M
ated with the scattering of acoustic or electromagnetic waves by
bounded obstacles has been a subject of great interest during
the last few decades. This is primarily due to the fact that particle scatter
ing analysis is encountered in many practical applications as, for example,
aerosol analysis, investigation of air pollution, radiowave propagation in the
presence of atmospheric hydrometers, weather radar problems, analysis of
contaminating particles on the surface of silicon wafers, remote sensing,
etc.   M any techniques have been developed for analyzing scattering prob
lems.   Each of the available methods generally has a range of applicability
that is determined by the size of the scattering object relative to the wave
length of the incident radiation. Scattering by objects that are very small
compared to the wavelength can be analyzed by the Rayleigh approxima
t ion, and geometrical optics methods can be employed for objects that are
electrically large. O bje cts whose size is in the orde r of th e wavelength of
the incident radiation lie in a range commonly called the resonance region,
and the complete wave nature of the incident radiation must be considered
in th e solutio n of th e sca tterin g problem . Classical m eth od s of solution
in th e reso nan ce region such as th e finite-difference m et ho d, finite-element
method or integral equation method, owing to their universality, lead to
computational algorithms that are expensive in computer resources. This
significantly res tricts their use in study ing m ultipara me tric boundary-value
IX
X PREFACE
problems, and in particular in analyzing inverse problems which are mul-
t iparametric by nature. In the last few years, the discrete sources method
and the null-field method have become efficient and powerful tools for solv
ing boundary-value problems in scattering theory.
The physical idea of the discrete sources method is linked with Huy-
gens principle and the equivalence theorem. The obstacle, being a source
of secondary (scattered) field, is substituted with a set of fictitious sources
which generate the sam e secondary field as does the actu al obstacle. These
global principles have led to a variety of numerical methods, such as the
m ultiple mu ltipole techniq ue (Hafner [66], [67]), discre te singu larity m etho d
(Nishimura  et al   [119]), m eth od of aux iliary sourc es (Zari dze [169]), Ya-
suu ra me thod (Yasuu ra and Ita ku ra [167]), spherical-wave expansion tech
nique (Ludwig [95]) and fictitious current models (Leviatan and Boag [92],
Leviatan  et al [94]). The difference between these approaches relates to
the ty pe of sources used. Essentially, the ap prox ima te solution to the sca t
tering problem is constructed as a finite linear combination of fields of
elementary sources. The discrete sources are placed on a certain support
in an additional region with respect to the region where the solution is re
quired and the unknown discrete sources amplitudes are determined from
the boundary condi t ion.
In the null-field method (otherwise known as the extended boundary
condition method, Schelkunoff equivalent current method, Eswald-Oseen
ext inct ion theorem and T-matr ix method) developed by Waterman  [155],
one replaces the particle by a set of surface current densities, so that in the
exterior region the sources and the fields are exactly the same as those ex
isting in the original scattering problem. A set of integral equations for the
surface current densities is derived by considering the bilinear expansion
of th e Green function. Th e solution of th e sca tteri ng prob lem is then ob
tained by approximating the surface current densit ies by the complete set
of pa rtia l wave solution s to the H elmh oltz (M axwell) equatio n in spherical
coordinates. A number of modifications to the null-field method have been
suggested, especially to improve the numerical stability in computations for
particles with extreme geometries. These techniques include formal modi
fications of the single spherical coordinate-based null-field method (Iskan-
der  et al   [76], Bo stro m [15]), different choices of bas is function s an d t he
application of the spheroidal coordinate formalism (Bates and Wall [11],
Hackman [64]) and the use of discrete sources (Wriedt and Doicu [165]).
The strategy followed in the null-field method with discrete sources is to
derive a set of integral equations for the surface current densities in a va
riety of auxiliary sources and to approximate these densities by fields of
discrete sources.
These considerations, combined with the continued cooperation be
tween the Dep artm ent of Process Technology at the Insti tute for M aterial
 
  CKNOWLEDGMENTS
would like to express my sincere thanks to Professor Klaus Bauckhage,
head of Department of Process Technology at the Insti tute for Material
Science Bremen for his constant help by providing me with the technical
sup por t necessary to complete this m anusc ript . During much of the w rit ing
of this book Professor Klaus Bauckhage was an incisive critic and a fertile
source of ideas. In fact, the developm ent of com pute r program s in the
framework of the null-field method with discrete sources was motivated by
practical problems: measurement of spheroidal part icles, agglomerates and
rough part icles using the Phase Doppler Anemometer. Without Professor
Klaus B auckh age s constan t encourag emen t t he w rit ing of this book would
not have been possible.
d r i a n D o i c u
XI I I
ANALYSIS
In this chapter we will recall some fundamental results of functional anal
ysis.
  W e firstly p resen t the notion of a Hilbert space and discuss some
basic propert ies of the orthogonal projection operator. We then introduce
the concepts of closeness and completeness of a system of elements which
belong to a Hilbert space. The completeness of the system of elementary
sources is a necessary condition for the solution of scattering problems in
th e framework of th e discrete sources me tho d. After this discussion, we will
briefly present the notions of Schauder and Riesz bases. We will use these
concepts when we will analyze the convergence of the null-field method.
We then consider projection methods for the operator equation
Au  = / ,
where A is a linear bounded and bounded invertible operator from a Hilbert
space  H  onto  itself.  We will consider the equivalent variational problem
B{u, x)  =  J^*{x)  for all  x e H,
where B is a bounded and strictly coercive sesquilinear form and J" is a
 
