H. T. Banks and J. K.Raye
Center for Research inScientic Computation
and
Departmentof Mathematics
North CarolinaState University
Raleigh, NC 27695-8205
January25, 2001;revised November5, 2001
Abstract
Weconsidernumericaltechniquesforanonsmoothacousticpressuresystemarisingin
elec-tromagneticinterrogationof dielectricmaterials. Werstdescribeseveralformulationsof
cou-pled electromagnetic/ acousticsystemsthat arisenaturally whentravelingacoustic interfaces
are used as reectors for pulsedmicrowaveinput signalsfor dielectric property and geometry
identication. Wethen developfullyGalerkinschemesinasomewhatnonstandardvariational
formulationofonlytheacousticsystemwithdistributionalinputs. Samplecomputational
nd-ingsusingtheresultinglargealgebraicsystemaregiven.
1 Introduction to electromagnetic/ acoustic problems
An increasingly important class of electromagnetic inverse problems entails using reected
mi-crowave inputimpulse responses to characterize material dielectricproperties aswellasgeometry
(e.g., see [5 ]). These reections satisfy a dissipative form of Maxwell's equations and one goal
involvesidenticationofthepolarizationmechanismrepresentedbyahysteresisterminMaxwell's
equations[8 ],[6 ],[11 ],[3 ]. Foranumberofpolarizationmechanisms(includingmodelssuch asthe
standardDebye andLorentz), these systems have theform(again see [5 ],pages 20-21)
~
r
E(t;z)+b _
E(t;z)+hE(t;z)
+ R
t
0
k(t s;z)E(s;z) ds c 2
E 00
(t;z)=J(t;z)
(1)
where the hysteresis kernel k is the second time derivative of a polarization susceptibility kernel.
For the problems of interest, this kernel is to be identied using the information contained in
reections of the inputpulse. As explained in[5 ], the formulation (1) is a useful conceptual and
theoreticalformulationforidentifyingtheinternalpolarizationdynamicsandtherebycharacterizing
thedielectricmaterial.
Incertain classesofelectromagneticinterrogation techniques,one mayemployaperfectly
ing for an underground nonmetal object or indetecting a braintumor. In such cases, it may be
possiblefor a travelingacoustic wave, perhapseven one occurring naturally, to serve as a virtual
interface. In [5 ], the authors describe models and applications for techniques which employ
per-fectly conductive metalbackings andstanding acoustic wavesas reectorsfortheelectromagnetic
waves. In addition, they suggest the possibilityof a technique inwhich a traveling acoustic wave
might beusedas avirtualinterfaceto reectan oncomingelectromagnetic wave.
An essentialfeature of the aforementioned modelsis theinteractionbetween the electromagnetic
and acoustic waves. This interaction can be modeled in various ways. Here we briey outline
several modeling approaches (not all equivalent) foundin the literature. We use interchangeably
the notation used by the original authors (e.g.,
E and @
2
@t 2
E are the same) to facilitate
cross-referencing. We begin bynotingthat in [5 ] theauthors assume that the dielectricmaterial obeys
thegeneralized pressuredependent polarizationrule
1
0 @
2
P
@t 2
=f
0
(p)E+f
1 (p)
@E
@t +f
2 (p)
@ 2
E
@t 2
and make thesimplication
f
0
(p)=0; f
1
(p)=0; f
2
(p)=
0
+p(t;z):
Thisreduces themodelto
1
0 @
2
P
@t 2
=(
0
+p(t;z)) @
2
E
@t 2
; (2)
which isused withstandingacoustic waves inboth[5 ] and [1 ].
Asan alternative to (1), Maxwell'sequationsmaybe writtenintheform
~
r
E(t;z)+b _
E(t;z)+e
P(t;z) c 2
E 00
(t;z) =J(t;z): (3)
We note thatif
P(t;z)= Z
t
0
g(t s;z)E(s;z) ds
for some (suÆcientlydierentiable) polarizationsusceptibilitykernel g, equations (1) and (3) are
equivalent,up to theform of thecoeÆcient functions. We seethat thepolarizationmodel(2) can
beusedto replace
P in(3)to create apressure-dependentelectromagneticsystem(e.g., acoupled
electromagnetic/acoustic system, given thedynamicsforp).
