Adrian Doicu, Yuri a. Eremin, Thomas Wriedt-Acoustic and Electromagnetic Scattering Analysis Using...
316
PREF CE M athematical modelling of the boundary-value problems associ 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. Many 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 tion, and geometrical optics methods can be employed for objects that are electrically large. Objects whose size is in the order of the 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 the solution of the scattering problem. Classical methods of solution in the resonance region such as the finite-difference method, finite-element method or integral equation method, owing to their universality, lead to computational algorithms that are expensive in computer resources. This significantly restricts their use in studying multiparametric boundary-value IX
Adrian Doicu, Yuri a. Eremin, Thomas Wriedt-Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources (2000)
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