(a )  \\u\\fj >  0, (positivity)
(t>) ll^ll // = 0 if an d on ly   \i u =  0/ /, (definiteness)
(c) | |aix | |^ =  \a\ \\u\\jj  , (homogeneity)
(d) | |u +  v\\f^ < \\u\\ff  -f
  | | i ; | |^
  , (triangle inequality)
for  all u,v e H  and all a € C ar e satisfied. Therefore any scalar prod
uct induces a norm , bu t in general, a norm | | | | ^ is gene rated by a scalar
product if and only if the parallelogram identity
ll« +  v\\l + \\u - v\f„  = 2 {\\u\\l  +  Ml)  (1.3)
holds.
Given a sequence   {un)  of elements of a normed space X, we say that
 —
 u\\^  -+ 0 as n —• oo. A se
quence  (un)  of elements in a normed space  X  is called a Ca uch y sequence
if  \\un
 — UmWx —*  0 as n ,m ^ oc .
A subset M of a normed space   X  is called com plete if every Ca uchy
sequence of elements in   M  converges to an element in  M.  A normed space
is called a Banach space if it is complete. An inner product space is called
a Hilbert space if it is complete.
A sequence  (un)  in a Hilbert space H  converges weakly  to u £ H if  for
any  v E H, {un,v)fj
  —>  (u , i ; )^ as n —>>  oo. Ordinary (norm) convergence is
often called strong convergence, to distinguish it from weak convergence.
T he term s 'stro ng ' and 'wea k' convergence are justified by the fact t ha t
strong convergence implies weak convergence, and, in general, the converse
imp lication doe s not hold. If a sequence is containe d in a compact se t,
then weak convergence implies strong convergence. Note that every weakly
convergent sequence in a Hilbert space is bounded and every bounded
sequence in a Hilbert space has a weakly convergent subsequence.
Two elements u and v of an inner produ ct space H are called orthogonal
if  {u,v)fj = 0; we the n write  u±v.  f an element  u  is orthogonal to each
element of a set M , we call it ortho gon al to the set M and w rite   u±M.
Similarly, if each element of a set   M  is ortho gon al to each element of the set ,
K,  we call these sets orthogonal, and write   M±K.  Th e Pytag ora theorem
states that
for any orthogonal elements   u  and  v.
A set in a Hilbert space is called orthogonal if any two elements of the
set are orthogonal. If, moreover, the norm of any element is one, the set is
called orthonormal.
CHA PTER I ELEMEN TS OF FUNCTIONAL ANALYSIS
A subset  M  of a normed space is said to be closed if it contains all
its limit poi nts. For any set  M  in a normed space, the closure of   M  is
the union of  M  with the set of all limit points of  M,  T he closure of  M  is
wr i t t en  M.  Obviously,  M  is contained in M , and  M  =  M  if  M  is closed.
Note the following properties of the closure:
(a) For any set M ,  M  is closed.
(b) If_A/ C  K,  t h en M C F .
(c)  M  is the smallest closed set containing A/; that is, '\{ M  <Z  K  and
K  is closed, then  JI C K.
Complete sets are closed and each closed subset of a complete set is
complete.
Next, we define the orthogon al projection op erato r. Let J/ be a Hilbert
space and  M  a subspace of  H  (i.e. a com plete vector subs pace of  H).
Let  u E H.  Sinc e for any  v  €  M  we have ||^/ —  ^| |// > 0, we see th at
the set  {\\u — vW^^  / t ' G A/} posses an infimum. Let  d  =  mi^^^M  || ^ ~ ^IIH
and let  {vn)  be a  minimizing sequence,  i.e. (f^,) C  M  and  \\u -  VUWH ~ ^
dasn  —V oo . Since  M  is a vector subspace,  ^{vn + v^n)  G Af, whence
11  L___ Ii|| >  d^  Using this and the parallelogram identi ty
H
Vn  4-1; , ,
  - Vr,,\\l <  2 (||U -  VnWl +  II" " ^m ll«)  ' ^d^\  (1-6)
wh ence , b y let tin g n, m —• oo, | |i;„  —  I'mll// —* 0 follows. T h u s ,  {vn)  is
a Cauchy sequence and since   M  is complete, there exists  w £ M  such
t h a t  \\vn — 'w\\f
  —>  0 as n  —>  oo; moreover  \\u
 — i^n||//  — ||^ - ?^||// = rfas
n —• 00. Sup pose now th at the re exists ano the r elemen t   w'  for which the
function  \\u
 —  u| |^ a t ta ins i t s minimum; then d = \\u -  if || ^ = ||w —
 vj'^n
d —  inf  \U - l^llrr <
w   +  w
H
\\W
w ^-w'
= 0, (1.8)
we find  w  =  w'.
The vector  w  gives the best approximation of  u  among all the vectors
of M. Note that  d  is called the distance from   u to M  and is also noted by
p(u , M ) . T he opera to r  P
  :
i.e.
where ||t/ —  t/;||j^
  =  d — miy^M
  11  "" ^11//  ' ^ ^ bo un de d Hnear o per ato r
with the proper t ies :  P^ = P  and  {Pu.v)jj — {u,Pv)ff  for any  u,v  €  H.
It is called the orthogonal projection operator from   H  onto M, and  w  is
called the projection of   u  onto M.
The following statements characterizing the projection are equivalent:
(a)  \\U-W\\H  <  \\u-v\\ff,
(b )  Re{u- w,v - w)fj < 0,
(c)  Re{u-v,w - v)fj > 0,
toT ue H, w ^ Pu e M  and any v  €  M,
Let M  be a subset of a Hilbert space //. The set of all elements or
thogonal to  M  is called the orthogonal complement of   M ,
M^ = {ueH/u±M}.
Clearly, M-^ is a subspace of   H,  To show this we firstly observe that A/-^
is a vector subspace, since for any scalars a and /? and any   u.v £  A/-^,
{au-\- (3v,(fi)^  = 0 for all ( G A/; when ce  au  -{-  f3v e  A/-^ follows. To
prove that Af-*- is complete, let us choose a Cauchy sequence {n„) C A/-^;
it converges to some   u £ H  because  H  is complete. We must show that
u  E  M ^.  Since for any  v e H,  and in particular for any  v e  A/, we have
{uny v)fj
  —>  (w,
 v)fj  as n  —>  oc and (un?  ^ ) H   = 0, n =  1,2,...,  it follows that
(w, i^)// = 0 for a ny v G A/. H enc e, tz G A/-'- a nd so Af- is com ple te.
Now, let if be a Hilbert space,   M  a subspace of if, and  P  the or
thogonal projection operator of   H  onto  M.  Let  u e H.  From the prop
ert ies of the projection we see that   Re{u - Pu.v  — Pu)fj <  0 for any
t; G Af. Ch oose   v = Pu ±  p with  p  being an arbitrary element of A/. Then
R e {u
 —
relat ion (f  by J V (J ^ = 1) we get
Re(w ~  Pu,j(f)ff ~  R e [ - j ( w -  Pu,(f)fj] = Im(w -  Pu,ip)ff  = 0. (LIO )
Thus, for a given   u e H  the projection u; =  Pu  satisfies  u  — w 1 M.
Therefore, any element  u £ H  can be uniquely decomposed as
u = w-^w^,  ( L l l )
 
6 CHAPTER   I  ELEMENTS  OF  FUNCTIONAL ANALYSIS
where  w e M  and w   G M-^.   This result  is  known  as the  theorem  of
orthogonal projection.
  :
Qu = u-Pu  (1.12)
is  the orthogo nal projection o pera tor from  H  onto  M .
2
  CLOSED
  ND
 M
  if for
any  u e X  and any £ > 0   there exist  u E M  such that  \\u — u^W  < e.
Equivalently,  M  is .dense  in X
  if and  only  if for any
 u e X  the re exists  a
sequence  {un) C M  such that  \\un — w||x —• 0 as n —> oo.
Every  set is dense in its closure, i.e. M  is dense in M. M is the  largest
set  in which M
 M
  is dense  in K,  then  K C M.li M is
dense   in a  Hilbert space  if,  then  M
  —
is dense   in  H.
Let  H he Si  Hilbert space.  If M  is dense  in H  and u  is orthogonal  to
M, then u = 6H- Indeed,  let  uJLM  and  choose  an  arb i t r a ry  v E H.  Since
M
  C
  —• (^?^) / /  as n  —>  oo. From  {u,Vn)ff
  =
0,  n = 1,2,..., it   follows that  (w, v) = 0 for any v e H.  T h u s , u ±H. The
element w
 is  orthogonal to any element of H and in par t icular  is orthogonal
to   itself, i.e.  {u,u)ff  =  | | u | | ^  = 0.  Hence, u
 =  OH-
Elements V^i,^2'  >'^N  ^f ^   vector space X  are   called linearly depen
dent  if  there exists  a  l inear combination  Yli^i  i' i  = 0 in  which  the co
efficients  do not  vanish,  i.e.  Yli=i  l^ l > - ^^^   vectors are   called linearly
independent   if  they  are not  l inearly dependent,  or  equivalently,  if  there
exists  no  non-trivial vanishing l inear combination.   If any  finite number of
elements  of an  infinite  set   { t / ^ J ^ i  is  l inearly independent,  the set  {V'J i^ i
is called linearly independent.
A system   of  elements {V^jj^i  is  called closed  in
  H
  if  there  are no
elements  in H  orthogonal to any element of the set except  the zero elem ent,
that means
 = 0H-
A system  of elements  {ipi}^i  is called complete  in H if the  linear span
C X
1 = 1
 
2.  CLOSED AND COMPLETE SYSTEMS IN HILBERT SPACES. BASES 7
is dense in H,  i.e. Bp {-^j,  ip2^ •••} =
  H.  Equivalently, if { t / ^J ^ i is complete
in  H  then for any  u e H  and any  e > 0  there exist an integer  N  =  N{e)
and a se t { t t r}^_i such that   \\u —  X) i=i ^f^^ i " •
Let us observe that the closure of the linear span of any set   { T / ^ ^ } ^ is
a subspace of  H.  It is a vector subspace by its very definition and it is also
complete as a closed subset of a complete set.
Obviously, if the system {V^J^i is complete in   H,  then the only ele
ment or thogonal to   {ipi}^i  is th e zero element of  H]  thus the se t {0i}i^i
is closed in H.  T he converse result is also true . To show this let { ^ J ^ i be
a closed system in  H.  Let us deno te by W  the linear span of  {ipi}^i  - Then
any element  u£ H  can be uniquely represented as w  =  Pu  4- Q u, w here  P
is the orthogonal projection operator from   H  onto  W ^ and  Q  is the orthog
onal projection operator from   H  onto  W  . Since  Qu  G W  and  \l)i € W,
2 =   1,2,..., we get  {Qu, il^^)u  = 0, i =  1,2,....  The closeness of {V^J^i in H
implies  Qu  =  6H->  and therefore for any element  u € H we  have  u = Pu  €
W .  T h u s ,  H C W ,  and since  W C H we get W = H;  therefore  W  is dense
in   H.  We summarize this result in the following theorem.
T H E O R E M   2 . 1 :  Let H  be  a Hilbert space. A systetn of elements  {'4 i} i
is complete in H if and only if it is closed in H.
A set {ipi
 }?=i is called a finite bas is for th e vector sp ace  X  if it is linea rly
independent and i t spans   X.  A vector space is said to be n-dimen sional if
it has a finite basis consisting of  n  elements. A vector space with no finite
basis is said to be infinite-dimensional.
Let  HN  be a finite-dimensional vector subspace of a Hilbert space   H
with orthogonal basis  {<f>i}^-i.  Then the orthogonal projection operator
from  H  onto  HN  is given by
N
t = l
For th e t ime being we note a simple but im porta nt result characterizing
the convergence of the projections. Let   {ipi}^i  be a com plete and linear
indepe nden t system in a Hilbert space /f , let  H^  sta nd for th e linear span
of  {ipi}i^i, an d let us den ote by   PN  the orthogonal projection operator
from  H  onto  HN.  We have
\\u -
  \\u
  - t; | |^
(1.13)
for any  u e H;  thus the sequence | |n -  PN' WH  ^^ convergent . Since { ^ J ^ i
 