r 2
E 1
c 2
@
@t 2
E =0:
We note that inone dimension thisis equivalent to (3) with~
r
=1, b=0, and no polarizationor
source term. They then suggest that a change in pressure will produce a change in the index of
refraction; they describe thisperturbation inthe refraction index interms of a variation Æ=
0 in
thedielectricconstant. Thisleadsthento thefollowingequation
r 2
E 1
c 2
@ 2
@t 2
E= Æ=
0
c 2
@ 2
@t 2
E: (4)
Reducedtoone dimension,equation(4)canbewrittenintheformof(3), with~
r
=1,b=0,J=0
and
e
P = Æ
0
E:
Thedielectricconstant canbethoughtofasafunctionofthepressureandentropyofthesystem,
~
P and S respectively. Thus
Æ=
@
@ ~
P
S Æ
~
P +
@
@S
~
P ÆS:
Ifthesystem isassumed to beat constant entropy,thisreducesto
Æ=
@
@ ~
P
S Æ
~
P
which can thenbeused in(4)to obtain
r 2
E 1
c 2
@ 2
@t 2
E= 1
c 2
1
0
@
@ ~
P
S p
@ 2
@t 2
E; (5)
wherep=Æ ~
P is thepressurevariation. Wenotethat if
@
@ ~
P
S
isconstant,thepolarizationmodel
in(5) isofthe same formas(2) with
0 =0.
An approach similar to that in [10 ] is found in [4]. The author considers the case where light is
scattered duetouctuationsinthedielectricconstant and assumesthatthese uctuationsarethe
result of uctuationsin thermodynamicvariables, such as pressure, withinthesystem. We follow
hisargumentstopresentamacroscopic viewoftheproblem. Thisbeginswiththeassumptionthat
thescattered eld ~
E is describedbytheequation (after conversionfrom gaussian to MKSunits)
r 2
~
E n
2
c 2
~
E =
0
c 2
~
n 2
c 2
E E
00
=
0
c 2
P: (7)
We then let be a uctuation in the dielectric constant and be a uctuation in electric
susceptibility. Since
=
0
(1+);
itfollows that
=
1
0 :
We next supposethatthepolarizationdueto theuctuationis given by
~
P =
~
E
0 =
1
0
~
E
0
(8)
where ~
E
0
isthe incident optical eld.
We further assume that density and temperature, and T, are the independent thermodynamic
variablesinorder to representthe dielectricconstant uctuationas
=
@
@
+
@
@T
T:
Under assumptionthatthedielectricconstant has astronger dependenceon densitythanon
tem-perature[4],wecan approximatethisrelationshipby
=
@
@
: (9)
If we then treat the density as dependent on pressure and entropy, p and s (which are now the
independent thermodynamicvariables),we ndthattheuctuation indensitycan be written
=
@
@p
p+
@
@s
s:
Finallysinceourmain interest isthe scatteringdueto variations inacoustic pressure, asopposed
to entropy,we neglectthesecond termand arrive at therelationship
=
@
@p
~
P = 1
0 @
@ @
@p p
~
E
0 ;
whichcanthenbeusedinequations(6)or(7). Wenotethatthisresultsinanequationverysimilar
to (5).
In ouralternate approach,we considertheideas developed in[2 ]. Here, the authorsproposethat
the coeÆcients in the polarization model (Debye, Lorentz, etc.) can be represented as a linear
functionof pressure. Forthe Debye polarizationmodel,thisleadsto the dierentialequation
_
P =
1
(
0 +
p)
P + (
0 +
p)
(
0 +
p)
E:
Similarly,theLorentzpolarizationmodelcanbeexpressedasthefollowingsecondorderdierential
equation
P + 1
(
0 +
p)
_
P +(
0 +
p)P =(
0 +
p)E:
Either of these two modelscan be coupledwith equation (3)through the
P term to describe the
electromagnetic/acoustic interaction.
Beforeanyofthese interactionmodelscanbeemployed,however,wemustbeginassessmentofthis
proposed interrogationtechnique by investigatingimpulsegenerated pressurewavesina
heteroge-neous medium. In particular,we considerhere an acoustic pressurewave initiatedbya windowed
sinewave impulsetraveling throughalayeredmediumand formulatethe equationsand boundary
conditions describing thesystem. We explore several approaches to solving theproblem withthe
nite element method and settle on a (somewhat nonstandard) fully Galerkin scheme. We then
discussnumericalndingsobtainedwiththismethod.
