Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems

✍ Scribed by Jean-Claude Nedelec

• Publisher: Springer
• Year: 2001
• Tongue: English
• Leaves: 329
• Series: Applied Mathematical Sciences, Volume 144
• Category: Library

✦ Table of Contents

Volume 144
Springer Science+Business Media, LLC
Applied Mathematical Sciences
1.1 Introduction
1.2 Physical Background
1.2.1 The acoustic equation
1.2.2 The Maxwell equations
The case of an axis invariance
1.2.3 Elastic waves
The P waves or pressure waves
The S waves or shearing waves
2.1 Introduction
+ W = °,
2.2 Harmonic Solutions
2.3 Fundamental Solutions
The unique radial outgoing fundamental solution of equation (2.3.1), if n = 2, is
2.4 The Case of the Sphere in ]R3
24.1 Spherical harmonics
if I is odd
Proof
Proof
(r27r) = w (2.4.30)
2.4-2 Legendre polynomials
Proof
Proof
Proof
Proof
2.4-3 Associated Legendre functions
Proof
Proof
Proof
Proof
Proof
Proof
Proof
2-4-4 Vectorial spherical harmonics
Proof
Proof
2.5 The Laplace Equation in ]R
2.5.1 The sphere
Definition
Definition
Proof
Proof
Definition
2.5.2 Surfaces and Sobolev spaces
Definition
Notation
Proof
Remark
Proof
Proof
Proof
Remarks
11^1 \h™(R3) < Cl\u\I, /2 5
Proof
Proof
Proof
JBrR 0 + R )
Proof
Remark
2.5.3 Interior problems: Variational formulations
Proof
Proof
Proof
Proof
Proof
Remark
General remark
2.5.4 Exterior problems
<y47“
Proof
Proof
Proof
Remark
Proof
Remarks
Proof
Remark
2.5.5 Regularity properties of solutions in Rn
Proof
Proof
2.5.6 Elementary differential geometry
|y-P(y)l = W
Notation
Proof
Proof
Proof
Proof
2.5.7 Regularity properties
Proof
Proof
+ Vr"w’’
Remarks
2.6 The Helmholtz Equation in IR3
2.6.1 The spherical Bessel functions
Proof
Proof
2.6.2 Dirichlet and Neumann problems for a sphere
00 1
Proof
] \VsYem\2da = W+l).
du
Proof
1 = gj(fe) < l^|2 k2 + \pe(k)\2 ~
2.6.3 The capacity operator T
2.6.4 The case of a plane wave
eik cosO = ^£ + T/2je(k)Ye°(e). (2.6.100)
Proof
Ve^ =~W+T)/_1
ve(k) = 2je(k).
Proof
Proof
°° , -Trje(k)h^\k)
2.6.5 The exterior problem for the Helmholtz equation
Proof
Remarks
Proof
Proof
Proof
Remark
End of Theorem 2.6.6
Proof
Comments
3.1 Integral Representations
Proof
Proof
Vu(y) = J VyE(x-y)q(x)dy(x).
+ - ifc-
((yo-z) ■ny„)q(x)dy(x)
pq(x)d^y(x).
Proof
3.2 Integral Equations for Helmholtz Problems
3.2.1 Equations for the single layer potential
3.2.2 Equations for the double layer potential
Jr
3.2.3 The spherical case
3.2.4 The far field
\y\J
(£ - m)!\
(si)r” '
Proof
Proof
Proof
3.2.5 The physical optics approximation for the sphere
e=o kh\ '(fc)
Lemma 3.2.3 It holds that
eikxY(x) = ^(i)e+1(£ + l/2)ve(k)Pe(x), e=o
Proof
Proof
Proof
3.3 Integral Equations for the Laplace Problem
i = uM'
Proof
Remark
G F1/2(r)/R.
> « IMlL/2(r)/R ; a > 0; Vy? e H1/2(r)/R.