8 CHA PTER I ELEME NTS OF FUNCTIONAL ANALYSIS
—> 0 as n —* oo. T he n, from 0 <   \\u — PN ' 'WH  ^ N  —  ^iVnll// we get
||u — PNa'^W n —  0 as n  —>•  oc; thus the convergent sequence  \\u —  PNU\\H
possesses a subsequence which converge to zero. Therefore, for any   xi  £ H
we have
  (1.14)
A map i4 of a vector space  X  into a vector space Y  is called linear if  A
t ransforms l inear combinations of elements into the same l inear combina
t ions of their images, i .e . i f ^ ( a i ^ i  -]-a2U2 +...) = aiA{ui)-ha2 A{u2 )-{-..
Linear m aps are also called linear operat ors. In the l inear algebra one usu
ally writes arguments without brackets,   A{u)  =  Au.  Linearity of a map, is
for normed spaces, a very strong condition which is shown by the following
equivalent statements:
(a)  A  transforms sequences converging to zero into bounded sequences,
(b)  A  is continu ous a t one poin t (for insta nce a t tx = 0),
(c )  A  satisfies the Lipschitz condition | | i4u| |y <   c||u||;^  for  all u e X
and  c  independent on tx,
(d)  A  is continuous at every point.
Each number c for which the inequality (c) holds is called a bound for
the operator  A.
Let  C{X, Y)  be the linear space of all linear continuo us m aps of a
normed space  X  into a normed space  Y.  The norm of an operator
uex,uy^0x \m\x  l|u|lx=i
satisfies all the axioms of the norm in a normed space, whence the linear
s pa c e £ ( X , Y)  is a normed space. Note th at th e num ber  \\A\\ is the smallest
bound for yl. It is not difficult to prove that the space  C{X, Y) is  complete
if the space  Y  is such.
A m ap of a vector space into th e space C of scalars is called a functional.
Th e above state m ents are valid for l inear func tio na l . Th e space   £{X^  C )
is called the conjugate space of   X  and is den oted by  X*.  It is always a
Banach space.
A system   {xpj}^^  is called m inim al if no elem ents of this sys tem belongs
to the closure of the linear span of the remaining elements. In order that
the system {V^l^i be minimal in a Banach space   X,  it is necessary and
sufficient that a system of linear and continuous functional defined on   X
exist forming with the given system a biorthogonal system; that is, a system
of Hnear and continuous functionals   {^j j^ j  such tha t  ^j  ( ^ J =  6ij^ where
6 ij  is th e Kronecker sym bol. If th e system {V ^ j ^ j is com plete and minim al,
 
2.  CLOSED AND CO M PLE TE SYSTEMS LN HILBERT SPACES. BASES 9
a Hilbert space  H,  by Riesz theorem (see section 1.3), there exists   ip j  such
t h a t  J'j (u)  =  ( , j)  for any u € / / ; therefore
  (tj^^j)^
= 6ij .  In this
case the system   ^^^^^  _ is called biortho gon al to the system
  {t' ' j}^_|.
A system { ^ J ^ j is called a Schauder basis of a Banach space   X  if
any element  u e X  can be  uniquel}^  represented as u = X]^^l ^?^^n where
the convergence of the series is in the norm of  X.  Every basis is a complete
minim al system . However, a complete minimal system may not be a basis in
the space. For example, the tr igonometric system   I/JQ  (t) =  1/2, 02n-i(^) =
sin(n;^),
  ^ 2 M
  (^)  — cos( ?if),n = 1,2, . . . , is a com plete minim al system in
the space   C([—TT,  TT])
  but it does not form a basis in it. In an arbitrarily
separable Hilbert space if, every complete orthogonal systems of elements
forms a basis. Thus, the trigonometric system of functions forms a basis in
L 2 ( [ - ; r , 7 r ] ) .
Th e sys tem { 0 j ^ j is cal led an uncondi t ional bas is in the Banach
space A' if it remains a basis for an arbitrary rearrangement of its elements.
Let T
  :
 X -^ X hea  bou nde d linear op era tor w ith a bou nded inverse. If the
system  {ipi}^i  is a bas is , then the sys tem { T ^ j j ^ j is a bas is . If  {u%}^^
is an uncondit ional basis, then   {Tu'i}^i  is an unco nditiona l basis. In a
Hilbert space, every orthogonal basis is unconditional. It can be shown that
an arbitrary uncondit ional basis in a Hilbert space is representable in the
form   { T 0 ^ } ^ j ,  where {0^}J^i is an orthonormal basis oi H.  Such bases are
called Riesz bases. If  { 0j^i  is a Riesz basis the n the biorthogon al system
\p i
 >  is also a Riesz basis. A com plete system   {i^i}^i  forms a Riesz
basis of  H  i f the Gramm matr ix G =  [Gij],  Gjj  =  { i ' 'j)ff  >  generates
an isomorphism on /^. The system {t'^jj^i forms a Riesz basis of   H  if the
inequalities
N
H   *=1
hold for an y co nst ant s QJ and for any iV, wh ere the positive const ants cj
and  C2  should not depend on   A  and a^. Equivalently, { ^ J ^ i forms a Riesz
basis of  H  if the re exist the positive cons tan ts ci an d C2 such t ha t
oo oo
? = 1 1 = 1
for arbitrary  u E H.  Note that if {t\}^i  is a Riesz basis, th en sup^ 11^/11//  <
{* }:,•
 
  PROJECTION METHODS 1 1
A2  such that  B{x^y) = {Aix,y)jj = {A2X,y)fj  for all  x,y  6  H.  Then, we
have  {Aix  — A2X , y)fj  = 0 for all x^y e H,  which implies Aix  =  A2X  for all
X E H.  Therefore, for any bounded sesquilinear form B  : H x H -^ C  there
exists an uniquely determined l inear and bounded operator   A : H  —   H
such that
B{x, y) = {Ax, y)fj  for all  x,y e H.  (1.23)
In a similar manner we can prove the existence of a linear and bounded
operator  A ^ : H -^ H  such that
B{x, y) = (x,  \ \  for all  x,y £ H.  (1.24)
The opera to r  A^  is called the adjoint operator of  A   Note that if  B  is
str ict ly coercive then   A  is strictly coercive, that is  Re{AXyX )ff >  c||rr||;^
for  sll X € H  and c > 0 .
The Lax-Milgram lemma states that if B is a bounded and str ict ly
coercive sesquilinear form on a Hilbert space   H,  then the strictly coercive
bounded operator  A : H —^ H  generated by  B  has a bounded inverse
A-^  :H -^ H.
As a consequence of Riesz theorem and Lax-Milgram lemma if 6 is a
bounded and strictly coercive sesquilinear form and /" a bounded linear
functional on a Hilbert space   H then  the variat ional problem
B{%x)  =  T*{x)  for all  x e H,  (1.25)
is unique solvable and the solution solves the operator equation
Au  = / , (1.26)
where  A  is the operator generated by  B  and / is the uniquely determined
element corresponding to   J^.
We are now well prepared to present the main result of this chapter,
namely the fundamental theorem of discrete approxim ation. This theo
rem is frequently used in the finite-element method for solution of various
boundary-value problems by discrete schemes.
T H E O R E M
  3.2:  f u n d a m e n t a l t h e o r e m o f d i s c r e t e a p p r o x i m a
t i o n )  Let H be a Hilbert space and B a bounded sesquilinear form on H
satisfying
\B{x, x)\>c \\x\\]j  for all  xeH,  (1.27)
Let T be a linear and continuous functional on H and  {^i}^x  ^  complete
and linearly independen t system in H. Then
 
(a) the algebraic system of equations
N
possess a unique solution^
is convergent; if
solution to the variational problem
^ * ( x ) =  B{u,x)  for all  x G H,  (1.30)
Proof:  Before we presen t the proof we no te th a t conditio n (1.27) is
weaker than the coerciveness condition (1.21). Coming now to the proof of
(a) we define the matrix B =   [Bij] by Bij  —
  B{'tl)^^ ipj)^  z, j  =  1,2,..., N.  Let
HN
= Sp {^i, . . . , i /^;v} ^^d let  PN  be the orthogonal projection oper ator
from  H  onto
  Hjsf.
 W i th  A  standing for the operator generated by the
sesquilinear form   B,  i.e.  B{x,y)  =  {Ax,y)fj  , we have
Bij  = 5 ( ^ i , ^ , ) = ( M , ^ i > H =  (A^i.PN'ipj),,  =  (PNAiPi.tlj^)^  .
(1.31)
(1.32)
to obtain | |P/v>la: | |^ > c | |x | |^ . Consequently, the operator   PjsfA  : H^  —
HN
is in ve rti ble. Since {t/ jl ^^x form a bas is of H^  we see th at th e vectors
(f i  =  PisfAtp^, i = . l,. .. ,iV , form a basis of f// . Le t us de no te by T =
[Tij]  , z, j =  1,2,..., A/', the nonsingular tran sit ion m atri x passing from the
basis {t/^j j l i to the basis  {iPi}^^i, i.e.  (p^  = Ylk^i^ik^k^^^ ^ =  1,...,A^.
Then, we have
k=l k=l
wh ere $ = [*ij] , * i j =  {' ii' j)fj  ^h j  = 1? 2, . . . , AT, is th e G ram m m atr ix
of the l inearly independent system   {ipi}^^i  - T he m atri x B is expressed as
 