2 Wave system formulation
We rst present the decoupled acoustic system of interest inthe problems formulated above. We
initiallywrite theequations that describe the behavior of the traveling acoustic wave in a strong
form. Then we develop a variational formulation for thesystem and discuss diÆcultiesthat arise
whiledoingso.
We considerthewave equationforacousticpressureinamaterialconsisting ofthreehomogeneous
layers. We assume that in the left and right layers of the material the wave propagates with the
same wavespeed,butthatthewavetravelsat adierentspeedinthemiddlelayer. The boundary
conditions are given by the input of windowed sine wave at z = 1 and a no reection, or total
absorbing, conditionat z =0. Since discontinuities (atz
1 and z
2
) are present in the propagating
medium, we also must introduce interfaceconditions. We do thisbyrequiringcontinuityof p(t;)
and c 2
p 0
(t;) at z =z
1
and z = z
2
continuity of c p will be a natural condition in our weak formulation below. A schematic of the
geometryis given inFigure 1. We supposethatthesystemis initiallyat rest. Then theequations
thatgovern thissystemaregiven by
~ p c
2
(z)p~ 00
=0 (11)
~
p(0;z)=0 p(t;~ 1)=f(t)
_
~
p(0;z)=0
_
~
p(t;0) c(0)~p 0
(t;0) =0
where
c(z)= 8
>
<
>
: c
1
0z<z
1
c
2 z
1
zz
2
c
1 z
2
<z1;
f(t)= (
0 0t;t2
sin( 2
(t )) <t<2:
for0<z
1 z
2
1 and 0<.
speed
c1
speed
c1
c2
no
reflection
speed
input
function
f(t)
Figure 1 Schematic diagram ofgeometry
Since ndinga solutionto thewave equation is normallyan easy exercise insolvingpartial
dier-entialequations, computinga numericalsolutionto thissystemwouldappearto bea simpletask.
However, uniquecharacteristics ofthissystem make solvingit asomewhat more challengingtask.
To treat the nonhomogeneous time dependent Dirichlet boundary condition at z = 1, we make
a change of variables which facilitates niteelement solutions. To obtain a new equation with a
homogeneousboundaryconditionat z=1;we introduceanew state variable pdened by
p(t;z)=p(t;~ z) zf(t): (12)
p c (z)p +z
f(t)=0 (13)
p(0;z)=0 p(t;1)=0
_
p(0;z)=0 p(t;_ 0) c
1 p
0
(t;0) c
1
f(t)=0
p(t;z
1
)=p(t;z
1
+) c
2
(z
1 )p
0
(t;z
1 )=c
2
(z
1 +)p
0
(t;z
1 +)
p(t;z
2
)=p(t;z
2
+) c
2
(z
2 )p
0
(t;z
2 )=c
2
(z
2 +)p
0
(t;z
2 +)
(14)
wherec(z)andf(t)areasdenedabove. We observe thatthischangeofvariabledoesprovidethe
desired boundaryconditionat z=1.
Since c(z) is only piecewise continuous in z, we do not expect solutions to the above equation
in strong form in space (i.e., C 2
or even only H 2
in z). Therefore, for both theoretical and
computationalpurposes,itis usefulto writethe systeminweakor variational form inthe spatial
variable. This approach is standard. However, we note that in our change of variables, we have
introduced the term
f(t) into the wave equation. If we recall that the function f is a windowed
sinewave, werealizethatitssecondderivative
f(t)includesadeltaimpulseintime(see Figure2).
One thusobserves thatwealso may notbeable to expect solutionsinstrong formin time. Thus,
we may expectdistributionalderivativesinbothtimeand space.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
f
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−30
−20
−10
0
10
20
30
t
fdot
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−2000
−1500
−1000
−500
0
500
1000
1500
2000
fddot
t
Figure 2(c) Second derivative off(t)
We tried two dierent approaches to deal with potential diÆculties due to lack of smoothness
of solutions. First, we ignored the lack of smoothness of
f and proceeded with a standard
semi-Galerkinniteelementmethod. Sinceweknowthatoursolutionshouldbeatravelingsinewave(at
leastifwe assumec
1 =c
2
),it wasclearfromtheresultingsimulationsthatthissolutiontechnique
was notadequate. For oursecond approach,weused molliersto smooththe \windowing"of the
functionf andagaincontinuedinthetraditionalwayusingastandardsemi-Galerkinniteelement
method. However, this approach led to solutions that failed to converge to the known solution.