Proof
/ dv t t Ou
p^p-d/y = (g - h)dx = pp^dl, (3.3.30)
Comments
3.4 Variational Formulations for
the Helmholtz Problems
3.4-1 The operator S
Proof
3.4-2 Fredholm operators
3.4-3 The operator N
Proof
Proof
Proof of Theorem 3.4.2 continued
Proof
Remark
3-4-4 Formulation with the far field
Proof
Proof
4.1 The Hilbert Transform
Proof
Definition
Proof
Proof
Proof
Definition
Proof
Proof of Theorem 4.1.2
Proof
Proof
Definition
P Q
Proof
4.2 Singular Integral Operators in IRn
9(y) = K(y,y-x)f(x)dx,
JR"
4.2.1 Odd kernels
Il#llLP(Rn ) — ll/lllJ’(Rn) • (4.2.9)
Proof
\9e II )
4-2.2 The M. Riesz transforms
Proof
L^M‘,2dx=cL^e~'('^ (42-26)
Proof
(JCe of(z) = [ [ Ke(z,z -y)1liS(y - x)f(x)dxdy. (4.2.32) Jr" Jr"
L£(z, w) - L^z, w) = K(z,w - t) dt. (4.2.36)
7n \W + 0(t — W)\
e<\w-t\<7)<^, (4.2.38)
Proof
2Mn+1) \Js^' J
I |l£(z, w) - Le(z, w)\dw < Cq [ \K(z,w)\q dw'. (4.2.63)
&(z) = [ Ke(z,y - z)f(y)dy (4.2.64)
9e(z) = ( Ke(z,y — z)(f(y) - f(z)h(y - z))dy. (4.2.65) Jr"
\9e(z)\<C.
y-z\ \y-z)
4.2.3 Adjoint operators
JR"
Proof
Proof
4.3 Application to Integral Equations
4-3.1 Introduction
4-3.2 Homogeneous kernels
Definition
Example 4. 1
Example 4. 2
Proof
< pi = atp.
9 = t^9^
gt= y K{x,y - x)tpi(x)d'y(x).
9i = y 0i(y)K(x,y - x)tpi(x)dy(x) (4.3.16)
9i~gz = J\0i(y)-l)K(x,y-x)pi(x)dy(x). (4.3.17)
JR"-1
dmh \
dy™'/
= x DyW ■
d
L (x, z) = L(x, z) 4- —L(x, z)(z - z) 4 F
oz
4-3.3 Pseudo-homogeneous kernels
Definition
Example 4. 5
Example 4. 6
Jk\z\
Example 4. 7
g (eik\x-y\ dnx |ar - t/|
Example 4. 8
Example 4. 9
4.4 Application to Integral Equations
Proof
Comments
5.1 Introduction
5.2 Fundamental Solution and
Radiation Conditions
Proof
Remark
Proof
Remarks
5.3 Multipole Solutions
5.3.1 Multipoles
and the transverse magnetic multipoles given by
Proof
=e r
Proof
Remarks
Proof
Proof
2 , z-oo + k+#r)l '
+ 4-? KI + n J J—4> ~dr
! K —dr
1 । 1 । ?2V<i । (2^+1)2
[ l^£dx<g £ £(£ + i)L(MKI2
Proof
5.3.2 The capacity operator
Definition
£=0 m=-£
Proof
3ft I (E Et ’ Et) dcr = E E e=om=-e^
Proof
Proof
5.4 Exterior Problems
5.4.1 Trace and lifting associated with the space H(curl)
Remark
Proof
Proof
Proof
Remark
Proof
Proof
Proof
5.4.2 Variational formulations for the perfect conductor problem
Proof
Proof
Proof
f a|IMv -cIIMh < 3[a(u£,u£)]
I < ll^llv Il'MIy ’
IIMv <c[||u£||H + llfflly.]. (5.4.87)
Remark
Proof
Proof
End of the proof of Theorem 5.4.6
Proof
Proof
* dq i r
\ dn
1 r)F
5.4- 3 Coupled variational formulations for impedance conditions
Proof
Proof
Remarks
Proof
The new coupled variational formulation is:
Proof
Remarks
5.5 Integral Representations
We denote by G(r) the outgoing fundamental solution
The interior value of E A n is
Proof
^(y) = — [ G(x-y)dwrj(x)d-y(x). (5.5.22)
< = k2vv -k2VV + k2E - curl curl E (5.5.23)
= -iaifijSr.