B{uN,i j r{i j ,  j-l,...,Ar.   (1.34)
Multiplying the above relat ions   by a^* and sum min g over  j  we obtain
B{UNJUN)  =  T*{UN).  Th en from
c| |«^| |^
we deduce that  {UN)
 a
(1.34) we get  B{uNk^'^j)  =  T*{tl)j),  j  =  l,,..,Nk.
  Since
  for
  j
the mapping  x  G i/ , x  «-* B{x^ ipj)  is a  linear and continu ous functional on
H^  and since u^^
for any  j
 an y a: e H, L et
  us
 unique. Assume
that there exis ts u' ^ u  such that  B{u\x) —  *{x) for  any x E H. Then
B{u
  — w', a:)
0
  (1.36)
th e conclusion readily follows. T hu s, all weak convergent subsequences have
the same weak l imit; whence   UN
 -^
Let
  us
 now prove th e m ore stronger result , nam ely th at   \\UN —
 u\^  —•
 0
and the identi t ies   B{U^UN)  = T*{UN)
  and  B{UN,UN)  =   J^*{UN)  we get
c \\UN -
 x 6 if,
defines
  a
 H  we obtain  B{UN^U)  —•
B{u^ u) as N —*  oo,  and the conclusion readily follows.
Evidently, the fundamental theorem
 a
 stri ctly coercive sesquilinear form  B.  In this context, the unique
solution  to  th e th e variation al problem (1.30) coincides w ith the unique
solution
  to
 the op erat or equa tion (1.26). Th e projection relat ions (1.28)
may be wri t ten  as
< ^ w j v ~ / , ^ ^ ) ^
or equivalently  as
PNAPNUN  = PN/^  (1.40)
where   P/v is the  orthogon al projection ope rator from  H  onto  Hjsf and
HN  =  Sp{V^i,. . ,V^^}.  The  above projection m ethod   is  also called  the
Galerkin method.   The  stronge st cond it ion which guaran tee   the  conver
gence of the  projection scheme  is the   strictly coercivity of the  sesquilinear
form B.  According  to   (1.32) we see th at this condit ion implies
WPNAPNUW^^  > c \\PNU\\H  for all ueH.  (1.41)
Let  us generalize  the   above results when  A is  a. l inear boun ded  and
boundedly invert ible operator from   a  Hilbert space H  onto a  Hilbert space
G.  Let HN  C  HN^I  with  dimHN
  =  AT be a  sequence  of  subsets limit
dense  in H,  i.e. for any u £ H,  P{U^HN)  —• 0 as iV —  OO,
 and let  PN
s tands   for the orthogonal projection ope rator on to  HN.  Analogously,  let
GN  C  Giv+i with dim GAT =  AT be a sequence of  subsets l imit dense  in G
and   let QN  s tands  for the orthogonal project ion operator onto  GN. The
projection method giving   the  approximate solut ion u^  of  (1.26) is
  = QN/-
  (1.42)
T h e n   we can  formulate  the   following result  (cf.  Ramm [128]).
T H E O R E M   3.3:  Let A : H -^ G be a  linear bounded and. boundedly
invertible operator. Equation (1.4^)  is  unique solvable for all  sufficiently
large N, and
if and only if
WQNAPNUW^  > c \\PNu\\fj
 NQ  is  some integer  and c > 0
  does not  depend  on N and u.
Proof:  Let us   prove  the   necessity. Assu me th at (1.43) holds and
(1.42)  is  unique solvable. Th en   for / G G we have  \\UN — u\\ff  —> 0 as
N  -^  oo, where  UN =  {QNAPN)~^
  QNf  and u = A~^f.  T h u s
sup
N
{QNAPNr'QN\\  < c < o o . (1.45 )
Since  QNAPNU  =  Q^QNAP^U  and   P^PN'^  =  PNU we have
QNQNAPMU
I I
H
an d (1.44) is prov ed. For prov ing th e sufficiency we assu m e (1.44). Con
sequently, the operator   Q^APN  \  P^H  -^  QNG  is an injective m appin g
betwe en tw o A/'-dimensional spaces and therefore th is m app ing is surjective.
Hence, (1.42) is unique solvable for   N > No.  From
QNA [PNU   -f (/ -  PN)  U]  = Qiv / ,
QNAPNUN  = QN/I
QNAPN  {U N  -  PNU)  =  QNA  ( / -  PN )  U.  (1.49)
Since  HN  is limit dense in il, it follows that  \\u
 — P /v^ | | //  —>  0, iV -* oo.
Then, from
  '\\QNAPN{UN-PNU)\\G^-\\QNA{I^PN)U\\C
c c
(1.50)
we see that
  \\UN —  -Pivw||^ — 0 a s A^ — oo; w hen ce, by th e tria ng le in
equality, we find that | |t* —  i^ivll//  —*  0 as iV —>  oo. This finishes the proof
of the theorem.
The following theorem will also be used many times in the sequel.
T H E O R E M  3.4:  Let A : H —   G he a linear hounded and houndedly
invertible operator satisfying (1.44)- Let B : H   —  G be a compact operator
and A-\- B he hounde d invertihle. Then,
^QN{A  + B)PN u\\ci>c\\PNu\\jj  f o r a l U G ^ a n d i V > i V o , (1.5 1)
where  NQ  is some integer and  c > 0  does not depend on N and u.
Proof:  For th e proof we refer to R am m   [128].
Theorems 3.3 and 3.4 show that the equation
QN{A-i-B)PNUN  =  QNf  (1.52)
 