We concludedthatanappropriatewayto solve theproblem mightbeto useafullyGalerkinnite
elementscheme. Forfurtherdiscussionof fullyGalerkinmethods,see, forexample, [7],[9 ].
We let <;>denote theusualL 2
innerproducton (0;1), i.e., <f;g>= R
1
0
f(z)g(z) dz,and we
let < ;>
(a;b)
denote the L 2
inner product on the specied interval(a;b). We dene the spaces
H 1
R
(a;b)=f2H 1
(a;b)j(b)=0g and H 1
L
(a;b)=f2H 1
(a;b)j(a)=0g:
We supposethatp satises(13), (14) and thefollowinghold:
p2H 1
L
(0;T;H 1
R (0;1))
p(;z)2H 2
(0;T) almost everywhere in(0;1)
p(t;)2 ~
Hf2C(0;1):2H 2
( ~
)g almost everywherein(0;T);
where ~
(0;z
1 )[(z
1 ;z
2 )[(z
2 ;1):
(15)
Then
Z
T
0
<p; > dt Z
T
0 <c
2
(z)p 00
;> dt + Z
T
0
f dt<z;>=0
holdsforall 2H 1
R
(0;1) and forall 2H 1
R (0;T):
R
T
0
<p;_ > _ dt + R T 0 c 2 1 <p 0 ; 0 > (0;z 1 ) +c 2 2 <p 0 ; 0 > (z 1 ;z 2 ) +c 2 1 <p 0 ; 0 > (z 2 ;1) dt R T 0 _ f _
dt<z;>+<p;_ > j T 0 R T 0 c 2 1 p 0 j z 1 0 +c 2 2 p 0 j z 2 z1+ +c 2 1 p 0 j 1 z2+ dt
+<z;> _
f j T
0 =0:
Wemaythensubstituteourboundary,interface,andinitialconditions(14),aswellastheconditions
on and ,into theabove equation toobtain
R
T
0
<p;_ > _ dt + R T 0 c 2 1 <p 0 ; 0 > (0;z 1 ) +c 2 2 <p 0 ; 0 > (z 1 ;z 2 ) +c 2 1 <p 0 ; 0 > (z 2 ;1) dt R T 0 _ f _
dt<z;>+c
1 R
T
0 _
p(;0)(0) dt c 2
1 R
T
0
f()(0) dt=0:
Thisimplies
R
T
0
<p;_ > _ dt + R T 0 <c 2 (z)p 0 ; 0 > dt R T 0 _ f _
dt<z;>
+c 1 R T 0 _
p(;0)(0) dt c 2
1 R
T
0
f()(0) dt=0:
Thissuggests that ourweak solutionwithp(0;z)=0and p(t;1)=0 shouldsatisfy
R
T
0
<p;_ > _ dt + R T 0 <c 2 (z)p 0 ; 0 > dt R T 0 _ f _
dt<z;>
+c 1 R T 0 _
p(;0)(0) dt c 2
1 R
T
0
f()(0) dt=0
forall 2H 1
R
(0;1) and forall 2H 1
R (0;T).
Thus,we seeksolutions p2H 1
L
(0;T;V),where V H 1
R
(0;1), thatsatisfy
R
T
0
<p;_ > _ dt + R T 0 <c 2 (z)p 0 ; 0 > dt R T 0 _ f _
dt<z;>
+c 1 R T 0 _
p(;0)(0) dt c 2
1 R
T
0
f()(0) dt=0
(16)
forall 2H 1
R
(0;1) and forall 2H 1
R (0;T).
