Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation

Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation Acoustic and electromagnetic scattering analysis using discrete sources II the scalar helmholtz equation Acoustic and electromagnetic scattering analysis using discrete sources II the scalar 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 analysis Discussing the Green representation theorems will enable us to derive some estimates of the solutions We will then analyze the general null-field equation for the exterior Dirichlet and Neumann problems 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

17

1 8 CHAPTER II THE SCALAR HELMHOLTZ EQUATION

1 BOUNDARYVALUE 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 fluid 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 mass and momentum are linearized in terms of the 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

^ + - V p = 0, (2.1) and the linearized equation of continuity

| ^ + P o V - v = 0 (2.2) From thermodinamics 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 linearized state equation

^ = g ; ^ ( P o , 5 o ) ^ (2.3) Defining the speed of acoustic waves via

c^ = f^{p„So) (2.4)

we see that the pressure satisfies the time-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)

Po

1 BOUNDARY-VALUE PROBLEMS IN ACOUSTICS 1 9

where f/ is a scalar field called the velocity potential We mention that the

above equation is a direct consequence of the assumption of a nonviscous

fluid Further, substituting (2.7) in (2.1) we obtain

and clearly the velocity potential also satisfies the time-dependent wave

equation

:^-^ = ^u (2.9)

For time-harmonic acoustic waves of the form

U{x,t) = Re {i/(x)e-^^*} (2.10)

with frequency a; > 0, we deduce that (2.9) can be transformed to the

well-known reduced wave equation or Helmholtz equation

Au-j-k^u^O, (2.11)

where the wave number k is given by the positive constant k = u/c If

we consider the acoustic wave propagation in a medium with damping

coefficients C» then 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 boundary-value problems for the Helmholtz

equation let us introduce some normed spaces which are relevant for

acous-tic scattering Let G be a closed subset of R^ By C{G) we denote the

linear space of all continuous complex-valued functions defined on G C{G)

is a Banach space equipped with the supremum norm

20 CHAPTER II THE SCALAR HELMHOLTZ EQUATION

induced by the scalar product

G

The 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 uniformly Holder continuous with Holder exponent 0 < a < 1 if

K x ) - a ( y ) | < C | x - y r

for all X, y G G Here, C is a positive constant depending on a but not on

x and y The Holder space C^'"(G) is a Banach space endowed with the norm

l | a | U =sup |a(x)H- sup '"^.^^ ° y

Going further, the Holder space C^'°'{G) of uniformly Holder uously differentiable functions is the space of all differentiable functions a

contin-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

surfaces (see, e.g MuUer [114] and Colton and Kress [32]) Let S be the boundary of a bounded domain Di C R^ (a bounded, open and connected subset of R^) We say that the surface S is of class C^ if for each point

x G 5 there exists a neighborhood V^ of x such that the intersection V^ fl 5 can be mapped bijectively onto some domain U C'R? and this mapping

is twice continuously differentiable We 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

1 BOUNDARY-VALUE PROBLEMS IN ACOUSTICS 2 1

class C^ we introduce the 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 limit

| ^ ( x ) = lim n (x) • Va (x /in (x)) xedG, (2.12)

a n /i-^o+

exists uniformly on dG, Here the unit normal 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 bounded domain with

boundary S and exterior Ds- In the scattering of time-harmonic acoustic

waves by a sound-soft obstacle, the pressure of the total wave vanishes on

the boundary This leads to the direct acoustic obstacle scattering problem:

given UQ as an entire solution to the Helmholtz equation representing an

incident field, find the total field u = Ug -{- UQ satisfying the Helmholtz

equation in Dg and the boundary condition

u = OonS (2.13)

In addition, the scattered field should satisfy the Sommerfeld radiation

condition

" V „ - , ^ o ( ± ) » | x H o o , (2.14, X|

uniformly for all directions x / | x |

The direct acoustic scattering problem is a particular case of the

fol-lowing Dirichlet problem

Exterior Dirichlet boundary-value problem Find a function Us €

C^iDs) nC{Ds) satisfying the Helmholtz equation in Ds, the Sommerfeld

radiation condition at infinity and the boundary condition

Us==fonS, (2.15)

where f is a given continuous function defined on S

The interior Dirichlet boundary-value problem has a similar

formu-lation but with the radiation condition excluded It is known that this

boundary-value problem is not uniquely solvable If this problem has an

unique solution, we say that k is not an eigenvalue of the interior Dirichlet

problem The countable set of positive wave numbers fc for which the

in-terior Dirichlet problem in A admits nontrivial solutions or the spectrum

of eigenvalues of the 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 up to the

boundary However, assuming / G C{S) means that in general the normal

derivative will not exists Imposing some additional smoothness condition

on the boundary data we can overcome this situation 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-tic wave vanishes on the boundary This leads to a Neumann boundary

condition du/dn = 0 on 5, where n is the unit outward normal to S and u

is the total field After renaming the unknown functions, we can formulate

the following Neumann problem

Exterior N e u m a n n boundary-value problem Find a function

Us G 5R(J?s) satisfying the Helmholtz equation in Da, the Sommerfeld

radi-ation condition at infinity and the boundary 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, the

boundary values Us on S are given by

Us = Bg, (2.18) where B : C^'^(5) - • C^'^(5) is the inverse of A Note that A and B are

bounded operators

It is known that the exterior Dirichlet and Neumann problems (with

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 restriction

of the analytic function UQ to S In particular we will assume sufficient

smoothness conditions on the surface S such that the scattered 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 G C^'^{Dg)