EQUATION
Thi s chapter is devoted to presenting th e foundations of obstacle scattering
problems for tintie-harmonic acoustic waves. We begin with a brief discus
sion of the physical background of the scattering problem, and then we
will formulate the boundary-value problems for the Helmholtz equation.
We will synthetically recall the basic concepts as they were presented by
Colton and Kress [32], [35]. However, we decided to leave out some details
in the analysis. In this context we do not repeat the technical proof for the
jump relations and the regularity properties for single- and double-layer
potentials with continuous densities. Leaving aside these details, however,
we will present a theorem given by Lax [90] which enables us to extend the
jump relations from the case of continuous densities to square integrable
densities. We then establish some properties of surface potentials vanish
ing in sets of R^. These results play a significant role in our completeness
analy sis. Discussing th e Gree n repr esen tation th eorem s will enable us to
derive some esti m ate s of th e solutio ns. We will the n analyze th e general
null-field equ ation for th e exterior Dirichlet and Ne um ann problem s. In
particular, we will establish the existence and uniqueness of the solutions
and will prove the equivalence of the null-field equations with some bound
ary integral equations.
1 BOU NDA RYVA LUE PROBLEMS IN ACOUSTIC THEORY
Acoustic waves are associated only with local motions of the particles of the
fluid and not with bodily motion of the fluid  itself.  The field variables of
interest in a f luid are the particle velocity v ' = v '(x ,^), pressure p ' = p '( x, t) ,
mass density p ' =   p\x.^t)  and the specific entropy 5 ' = 5 '( x , t ) . To derive
the diff^erential equations describing acoustic fields we assume that each
of these variables undergoes small fluctuations about their mean values:
Vo = 0, P05   Po and  SQ.  Generally, quadratic terms in particle velocity,
pressure, density and entropy fluctuations are neglected and conservation
laws for m ass and m om en tum are linearized in ter m s of th e fluctuations
V = v(x,^) ,   p   = p(x , t ) ,  p   =  p{x^t)  and  S  = 5(x,t) . In this context the
motion is governed by the linearized Euler equation
dv  1
| ^ + P o V -v = 0 . (2.2 )
From th erm odina m ics we can write the pressure as a function of density and
entropy. If we assume that the acoustic wave propagation is an adiabatic
process at constant entropy and the changes in density are small , we have
the l inearized state equation
^ = g ; ^ ( P o , 5 o ) ^ . (2.3)
Defining the speed of acoustic waves via
= f^{p„So (2.4)
we see that the pressure satisfies the t ime-dependent wave equation
Taking the curl of the linearized Euler equation we get
V X  V = 0 (2.6)
and therefore we can take
V = — V [/ , (2.7)
where
 f/
  is
 a
 scalar field called th e velocity poten tial. We m ention t ha t th e
above equation
  is a
 direct consequence
  in
 (2.1) w e o bta in
and clearly  the velocity po ten tial also satisfies  the t ime-depen dent wave
equat ion
:^ ^ = ^u.  (2.9)
 of
 be
 transformed
  to
  the
Au-j-k^u^O,
  (2.11)
where  the wave num ber  k  is given  by the posit ive constant  k = u/c. If
we consider  the acoustic wave p ropag ation  in a  medium with damping
coefficients C» th en  the  wave number  is given by fc  = a; (a; +  jQ  /c^- We
choose the sign of k  such that  Im fc > 0.
Before
  we
 consider  the bound ary-value problems  for the Helmholtz
equation let us introduce some normed spaces which are relevant for acous
t ic scattering.  Let G
 be a
  closed subset
linear space of all continuous complex-valued functions defined on
 G.  C{G)
ll«llcx),G = S^P I ^ W I -
x € G
By  L^{G)  we den ote th e Hilb ert space of all squa re integrable functions
on G, i.e.
  \a\
G
L^{G)  is the comp letion of  C{G)  with respect  to the squ are-integral no rm
IHl2,G=(/N dG
induced by the scalar product
G
Th e H5lder space or the space of uniformly Holder continuous functions
C^' (G) is the linear space of all complex-valued functions defined on
  G
which are bounded and uniformly Holder continuous with exponent a. A
function a : G —> C is called un iformly Holder continu ous w ith H older
expon ent 0 < a < 1 if
K x ) - a ( y ) | < C | x - y r
for all X, y G G. Here, C is a positive constant depend ing on
 a
 but not on
x and y. The Holder space C^' (G) is a Banach space endowed with the
norm
l | a | U = s u p | a( x )H - s up ' . ^ ° y .
Going further, the Holder space C^'°'{G)  of uniformly Holder contin
uously differentiable functions is the space of all differentiable functions  a
for which Va (or the surface gradient V^a in the case G is a closed surface)
belongs to
 C^'^{G). C^'^{G)  is a Banach space equipped with the norm
IHIl ,a ,G = l l«lloo,G + l | V a |U G .
For vector fields the above definitions remain valid but instead of ab
solute values we take Euclidian norms.
Actually, G stands for the bounded set Dj, the unbounded set
  Dg —
R^ -
  Di
  Th ere are a few comm only considered
classes of surfaces, which are general enough to be a versatile and useful
tool in constructing physically relevant model surfaces, and which at the
same time are restricted enough to yield useful results, such as Lyapunov
surfaces (see, e.g. Smirnov [135]) and twice contin uou sly differentiable
surfaces (see, e.g. MuUer [114] and C olton an d Kress [32]). Let
  S
 Di
subset of R^ ). We say that the surface
  S
  if for each point
x G 5 there exists a neighborhood V of x such that the intersection
  V^
  U C'R?
  and this mapping
is twice continu ously differentiable. W e will express this property also by
saying that  Di  is of class G^.
In order to guarantee the validity of Green's theorems it is necessary to
define an additional linear space. For any domain G with boundary
  dG
  of
  BOUNDARY-VALUE PROBLEMS IN ACOUSTICS  21
class C^ we introd uce th e linear space 9?(G) of all complex-valued functions
a e  C^{G)nC{^)  which possesses a normal derivative on the boundary in
the sense that the l imit
| ^ ( x )  =  lim  n (x)  • V a (x -- / in (x))  .xedG,  (2.12)
a n /i-^o+
exists uniformly on  dG,  Here the unit norma l vector  to the boundary  dG
is directed into the exterior of G.
Next we will consider formulations of acoustic scattering problems for
penetrable and impenetrable objects . Let Di be a boun ded dom ain with
b ou nd a r y  S  and exterior  Ds-  In the scat ter ing of t ime-harm onic acoustic
waves by a sound-soft obstac le, the p ressure of th e to tal wave vanishes on
the boun dary. Th is leads to the direct acoustic obstacle scattering p roblem:
given   UQ as an entire solution  to the Helmholtz equation representing an
incident field, find the total field  u =
  Ug -{-  UQ satisfying th e Helm holtz
equat ion  in Dg  and the boundary condi t ion
u = OonS.
  2.13)
In addition, the scattered field sho ld satisfy the m erfeld radiati n
condition
X |
uniformly for all directions x/ | x | .
The direct acoustic scattering problem  is a  particular case of  the fol
lowing Dirichlet problem
E x t e r i o r D i r i c h l e t b o u n d a r y - v a l u e p r o b l e m .   Find  a function  Us €
C iDs)  nC{Ds) satisfying the Helmholtz equation
  in s,
Us==fonS,  (2.15)
  S.
The interior Dirichlet boundary-value problem has  a  similar formu
lat ion  but with  the  radiation condition excluded.  It is known th at this
boundary-value problem  is no t unique ly solvable.  If this problem has an
unique solution, we say that k  is not an eigenvalue of the interior Dirichlet
problem. The countable se t of posit ive w ave num bers fc for w hich t he in
terior Dirichlet problem  in A  admits nontrivial solutions or the spectrum
of eigenvalues o  th e interior Dirichlet problem  in Di will be denoted by
P ( A ) .
2 2 CHAPTER II THE SCALAR HELMHOLTZ  EQUATTON
In the above formulations we require  Us  to be continuous u p to t he
boundary. However, assuming / G  C{S)  me ans th at in general the norm al
derivative will not exists . Imposing some addit ional smoothness condition
on the bou nda ry da ta we can overcome this si tuatio n. Taking / G  C^ ^{S)
we guarantee that  Ug €  C^ ^{Ds) In particular, the normal derivative
dus/dn  belongs to C ^'^ (5 ), and is given by
^ ^ A f .  (2.16)
where  A  : C^ ' (5 )  -^   C^^ iS)  is the Dirichlet to Neumann map.
In the case of a sound-hard obstacle, the normal velocity of the acous
t ic wave vanishes on the bound ary. Th is leads to a Ne um ann bou nd ary
condition  du/dn  = 0 on 5 , where n is th e unit outw ard norm al to  S  and  u
is the total field. After renaming the unknown functions, we can formulate
the following Neumann problem.
E x t e r i o r N e u m a n n b o u n d a r y - v a l u e p r o b l e m   Find a function
Us G 5R(J?s)  satisfying the Helmholtz equation in Da, the Somm erfeld radi
ation condition a t infinity and the bounda ry condition
^=gonS,  (2.17)
where g is a given continuous function defined on S.
For the interior Neumann problem in  Dt , there also exists a countable
set  r]{Di) of positive wave numbers  k  for which nontrivial solutions occurs.
In the case when  g  G C^ ' ( 5 ) then  Ug G C^ ^iDg).  In particular, th e
boundary values Us on  S  are given by
Us =  Bg,  (2.18)
where  B  : C^ '^(5 ) - • C ^'^( 5) is the inverse of  A.  Note tha t  A  and  B  are
bounded operators .
I t is known tha t the exterior Dirichlet and Neum ann problem s (w ith
continuous boundary data) have an unique solution and the solution de
pends continuously on the boundary data with respect to uniform con
vergence of the solution in  Dg  and all its derivatives on closed subsets of
Ds.
For the scattering problems under examination the boundary values
are   as  smooth as the boundary since they are given by the restrict ion
of the analytic function  UQ t o  S.  In particular we will assume sufficient
smoothness conditions on the surface  S  such tha t t he sca ttered field  Ug
belongs to C^' (Z^s). The regularity analysis given by Colton and Kress
[35] shows that for domains  Di  of class  C^  we have
 Ug
The above boundary conditions are ideal boundary conditions, which
may not be reahzed in practice. But in many instances it may be possible
to relate the pressure on the surface at any location to the normal velocity
at the same location via a parameter, referred to as the acoustic impedance
7 .  This acoustic impedance is a particularly useful concept when we are
dealing with thin walls, screens, etc. on which acoustic waves are incident.
In these cases we are not interested in studying the details of the acous
tic field inside the thickness region. The interaction of such a surface with
acoustic waves is part icula rly simple and can be described by the boun dary
condition   u
  ^du/dn  = 0. M ore generally, we will consider the following
formulation of the exterior impedance boundary-value problem.
E x t e r i o r i m p e d a n c e b o u n d a r y - v a l u e p r o b l e m   Find a function
Us €  ^{Ds) satisfying the Helmholtz equation in Ds, the Somm erfeld radi
ation condition at infinity and the bounda ry condition
7 ^  = / o n 5 , (2.19)
where f and  7  are given continuous functions defined on S.
The exterior impedance boundary-value problem possesses a unique
solution provided Im(fc7) > 0.
We note here th at one can pose and solve the boun dary-value problems
for the boundary conditions in an L^-sense. The existence results are then
obtained under weaker regulari ty assumptions on the given boundary data.
On the other hand, the assumption on the boundary to be of class   C^  is
connected with the integral equation approach which is used to prove the
existence of solutio ns for sca tterin g proble m s. Ac tually it is possible to
allow Lyapunov boundaries instead of C^ boundaries and sti l l have com
pact operators. The si tuation changes considerable for Lipschitz domains.
Allowing such nonsmooth domains and ' rough' boundary data drastically
changes the nature of the problem since it affects the compactness of the
boundary integral operators. In fact , even proving the very boundedness
of these operators becomes a fundamentally harder problem. A basic idea,
going back to Rellich [130] is to use the quantitative version of some ap
propriate integral identities to overcome the lack of compactness of the
boundary integral operators on Lipschitz boundaries. For more details we
refer to Brown [17] and Dahlberg  et al.  [37].
2 SINGLE- AND DOUBLE-LAYER POTENTIALS
We briefly review the basic jump relations and regularity properties for
acoustic single- and double-layer potentials.
 