If we assume that our solutions have enough smoothness, i.e., p(t;) 2 ~
H and p(;z) 2 H 2
(0;T);
we can verify thatthisis,infact, adesired weak formof ourequation. Assumingthissmoothness
R
T
0
<p; > dt R T 0 c 2 1 <p 00
;>
(0;z 1 ) +c 2 2 <p 00
;>
(z 1 ;z 2 ) +c 2 1 <p 00
;>
(z
2 ;1)
dt
+<z;> R
T
0
f dt+c
1 R
T
0 _
p(;0)(0) dt c 2
1 R
T
0
f()(0) dt
<p;_ > j T
0
<z;> _
f j T 0 + R T 0 c 2 1 p 0 j z1 0 +c 2 2 p 0 j z2 z 1 + +c 2 1 p 0 j 1 z 2 +
dt=0
forall 2H 1
R
(0;1) and forall 2H 1
R
(0;T) withp(0;z)=0and p(t;1)=0.
Then
R
T
0
<p; > dt R T 0 <c 2 (z)p 00
;> dt+<z;> R T 0 f dt +c 1 R T 0 _
p(;0)(0) dt c 2
1 R
T
0
f()(0) dt+<p(0;_ );> (0)
R T 0 c 2 1 p 0
(;0)(0) dt
+ R T 0 (z 1 ) c 2 1 p 0 (;z 1 ) c 2 2 p 0 (;z 1 +) (z 2 ) c 2 1 p 0 (;z 2 +) c 2 2 p 0 (;z 2 ) dt =0 (17)
forall 2H 1
R
(0;1) and forall 2H 1
R
(0;T) withp(0;z)=0and p(t;1)=0.
If we choose 2 H 1
I
(0;1) = f 2H 1
(0;1)j(0) =(1) =0; (z
1
) = (z
2
) =0g H 1
R
(0;1) and
2H 1
0
(0;T)=f 2H 1
(0;T)j (0)= (T) =0gH 1
R
(0;T), thenwehave
Z
T
0
<p; > dt Z T 0 <c 2 (z)p 00
;> dt <z;> Z
T
0
f dt=0
forall 2H 1
I
(0;1) and forall 2H 1
0 (0;T).
Since 2H 1
0
(0;T) is arbitrary,thisimpliesthat
<p; > <c 2
(z)p 00
;>
f <z;>=0
forall 2H 1
I (0;1).
Hence,sinceH 1
I
(0;1) is denseinL 2 (0;1), p c 2 (z)p 00
fz=0
intheL 2
c 1 R T 0 _
p(;0)(0) dt c 2
1 R
T
0
f()(0) dt+<p(0;_ );> (0)
R T 0 c 2 1 p 0
(;0)(0) dt
+ R T 0 (z 1 ) c 2 1 p 0 (;z 1 ) c 2 2 p 0 (;z 1 +) (z 2 ) c 2 1 p 0 (;z 2 +) c 2 2 p 0 (;z 2 ) dt =0 (18)
forall 2H 1
R
(0;1) and forall 2H 1
R (0;T).
Ifwe againchoose2H 1
I
(0;1), wehave
<p(0;_ );>=0
forall 2H 1
I
(0;1), since (0) isarbitrary.
Thus,sinceH 1
I
(0;1) isdense inL 2
(0;1);
_
p(0;z)=0
almost everywhere inz2[0;1]:
Returning to (18), we have
c 1 R T 0 _
p(;0)(0) dt c 2
1 R
T
0
f()(0) dt R T 0 c 2 1 p 0
(;0)(0) dt
+ R T 0 (z 1 ) c 2 1 p 0 (;z 1 ) c 2 2 p 0 (;z 1 +) (z 2 ) c 2 1 p 0 (;z 2 +) c 2 2 p 0 (;z 2 ) dt =0 (19)
forall 2H 1
R
(0;1) and forall 2H 1
R (0;T).
We choose2f2H 1
R
(0;1) :(z
1
)=(z
2
)=0g andnote thatsince(0) is arbitraryin thisset
Z T 0 c 2 1 p 0
(;0)+c
1 _
p(;0) c 2
1 f()
dt=0
forall 2H 1 R (0;T). Since H 1 R
(0;T) is denseinL 2
(0;T),
_
p(t;0) c
1 p
0
(t;0) c
1
Finally,we nowhave that(19) becomes
Z
T
0 h
(z
1 )
c 2
1 p
0
(;z
1 ) c
2
2 p
0
(;z
1 +)
(z
2 )
c 2
1 p
0
(;z
2 +) c
2
2 p
0
(;z
2 )
i
dt=0
forall 2H 1
R
(0;1) and forall 2H 1
R (0;T):
Ifwechoose2H 1
R
(0;1) such that(z
2
)=0; thenwe have
Z
T
0 (z
1 )
c 2
1 p
0
(;z
1 ) c
2
2 p
0
(;z
1 +)
dt=0
forall 2H 1
R (0;T):
Since (z
1
)is arbitrary andH 1
R
(0;T) isdense inL 2
(0;T),wehave
c 2
1 p
0
(t;z
1 )=c
2
2 p
0
(t;z
1 +)
almost everywhere in[0;T],and bysimilararguments
c 2
1 p
0
(t;z
2 +)=c
2
2 p
0
(t;z
2 )
almost everywhere in[0;T].
