.6) and (3.7) Then hs

solves the general null-field equations (2.83) and (2.87), and conversely

Proof: This is proved in the same manner as theorem 1.4 with the

roles of single-layer potential and scalar fields (3.8) and (3.9) interchanged

As we will see in the electromagnetic case an alternative proof of

the-orems 1.4 and 1.5 can be given by using the translation addition theorem

for spherical wave functions

The above theorems deal with radiating solutions to the Helmholtz

equation These functions are suitable for analyzing exterior scattering

problems However, the conventional scheme can be used to treat interior

scattering problems as well Next, we will present completeness results

for systems of functions consisting in entire solutions to the Helmholtz

equation We begin by considering the function

sinffclx — yl) ^ o /«^^x

which is an entire solution to the Helmoltz equation Let us define the

modified single-layer potential Ua by

Ua{x) - y a(y)x(x,y,A:)dS(y), x € R ^ (3.33)

s

Then the following theorem is valid

T H E O R E M 1.6: Consider Di a bounded domain with exterior Ds and

let the modified single-layer potential Ua y^ith density a be given by (3.33)

Then from Ua = 0 in Di it follows that the single-layer potential Ua with

density a vanishes in Dg, that is Ua =0 in Dg^

Proof: Let 2a(x) = 0, x € A - For x E 5^, where S^ is an inscribed

sphere, we use the expansion

x{y.,y,k) =-J2 XI ^mnttL„„(y)«LW (3.34)

and the orthogonality of spherical wave functions to obtain

a{y)y}_^^{y)dS{y) = 0, n = 0,1, , m = - n , , n (3.35)

/ •

Let S^ be a sphere enclosing Di and let us denote its exterior by D^

Multiplying (3.35) by n ^ ^ ( x ) , where x G D ^ , summing over m and n, and

using the spherical waves expansion of the Green functions for |y| < jx|,

we see that the single-layer potential Ua with density a vanishes in D^

Consequently, by the analyticity oi Ua, Ua = 0 in Ds follows The theorem

is proved

We note that similar results can be proved for the modified double-layer

potential Va and the modified combined potential

Wa-As next we will consider systems of functions consisting of entire

solu-tions to the Helmholtz equation A specific scheme for complete systems

construction can be given as follows:

(a) Let { x n } ^ i C E represents the position vectors of the discrete

sources

(b) Let Ua be the modified single-layer potential with density a and let

us consider the set of conditions

(£ua)(xn) = 0, n = l,2, , (3.36) which provide that 2^ = 0 in Df

In this context, let us define the scalar functions Qn by

^n(y) = (£xx)(xn,y) (3.37)

Since

(Cua) (Xn) = J a{y)9n{y)dS{y) = (^n, a*)2,5 (3.38)

dD

we see that the vanishing conditions (3.36) represents the closure relations

for the system{pn}^i- The following theorem is the analog of theorem 1.3

for the system of functions (3.37)

T H E O R E M 1.7: Let S be a closed surface of class C^ Then the system

{^n}^i is complete in L'^{S)

We define the distributed spherical wave functions which form a set of

entire solutions to the Helmholtz equation by

(3.39)

= J|m|(fern)P,5'(cosen)e^-^^, n = 1,2, , m G Z

Analogous to theorem 1.4 we can state the following result

T H E O R E M 1.8: Consider the bounded sequence (zn) C F^, where Fz

is a segment of the z-axis Assume S is a surface of class C^ enclosing

Fz' Replace in theorem 1.1, part (b), the localized spherical wave functions

'^mn^ n = 0,1, , m = —n, , n, by the distributed spherical wave functions

^mn^ ^ == 1,2, ,m G Z Then, the resulting systems of functions are

complete in L'^{S)

Proof: For proving the first part of the theorem it suffices to establish

the explicit form of the vanishing conditions

l i m 2 ^ ( ^ ) / ( M ' ' " ' = ^ ( ^ n ) = 0,n = l , 2 , , m e Z (3.40)

Using the representation (cf Eremin and Sveshnikov [55])

oo

lk?00 "l^Z+lml SA'n,riy) = g g ^.(i 4 H ) ! 2 ^ ' + I " ' I (fe^)"^"''""" J2iMmm)

(3.41)

where Sm are the ring currents corresponding to x? we see that

lini 5^(17,y)/ ( M ' " ' = M m | ( ^ r , ) P g ( c o s 0 , ) (3.42)

The proof can now be completed as in theorem 1.4

1.3 Distributed point sources (fundamental solutions)

In this section we will present completeness results for systems of functions

with singularities distributed on closed or open surfaces We debut with

the fundamental solutions

<^n W = Sf(xn, x,A:), n = 1,2, , (3.43) where { x n } ^ i is a set of points distributed on a closed surface of class

C^ Let {<Pn}n=i denote the system of functions with the singularities

{ x ~ } ^ j distributed on the interior surface 5 " and let {v^n}^i denote

the system of functions with the singularities {^n}^=i distributed on the

exterior surface S^ The auxiliary surfaces are shown in Figure 3.3

Com-pleteness results for the system of distributed sources (3.43) are given by

the following theorem

T H E O R E M 1.9: Consider Di a bounded domain of class C^ Let the set

{ x ~ } ^ j be dense on a surface S~ enclosed in Di and let the set {'^t}^=i

be dense on a surface S'^ enclosing Di Assume k ^ p{D^)^ where D^

is the interior of S" Replace in theorem 1.1 the radiating spherical wave

FIGURE 3.3 Illustration of the support of distributed point sources

functions u^^, n = 0, l, ,m = —n, ,n, by the functions ip~, n= 1,2, , and the regular spherical wave functions u}^^^^ n = 0, l, ,m = —n, ,n,

by the functions (^^, n = 1,2, Then, the resulting systems of functions are complete in L^{S),

Proof: Let us consider (a) It has to shown that for a G L^{S) the

set of closeness relations

/ a*{y)<p-{y)dSiy) = 0,n = l,2, (3.44)

implies a ^ 0 on 5 Prom (3.44) it follows that the single-layer potential

Ua' with density a' = a* satisfies Uaf{x~) = 0, n = 1,2, Since { x ~ } ^ j

is dense on S" we get Ua' = 0 on S" The assumption k ^ p{D^) gives

Ua' = 0 in D r , and going further, the analyticity of Ua' yields Ua' = 0 in

Di Theorem 2.2 of Chapter 2 may now be used to conclude Repeating

the above arguments for the double-layer potential Va' and the combined potential Wa* = ««' — ^Va' we see that (a) is proved The proof of the

second part of the theorem is similar

The system of functions given by theorem 1.9, part (b), is suitable for approximating internal fields However, for such kind of application we can also use a system of discrete sources distributed on an interior surface These functions are given by

^ n W = x(Xn,x,fc), n = l , 2 , (3.45)