THE SCALAR HELMHOLTZ EQUATION
Let 5 be a surface of class  C^  and let  a be an integrable function. Th en
U a(x ) = y ^ a ( y ) p ( x , y , f c ) d 5 ( y ) , x
 G  R^ -  5 , (2.20)
s
are called the acoustic single-layer and acoustic double-layer potentials,
respectively. Th ey satisfy th e He lmh oltz equ ation in Di  and in  Dg  and
the Sommerfeld radiation condition. Here g is the Gre en function or t h e
fundamental solution defined by
ff(x,y,fc) = ^ ^ ^ ^ _ y ^ , x ^ . (2.22)
 a
||walL,R3 <Ca  | | a |U ,5 , 0 < a < 1. (2.23 )
For densities  a  G C^ ' (S) , 0 < a <  1, the first derivatives of the single-
layer potentia l Ua  can be uniformly extended in a Holder continuous fashion
from   Dg  into  Dg  and from  Di  into  Di  with boundary values
(Vt/a)± (x) =   ja(y)Vx^(x,y,fc)d5(y)  T| a ( x ) n ( x ) , x € 5 , (2.24)
s
where
{Vua)^  (x ) =  lim V u( x ± ftn(x)) (2.25)
in the sense of uniform convergence on S  and where the integral exists
as improper integral . The same regulari ty property holds for the double-
layer potential
 C ^ ^{S),  0 < a <  1. In add ition, th e
first derivatives of the double-layer potential
  Va
 with density  a    C^ ' ( 5 ) ,
0 < a <  1, can be uniformly Holder continuously extended from  Dg  into
Dg  and from  Di  into  Di.  The es t imates
IIV Ua lU .D . < C a | | a | U , s , ( 2 .2 6 )
\\VaL,D,  Ca  | | a | L , s ( 2 - 2 7 )
 
and
(2.28)
hold, where t stands for s and  i.  In all inequalities the constant  Ca depend
on  S and a.
For the single- and double-layer potentials with continuous density we
have the following jump relations:
(a)  lim
 =0,
= 0,
(2.29)
where x € 5 and the integrals exist as improper integrals. The single- and
double layer opera tors 5 and  /C, and the normal derivative operator K. will
be frequently used in the sequel. They are defined by
(5a) (x)  =  jaiy)gix,y,k)dS{y),xeS,
(2.30)
and
{IC a) (x) = I a{y) i^ dS{y),  x 6 5. (2.31)
The operators 5,  K  and  K are compact in   C{S)  and C^ °^{S)  for 0 <
a <1. 5, /C and /C' map
  C{S)
CHA PTER II THE SCALAR HELMHOLTZ EQUATION
Then  a ~ 0 on 5  (a vanishes almost everywhere on S).
Proof:
  From the jum p relations for the normal derivative of the single-
layer potential with square integrable density
lim
dug
•hn{.))-
1
2» +
/ « ( y )
d9i.,y,k)
an(.)
  a
  satisfies
0
(2.43)
almost everywhere on 5. Th e above integral equation is a Predholm in
tegral equation of the second kind. Th e operator in the left-hand side of
(2.43) is an elliptic pseudodifferential operator of order zero. According to
M ikhlin [101] we find tha t a - ao 6
  C{S).
  C^' (5). Using the regularity results for the derivative of
the single-layer potential we conclude that   Uao  belongs to  C^' (JDS) . NOW
the jump relations for the single-layer potential with continuous density
show that  Uao  solves the homogeneous exterior Dirichlet problem. There
fore
  Uao
  «« ao ^ 0 we
get a '^ 0 on 5. T he theorem is proved. We mention th at th e equivalence
a ^ ao € C^'^{S) can be obtained directly by using th e following regularity
result: if
  is an elliptic pseudodifferential operator of order zero then any
solution in C^'^{S)  of the inhomogeneous equation
  Aa
tional smoothness from / , so that /  G C'^^'^iS)  implies that
  a
where m > 0 and 0 < a < 1; in particular, if
  a
  nm>oC^''*(S') C C ^ ( 5 ) .
A similar result holds when  Ua  vanishes in the unbounded domain
  Dg.
T H E O R E M  2 . 3 :  Consider Di a bounded domain of class C^ with  bound
ary S and exterior Dg, Assume k  ^ p ( A )  Ol^d  let the single-layer poten tial
Ua with density a  G L^{S) satisfy
Ua
Proof:
  Rep eating the arguments of the previous theorem we see that
Uao y
problem. The assumption
can be completed as above.
For the double-layer potential we can state the following results.
T H E O R E M   2 .4 :  Consider Dt a bounded domain of class C^ with  bound
ary S. Let the double-layer potential  Va with d ensity a  G L^{S) satisfy
Va =0 in Di,  (2.45)
 