Hence, we have shown that if p 2 H 1
(0;T;V) (which implies p(t;) is continuous at z
1 and z
2 )
satises the weak form (16) and p possesses the additional smoothness p(t;) 2 ~
H and p(;z) 2
H 2
(0;T), then
p c
2
(z)p 00
+z
f(t)=0 almost everywhere
p(0;z)=0 p(t;1)=0
_
p(0;z)=0almost everywhere p(t;_ 0) c
1 p
0
(t;0) c
1
f(t)=0almost everywhere
p(t;z
1
)=p(t;z
1
+) c
2
(z
1 )p
0
(t;z
1 )=c
2
(z
1 +)p
0
(t;z
1
+) almost everywhere
p(t;z
2
)=p(t;z
2
+) c
2
(z
2 )p
0
(t;z
2 )=c
2
(z
2 +)p
0
(t;z
2
+) almost everywhere.
Thusthevariational form (16) hasbeenveried withthe pointwise interfaceconditions at z =z
1
and z=z
2
In view of the weak formulation of the previous section for the system, we develop a dual nite
element approximation to the solution. Since we have written the equations weakly in bothtime
and space,itis naturalto use afully(time and space) Galerkinscheme.
Since we seek solutions p 2 H 1
L
(0;T;V) with V H 1
R
(0;1) that satisfy the weak form, it is also
natural to approximate p by a linear combination of piecewise linear basis elements in bothtime
and space. That is,
p(t;z)p MN
(t;z)= M X i=1 N 1 X j=0 a ij i (t) j (z); (20) where i 2H 1 L
(0;T) and
j 2H
1
R
(0;1) are thestandardpiecewiselinear splinefunctions.
We may substitute this approximation into our weak form to obtain dening equations for the
coeÆcients a ij givenby P M i=1 P N 1 j=0 a ij n R T 0 _ i _ dt< j
;>+ R
T
0 i
dt<c 2 (z) 0 j ; 0 >+c 1 R T 0 _ i dt j (0)(0) o n
<z;> R T 0 _ f _
dt+c 2
1 R
T
0
f dt (0) o
=0
forall 2H 1
R
(0;1) and for all 2H 1
R
(0;T). However thisresults intoo many equations, butas
usual,onerestrictsthefamiliesofand forwhichwerequirethesystemtohold. Inparticular,we
requireitfor=
l 2H
1
R
(0;1);l=0;1;:::;N 1andfor =
m
(t)2H 1
R
(0;T);m =0;:::;M 1;
where
l and
m
arepiecewiselinearsplinefunctions. Thisyieldsthereducedsystemofequations
P M i=1 P N 1 j=0 a ij n R T 0 _ i _ m dt< j ; l >+ R T 0 i m
dt<c 2 (z) 0 j ; 0 l >+c 1 R T 0 _ i m dt j (0) l (0) o n <z; l > R T 0 _ f _ m dt+c
2 1 R T 0 f m dt l (0) o =0
foreach l=0;:::;N 1;and foreach m=0;:::;M 1.
For each j = 0;:::;N 1, for each i = 1;:::;M, for each l = 0;:::;N 1; and for each m =
0;:::;M 1,we dene
G ij l m = R T 0 _ i _ m dt< j ; l >+ R T 0 i m
dt<c 2 (z) 0 j ; 0 l > +c 1 R T 0 _ i m dt j (0) l (0)
H
l m
=<z;
l >
T
0 _
f _
m dt+c
2
1 T
0 f
m dt
l (0):
Thus,we can writeouralgebraicsystemof deningequationsas
M
X
i=1 N 1
X
j=0 a
ij G
ij
l m =H
l m
foreach l=0;:::;N 1;m=0;:::;M 1.