A(y) = iwny G(x - y)j(x)drf(x).
E(y) = iwy, G(x - y) j(x) d-y(x)
H V G(x - y) divp j(x)dy(xY
H(y) = curl j G(x - y)j(x)dr/(x). (5.5.25)
H(y) = —iwe G(x — y) m(x)d^(x)
d
( curly(G(x - y)j(x)) /\ny = (yyG(x - y) A j(x)) A ny
I = $n~(x ~ ~ VyG(x - y) (j(x) • (ny - nx)).
= ((ny - nx) AV yG{x - y)) p(x)dy(x)
< Jr
= y ((ny-nx)/\/yG(x-y))p(x)d'y(x)
- y G(x - y)cuArp{x)d‘y(x),
^ny • curl y G(x — y)m(x)d/y(x)^
= y (ny-(VyG(i-y)hm(i)))di(x)
+ y ((VxG(x - y) A nx) ■ m^x))d'y(x)
5.6 Integral Equations
5.6.1 The perfect conductor
Proof
Proof
Comments
ir G^x ~
- k2 j j G(x - y) ■ \7rq\y)) d'y(x)d'y(y)
~k2 / / ~ ' Vr^(y)) d'y(x')d'y(y) (5-6-25)
= iive / divr E™cqtd'y,
5.6.2 The zero frequency limit
5.6.3 The dielectric case
f ji = Hihn,
me = —Ee A n.
x - yjjiWd^x)
+ uTiV Gi(x~ y) divr Mx)d~/(x)
Hi(y) =
x - y)mi(x)d'y(x)
x — y) divr mi(x)d-'/(x)
f Gi(x - y)ji(x)d'y(x
- VyGi(x - y) (ji(x) ■ (ny - nJ)
-y) (me(x) hny^d'ytx')
= iujyi y y Gi(x - y) (ji(x) ■ j\y)) d^(x)d^(y)
+ (curir^GiCr-y) ■ j^y^m^x) ■ (ny-nx)) d'y(x)d'y(y);
= -iu£i y y Gi(x-y) ■ m^y)) d'y(x)d'y(y')
+ ’dy~i y j Gi(x-y)divr rrii(x) divr mt(y)dy(x)d~f(> d'i(x')d'y(y')
(j/)) dy/(x)dy/(y)
Jr
Proof
Remarks
5.6.4 The infinite conductivity limit: The perfect conductor
Proof
Proof
1 f d (e~k\x~y\
d2 dnxdny
,2 r r e-k\x-y\
,2 r r e-k\x-y\
+ Itt JrJr (nx ' nv)
Proof
t?e-M I - 91 <! _ 9| > iqrn.
D(1 Df:- y -x
j /r G^x~y^lfii^j^d^^d'y^
j Jr G^x~y^<pi^j^d'y^d^
i JrJr W()
dqd/y.
Proof
+ J j (yyGe(x - y) ■ ((E,nc An(i)) A d'ydy,
+ J^ y (yyGe(x - y) • ((tf,nc An(x)) Am(y))) d'yd'y.
Proof
aa
+ c<7 ||curirv||TH_1/2(r) l|Vrw||TH-i/2(r) A:i
+ l|Vrw||TH_3/2(r)
I / (yyGt(x-y) ■ A j(?/))) d-y(x)dy/(y)
~u'hJr]rGe(x~y') divr Jo(z) divr jt(y)d'y(x)dy(y)
Comment
5.7 The Far Field
5.1.1 Far field and scattering amplitude
1 eik\y\ lz/|
+ ik [ e~ik(x • y)/^l A j(x)) d-y(x) Jr \M / .
Proof
= -J
JsR
13/1
Proof
—ikRcos 9
A(0,<p;y,0)
= (7 • A(0,tp-,y, (3))
= (l+cos0) (7 • A(6,<p:,y,0))+cos0cos(p(z • A(0,(p;y, 0)),
<
x [ r e~ikRcose^^de + \e~ikRvW ~ eikR^-
Proof
\ 11 /
Proof
Proof
5.1.2 Integral equations and far field
lseik{y'z}
Proof