29
Then a^O on S.
Proof:  Th e ju m p relation for the double-layer poten tial with square
integrable density
= 0 (2.46)
0
(2.47)
almost everywhere on  S,  Using the same arguments as in theorem 2.2 we
obta in  a--ao e C{S).  Th en , since /C m aps  C{S)  into C^^ iS) and C^ ' (5)
into  C^ ^ iS)  we deduce tha t  ao E  C ^' (5) . Using the regulari ty results
for the derivative of the double-layer potential we see that  Vao  belongs to
C^ ^{Ds)^
  Th e ju m p relations for th e norm al derivative of th e double-layer
potential with continuous density shows that Vao solves the homogeneous
exterior Neumann problem, and therefore  Vao = 0 in Ds.  Finally, from
^ao+ ~   ^ao-  = ao = 0 we get a '^ 0 on 5.
T H E O R E M   2 . 5 :  Consider Di a bounded domain of class C^ with  bound
ary S and exterior Ds- Assume k  ^  n{Di) and let the double-layer poten tial
Va with density a  €  L ^{S) satisfy
Va
Then  a ^ 0 on 5.
Proof:  T he proof proceed s as in theo rem 2.4.
Next we will consider combinations of single- and double-layer poten
tials.
T H E O R E M   2 .6 :  Consider Di a bounded domain of class C^ with  bound
ary S. Let the combined potential Wa =   Ua —  Xva with density a   €  L ^{S)
and Im{Xk) > 0  satisfy
Wa = 0 in Di.
  a ~ 0
Proof:  T he idea of th e proof is du e to Ha hne r [71]. T he ju m p relations
for the surface potentials with square integrable densities gives
lim   \\wa{.+hn{,)) + Aa|(2 c
h—^ •
0+
lim
dwg
dn
0,
0.
(2.50)
2,S
Sh
 = { y / y =  X  -h /in (x ), X G  5 , /i > 0} .
Then, we have
\f\a\^dS
J
  K
  Let us now consider a spherical surface 5/? of radius
  R
  enclosing
  Di.
 DhR
-Im(A;)
s \
  SR
DR
where
^lim^
 dV.-
  (2.54)
DhR
  DR
^^ ' (2.53)
Then, taking into account the radiation condition (weak form)
31
im /  { ^ - jkwal dS  = lim / <
DR  )
(2.56)
Now, if Im(AA:) > 0 the conclusion a ~ 0 on iS readily follows. If Im(Afc) =
0 and Im(A:) > 0 we obtain
  W a
Im(A:) = 0 we get
  follows.
Application of the jump relations (2.50) finishes the proof of the theorem.
It is noted that the same strategy can be used for proving theorems
2.2 and 2.4. For instance, let
  Ua
a
Um ^{.+hni.)) +
 being a parallel exterior sur
face we find that
SH
(2.58)
follows. Consequently,
and the conclusion follows as in theorem 2.6.
T H E O R E M   2.7:  Consider Di a bounded domain of class C^ with  bound
ary S  and exterior Dg, Let the combined potential Wa  =^   Ua \ Afa   ^ith
density a
(2.61)
Then
  on
 S.
Proof:
  From the ju m p relations for the single- and double-layer po
tential we get
  D^h
  bound ed by
the parallel interior surface 5_/i,  S-h  = { y / y = x -  / i n ( x ) , x G 5, /i > 0} ,
letting /i —>  0-1-  and using
-A /  \a\^ dS = lim /  Wa dS,  (2.63)
J  /1-.0+  J an
  -f  \Vwaf) dV.  (2.64)
S  Di
Since Im(Afc) > 0 and Im(fc) > 0 we conclude th at a ~  0 on 5.
3 GREEN'S FORMULAS AN D SOL UTION ESTIMATES
A basic tool in studying the boundary-values problem for acoustic scatter
ing is provided by G ree n's formulas. Con sider  Di a bou nded doma in of
class
  C^
  with boundary S  and exterior £>«, and let n be the u nit norma l
vector to S  directed into D^. Let u € ^{Ds)  be a radia ting solution to the
Helmholtz equation in  Dg.  Then we have the Green formula
5
(2.65)
33
A similar result holds for solutions to th e Helmholtz equation in boun ded
domains. With u £  3?(I>t) standing for a solution to the H elmholtz equ ation
in
 Di
(2.66)
In the literature, Green's formulas are also known as the Helmholtz
representations.
Ne xt we will derive some estima tes for the solutions to th e Dirichlet and
Neumann problems. We begin with the exterior Dirichlet boundary-value
problem. Th e departure point is the associated boundary-value problem
for the Green function
  k^G'{x,y)
the boundary condition
and the radiation condition
1 ^ . V x G H x , y ) - j f c G n x , y ) =   o (J^A  as |x | -^ oo,
(2.69)
uniformly for all directions x / |x | . T he superscript ind icates that w e are
dealing with the G reen function satisfying the D irichlet boundary condition
on
  S,
 Ds
ll^«lloo,Gs = s u p |tx(y) | = sup
y € G s
where
 Gs
 Dg
  and
C
  = sup
i
I
d 5 ( x ) . (2.72)
In order to derive a similar estimate for the solution to the exterior
Neumann problem we consider the Green function G^ = G^(x,y), y e £)«,
satisfying the Helmholtz equation in
 Da,
0, X e 5 ,
  ^ ( x ) G 2 ( x , y ) d 5 ( x ) , y € £>,; (2.74)
thus the estimate
 Gg
 of
 Dg
Finally, for the exterior impedance boundary-value problem we con
sider the Green function  = G^ (x ,y ) ,  Dg^  satisfying the Helmholtz
equation in JD^, the boundary condition
G ( x , y ) - 7  Q^^^^  - 0 , x € 5 ,
(2.76)
and the radiation condition. As before, app lication of Green's theorem in
the domain
 Dg.
  (2.77)
  Gg
 of
 Dg
Ug  7
3.  GENERAL NULL-FIELD EQUATIONS IN ACOUSTICS 3 5
The above estimates show that small deviations in the boundary data
in the square-integ ral n orm ensure small deviations in the scattere d field U s
in the maximum norm on closed subsets of  Da.  In fact the solution of the
boundary-value problems for the Helmholtz equation in the framework of
the discrete sources method is based on the representation formulas (2.70),
(2.74) and (2.77).
4 GENERAL NULL-FIELD EQUATIONS IN ACOUSTIC THEORY
In this section we will discuss the general null-field equations for the Dirich-
let and Ne um ann bo undary-v alue problems. We will prove the unique
solvability of these equations and will show their equivalence with some
boundary integral equations.
Let  Us  G C^{Ds)  n  C{Ds)  be a radia ting solution to the H elmholtz
equation which possesses a normal derivative on the boundary. The repre
sentation formula in the region  Di  gives
/[
d5 (y ) = 0 , x € A . (2 .79)
Let no be an ent ire solution to the Helm holtz equatio n. Gre en's formula
for the incident field  UQ in t he region
  Di
  yields
s
dS(y),
/h
(2.81)
Let us consider the direct acoustic scattering problem with boundary
condition  u = Us  -\-  uo = 0 on S.  For this boundary-value problem we can
formulate the following general null-field equations.
G e n e r a l N u l l - F i e l d E q u a t i o n s f o r t h e D i r i c h l e t P r o b l e m .   Con
sider Di a bounded domain of class C^ with boundary S and unit normal
vector  n  directed into the exterior of Di. Given   UQ  as an entire so lution to
the Helm holtz equation find a surface field h satisfying the integral equation
l io(x)
  = 0, x € A , (2.82)
3 6 CHAPTER II THE SCALAR HELMHOLTZ EQUATION
or equivalently, find a surface field hg satisfying the integral e quation
dg{x,y,ky
J  U . ( y M x , y , f c ) - f - w o ( y ) -  g , .
s
dS{y)  = 0, k G Du  (2.83)
Once the surface fields  h  or  hs  have been determined the solution to
the boundary-value problem can be constructed as
us{x)  = - y  h{y)g{x,y,fc)d5(y), x G £>., (2.84)
s
^^ (x) = - /  \hs
5 n ( y )
for   hs  solving (2.83).
For the Neumann boundary condi t ion  du/dn  = 0 on 5 , th e general
null-field equations cah be formulated as follows.
Q ^ ii eM l N u l l - F i e l d E q u a t i b n s fo r N e u m a n n P r o b l e m .  Consider
Di a bounded domain of class C^ with boundary S and unit normal vector
n directed into the exterior of Di. Given   UQ  as an entire solution to the
Helm holtz equation find a surface field h satisfying the integral e quation
« o ( x ) +  j Hy)^^^ ^dS{y)  = 0, X € A , (2.86)
S
or equivalently, find a surface field hs satisfying the integral equation
n
where hg = h —  UQ.
The solut ion to the Neumann boundary-value problem can be wri t ten
as
  D,,
  (2.88)
for   h  solving the general null-field equation (2.86) and as
d S ( y ) ,  x e D . , (2.89)
^ x) = / U.
  solving the general null-field equation (2.87).
We note that the existence of solutions to the general null-field equa
tions is guaranteed by the existence of solutions to the Dirichlet and Neu
ma nn bou ndary-value problems. W hen th e bounda ry values are the res tr ic
tion of the analytic function
  UQ
  to  S  we see that  hg =  dug/dn  € C° ' (5 )
solves the null-field equation for the Dirichlet problem, while for the Neu
mann problem the solution is  hs   =^   Ug £ C^ ^{S).  In thi s case from th e
regularity results for the derivatives of single- and double-layer potentials
we see tha t
  Ug
  given by (2.85) and (2.89) belongs to  C^ ^{Ds),  Th e unique
ness of solutions follows from theorems 2.2 and 2.4.
The equivalence between the general null-field equations and some
boundary integral equations is given by the following theorem
T H E O R E M   4 . 8 :  Consider Di a bounded domain of  class C^  with  bound
ary S and unit normal vector  n  directed into the exterior of Di.
(a) Let h solve the general null-field equation for the Dirichlet problem
(2.82),
  Then, h solves the integral equation
The converse theorem is also valid provided that k  ^  rj{Di).
(b) Let h solve the general null-field equation for the Neumann problem
(2.86).   Then, h solves the integral equation
{\i->c)k =
uo .  (2.91)
The converse theorem is also valid provided that k   ^  p{Di).
Proof:
  Le t us define th e fields
i^(x) = ixo(x) - / / i (y )^ (x ,y , fc )d 5( y) , x € A ,  h  € C «' (5 ), (2.92)
s
for the Dirichlet problem, and
tx(x) = txo(x) + y ft(y)M^lMdS(y), X € A , ft G C i' « ( 5 ) , (2.93)
for the Neumann problem. Lett ing x approach   S  along a normal direction
we obtain the direct imphcation of the theorem.
 