2 SINGLE- AND DOUBLE-LAYER POTENTIALS 2 3

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 particularly simple and can be described by the boundary

condition u — ^du/dn = 0 More generally, we will consider the following

formulation of the exterior impedance boundary-value problem

Exterior impedance boundary-value problem Find a function

Us € ^{Ds) satisfying the Helmholtz equation in Ds, the Sommerfeld

radi-ation condition at infinity and the boundary 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 that one can pose and solve the boundary-value problems

for the boundary conditions in an L^-sense The existence results are then

obtained under weaker regularity 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 solutions for scattering problems Actually it is possible to

allow Lyapunov boundaries instead of C^ boundaries and still have

com-pact operators The situation 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

24 CHAPTER II THE SCALAR HELMHOLTZ EQUATION

Let 5 be a surface of class C^ and let a be an integrable function Then

are called the acoustic single-layer and acoustic double-layer potentials,

respectively They satisfy the Helmholtz equation in Di and in Dg and

the Sommerfeld radiation condition Here g is the Green function or the

fundamental solution defined by

ff(x,y,fc) = ^ ^ ^ ^ _ y ^ , x ^ y (2.22)

The single-layer potential with continuous density a is uniformly Holder

continuous throughout R^ and

||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 potential 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 Vu(x ± ftn(x)) (2.25)

in the sense of uniform convergence on S and where the integral exists

as improper integral The same regularity property holds for the

double-layer potential Va with density a £ C^'^{S), 0 < a < 1 In addition, the

first derivatives of the double-layer potential Va with density a 6 C^'"(5),

0 < a < 1, can be uniformly Holder continuously extended from Dg into

Dg and from Di into Di The estimates

IIVUalU.D < C a | | a | U , s , (2.26)

\\VaL,D, < Ca ||a|L,s (2-27)

2 SINGLE- AND DOUBLE-LAYER POTENTIALS 25

and

\\^VaL,Dr^Ca\\a\\,^^^S (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 [ua (x ± hn{x)) - Ua{x)] = 0,

where x € 5 and the integrals exist as improper integrals The single- and

double layer operators 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,

s {Ka){x) = |o(y)^^^l^d5(y),xe5,

(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) into C^^'^iS), and 5 and K map C^^'^iS)

26 CHAPTER II THE SCALAR HELMHOLTZ EQUATION

into C^'^{S) We note that S is self adjoint and /C and /C' are adjoint with respect to the L^ biUnear form

two bounded linear operators such that T and T^ are adjoint to each other

with respect to the scalar products of H\ and if2- Then T can be extended

to a bounded operator from if 1 into ii2 and

\\n C(Hi,H2) < \\n C{Xt,Xj) r^ii C(X2,Xi) (2.35)

We are now in position to formulate the following jump properties for the single- and double-layer potentials in terms of L^-continuitỵ

THEOREM 2.1: Let S be a closed surface of class C^ and let n denotes

the unit outward normal to S Then for square integrable densities the behavior of surface potentials at the boundary is described by the following jump relations :

(a) lim ||uặ±/in(.))-tiặ)ll2,5 = 0'

2 SINGLE- AND DOUBLE-LAYER POTENTIALS 2 7

Proof: The proof of the theorem was given by Kersten [79] We

outhne the proof for easy reference For a continuous density ao we rewrite

the jump relations in compact form as

lim ||Thao|U,5 = 0, (2.37)

where (7^)o</i<ho » -^ ' ^C*^) "^ ^('5')) is a family of bounded linear

opera-tors The adjoint operators T^ can formally be obtained by interchanging

the arguments in the kernel of the corresponding integral representations

Direct calculations show that

Prom (2.37) it follows that for each ao there exist a constant c, depending

on ao, such that

r h « o l L , s < c (2.39)

for all h with 0 < h < ho Using the uniform boundedness theorem we find

that Th are uniformly bounded with respect to the H-H^^^-norm Similar

arguments hold for the adjoint operators TJj Hence, from Lax's theorem

we deduce that Th are uniformly bounded with respect to the ||.||2 5-norm,

i.e for each a € L^{S) there exists M > 0 such that HT^aJIg 5 < M ||a||2^

for all h with 0 < h < ho- Now, let a € 1/^(5), and choose e > 0 Since

C{S) is dense in L'^{S) we find ao € C{S) such that \\a — aoHa 5 ^ ^- ^^^

us choose /IQ such that | | ^ o o | | ^ 5 < e for all h with 0 < h < HQ, Then,

< C||T^aom5-hM||a-ao||2,5 (2.40)

< (C -f M)£

where the constants C and M do not depend on e and /IQ Consequently,

lim;i_^o+ 11^^112 5 = 0 and the theorem is proved

The analysis of the completeness of different systems of functions in

L^(S) relies on the results of following theorems

T H E O R E M 2.2: Consider Di a bounded domain of class C^ with

bound-ary S Let the single-layer potential Ua with density a € L'^{S) satisfy

Ua = 0 in Di, (2.41)