For proving the converse theorem we debut by showing that
  \i h E
 h
 h e C{S),
that is /i G
operators /C' :
 L^{S)
the equation (^/ + /C')
Fredholm alternative that
C ( 5 ) —•
 C{S)
  is also compact we may employ th e Fredholm alternative in
the dual system  {C{S),  ^ ^( 5) ) and deduce the existence of some ho G  C{S)
such tha t (^ / + /C') ho = / . Con sequen tly, from
  h = ho + {h--
  /C'), we see that
  h e
  du/dn
  We note that the idea to employ
the Fredholm alternative in two different dual sy stem s for investigating the
smoothness of a solution if the right side of the equation has a certain
smoothness is due to Hahner [71]. In the case of the Neumann problem we
may employ the same arguments to show that if /i G  L'^{S)  is a solution
of (2.91) then
  given by (2.93) belongs to
C^' (jDi) and (2.91) gives u = 0 in A provided tha t
  k i p{Di).
5 NOTES AND COMMENTS
M artin [99] showed th at th e null-field equation s (2.82) and (2.86), are equiv
alent with some integral equations of the second kind, which possess an
unique solut ion for all frequencies. Th ese integral equa tions are similar to
(2.90) and (2.91), but they contain a new symmetric fundamental solution
gi
 y,k).
 (x , y,A;) differs from p(x, y,fc) by
a finite linear combination of produ cts of radiating spherical waves. Th is
equivalence allows Martin to conclude the unique solvability of the null-
field equa tions. An approach similar to that given in Section 4 was taken
by Colton and Kress [33].
 
t ion using radiating solutions
 to
  In
 distr ibut ed radia ting spherical wave func
t ions .  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
 on
  In
 we
will analyze the completeness of distr ibut ed plane waves. T he last section
of this chapter deals with the l inear independence of these systems.
1 COMPLETE SYSTEMS OF FUNCTIONS
T he com pleteness p rop ertie s of th e sets of localized spherical wave functions
and point sources have been studied exhaustively
  by
 means
 of
  different
represe ntations th eorem s. In this cha pter we will present these basic results
but our main concern is to enlarge th e class of com plete system s.
1.1 Localized spherical wave functions
We begin
  the spherical
wave functions  in L'^{S).  Th ese functions form a set of characterist ic so
lutions to the scalar wave equation  in spherical c oor dina tes an d are given
by
ul;,lM   = zi^^{kr)PJr^{cose)  e^'^^, n = 0 , 1 , . . . , m = - n , . . . , n .  (3.1)
Here, (r, 0,
 (p)
 are th e spherical coord inates 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 polynom ials. No te th at
ulnn  is an entire solution to the H elmholtz equ ation an d u ^ is a radia t ing
solution to the Helmh oltz equation  in R^ —  { 0 } .
The expansion of th e G reen function  in t e rms of spherical wave func
tions will frequently used in the sequel. It is
,A.  L f  ^-mn(y)^mnW,  IYI > |x|
^(x,y,A:)
where the normalization constant  Vmn  is given by
_ 2 n - f l ( n - | m | )
^ " ^ " " "  4  ( n - f H ) '  ^ ^
 
1. COMPLETE SYSTEMS OF FUNCTIONS 4 1
T H E O R E M  1,1:  Let S  be a closed surface  of class C*   and let n denote
the unit outward norma l to 5 .  Then each of
  the
  systems
| t / ^ n ~ A ^ ^ , n = 0, l,...,m  = ~n ,. .. ,n / Im(AA:) > o | ,
(b) {wj ni n = 0 , l , . . . ,m = -n , . . . ,n / f c ^ p (D i) } ,
I ^ I S ^ '
{<
-h A - ^ , n =: 0 ,1 , . . . , m = -n,..., n/ Im(Afc) > 0
is  complete in L^{S)  .
Proof:  For proving th e first par t of (a) it suffices to show the closeness
of the system
in L^{S),  Let
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 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'
  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» ^ =  m = - n , . . . , n }
is not complete in L^ (5). The completeness can be preserved if a finite set
of functions represen ting a bas is of iV (^ /  —  /C') is added to the original
 
4 2 CHAPTER III SYSTEMS OF FUNCTIONS IN ACOUSTICS
Th e null-space of th e ope rator ^ J  —/C'  corresponds to solutions to the
homogeneous interior D irichlet problem , th at me ans iV (^ J  —  /C') = V,
where V stands for the linear space
V = i | ^  /ve^{Di),Av-^k^v = OmD uv = Oons\  .
In addit ion
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 ba sis for  AT  ( ^ I  —  /C) and  Vj  s tands
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') .
Fur therm ore, the ma tr ix T ^ = M^^L  Tj^j = (Xfc»<5j),  /c, j =  l,...,mD,  is
nonsingular. C oming to the proof we observe th at the closeness relations
Jg a'uln^
  = — n,... , n, leads to the vanishing of th e
single-layer potential
  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,  G Q  =  Yljl^i^jXj-  Now, condit ions / ^ a o X j d 5 = 0 , j =
1, . . . , m£), gives ag = 0 on 5, wh ence a  ~ 0 on 5 follows. On the other h an d
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
^otkiXky^j)  = 0 , j = l , . . . , m D . (3.5)
fc=i
Using the fact that the matrix  To  is nonsin gular we obtai n a/c = 0 , fc =
1, . . . , m£), when ce  a  ~ 0 on 5 follows.
The same strategy can be used to preserve the completeness of the
system
an
when  k e r]{Di).  In this case a finite set of functions representing a basis for
iV ( ^ / -h /C) should be added t o the original system . N ote tha t analogously
to th e interior Dirichlet problem if {0 j} J is a basis for iV (^ I- |- / C ')
and
  Uj
 
4 4 CHAPTER III  SYSTEMS OF FUNCTIONS IN ACOUSTICS
(b)  Let Ua be the single-layer pote ntia l wi th density a and let us con sider
th e set of  conditions
(£iXa) (xn) = 0,  n = l , 2 , . . . , (3.1 0)
which provide that  Ua = 0 in Df.  Here C is  some operator whose
significance will be clarified la tt er .
Let us define  the  scalar functions  fn  by  sett ing
/ 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 (3.1 2)
s
we see t h a t  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
suppor t  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  e can use as  su pp o r t  a  point ,  a  curve,  a  surface,
etc.
Let  E  consists  of the  point  O  which coincides with  the  origin of a
Cartes ian coordinate system. Then,  the  corresponding complete system is
the system
 will
assume that O is the center of a sphere S^  enclosed in Di and will deno te 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
 given
 by
n = | m j
 
1. COMPLETE SYSTEMS OF FUNCTIONS  45
Let us form ulate th e vanishing cond itions in term s of the F ourier coefficients
l i m - 5 - i - l r i  = 0  for all m G Z , / >  \m \  (3.15)
^-*o  {kry
and any  6 G [0, TT]  . No te, th at similar conditions can be provided by impos
ing that  Ua vanish at the point x = 0 tog ethe r with all derivatives. Ne xt,
we will prove that the set of vanishing conditions (3.15) implies
/Sr" = 0  for all m G Z a nd / >  |m |. (3.16)
F ix m and construct
^  = t
  ^n'i^Pl- i^^os ).  (3.17)
For / =  |m| we pass to the l imit w hen r —» 0 and use th e asym ptotic 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 . 1 9 )
Since the last relation  is valid for an y 0 £  [0,7r], it  follows that  0 ^^ = 0.
Taking / =  |m | + 1 we arrive at /3|^j_j.i  = 0 and t he sam e technique can be
used to conclude. Th us, condition (3.15) yields t i ^ = 0 in E fi Z)[ for all
m €  Z; whence ita = 0 in Di follows. T h e converse resu lt is imm ediate.
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 at tent ion 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>[ an d choose a sequence of points
  {zn) C F^. The support
of discrete sources  is depicted  in Fig ure 3.1 . Before  we s ta te  our next
results, let us prove some auxil iary lemmas.
LEMMA  1 . 1 :  Let u G 5ft(Di)  he a solution  to the H elmholtz equation in
Di and let u^  be its Fourier coefficients with respec t to (f. Then, the limit
limw^(r;)/(M''"^^^Z, (3.20)
4 2
F IG U R E 3 .1 Illustration of the support of discrete sources.
exists an d 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
oo
 d5(y).
s
Let us express the associated Legendre polynomials in terms of the hyper-
geometric function  F:
^ ^  (n- |m |) |m| r l^l
T (X \  I I . I I . 1 ~ C O S ^ \
cF f |m | - n, |m | + n -f
  1,  \m\  -f 1,  1  .
(3.22)
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-C r/)/ (M '* "' =  v^{z)  = f ; a - ^ M , (3.23)
'' n= |m | ( '^^ )
  COMPLET E SYSTEMS OF FUNCTIONS  47
where a * =  2 ^^^^-^—,  .M.  . . /^r-  Now, accounting of the  series repre-
"" (n - ~ |m | ) |m | "
senta t ion of the  spherical Bessel function  we see t h a t  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
 0 in Di.
Proof:  The boundedness of the sequence {zn) implies the existence of
a convergent subsequence  {zn^)
  f = 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 ) [ , i.e.
y  a^^^^^  = 0, for  zeOzn  D [ . (3.24)
Using the  technique previously described we arrive at a^  = 0 for all m € Z
and  n > |m|.  Th u s ,  u^  = 0 in E fl -D[ for all m G Z;  whence by the
analyticity of u the  conclusion readily follows.
W e pay now a t ten t ion to the system of dis trib ute d spherical wave func
tions which form  a set of  radiating solutions  to the  Helmholtz equation.
T he y are defined by
d n W  =  < | m | ( x ~ ^ n e 3 )
(3.25)
  G
 Z,
 6nr
  )  are the  spherical coord inates of