Next,for alli=1;:::;M,we let
~a
i =[a
i0 a
i1
a
iN 1 ];
and forall i=1;:::;M;l=0;:::;N 1;and m=0;:::;M 1,welet
~
G i
l m =
2
6
6
6
6
4 G
i0
l m
G i1
l m
.
.
.
G iN 1
l m 3
7
7
7
7
5 :
Then forall i=1;:::;M;l=0;:::;N 1;and m=0;:::;M 1,
N 1
X
j=0 a
ij G
ij
l m =~a
i ~
G i
l m :
So,forall l=0;:::;N 1and m=0;:::;M 1,
M
X
i=1 N 1
X
j=0 a
ij G
ij
l m =
M
X
i=1 ~a
i ~
G i
l m
= ~a
1 ~
G 1
l m +~a
2 ~
G 2
l m
+:::+~a
M ~
G M
l m
= H
l m :
Furthermore, wedene
=[~a
1 ~a
2
~a
G= 6
6
6
6
4 ~
G 1
00
~
G 1
N 10 ~
G 1
01
~
G 1
N 11
~
G 1
0M 1
~
G 1
N 1M 1
~
G 2
00
~
G 2
N 10 ~
G 2
01
~
G 2
N 11
~
G 2
0M 1
~
G 2
N 1M 1
.
.
.
~
G M
00
~
G M
N 10 ~
G M
01
~
G M
N 11
~
G M
0M 1
~
G M
N 1M 1 7
7
7
7
5
and
H=[ H
00 H
N 10 H
01 H
N 11
H
0M 1 H
N 1M 1 ]:
Thuswe ndthat ourniteelement scheme can bewritten
G=H (21)
and hence
=HG 1
:
Asusualin niteelement approximations, ifwe chooseN;M suÆcientlylarge, we expectthat by
computing from (21), we can obtaincoeÆcients a
ij
suchthat
p MN
(t;z)= M
X
i=1 N 1
X
j=0 a
ij i (t)
j (z)
isagoodapproximationforponthetimeinterval[0;T]. Thusthecorrespondingp~denedvia(12)
suÆcientlyapproximatesthebehaviorofthesolutionto(11). However,wendthatinpracticeitis
diÆculttoaccuratelyapproximatepoveragiveninterval[0;T
F
]inonestepduetotheconditioning
of the system (21) whenever T
F
is large. (We discuss the details of the implementation later in
thissection.) Instead we rst approximate p over a shorter interval[0;t
1
], where t
1 <T
F
and the
windowedsinewave isentirelywithinthe material by thetime t=t
1
. Notethatwecan changethe
time T in the weak form to accommodate any interval over which we wish to solve. Then, since
we can nd a suÆcient approximation for p over this smaller interval, we are able to accurately
approximatep~and describethe pressureoverthe interval[0;t
1 ].
In order to determine the behavior of the pressureon theentire given interval[0;T
F
], we needto
describe the pressure on the interval (t
1 ;T
F
]. This is equivalent to considering the originalwave
equation forpressurewith boundaryconditions given bya zero inputat z=1 and a noreection
condition at z = 0 but now initiallythere is a windowed sine wave already propagating through
~
y c (z)y~ =0
~
y(0;z)=g(z) y(t;~ 1)=0
_
~
y(0;z)=h(z)
_
~
y(t;0) c
1 ~ y 0
(t;0)=0
~ y(t;z
1
)=y(t;~ z
1
+) c
2
(z
1 )~y
0
(t;z
1 )=c
2
(z
1 +)~y
0
(t;z
1 +)
~ y(t;z
2
)=y(t;~ z
2
+) c
2
(z
2 )~y
0
(t;z
2 )=c
2
(z
2 +)~y
0
(t;z
2 +)
where
g(z)=p(t;~ z)j
t=t
1
; h(z)= _
~ p(t;z)j
t=t
1 :
Observe that we have a nonhomogeneous initial conditionat t =0. When using a semi-Galerkin
niteelement scheme, nonhomogeneousinitial conditionsare of littleconsequence, butthis isnot
the case for fully Galerkin schemes. To treat thiscase, it is desirable to make another change of
variables. Tothisend, we let
y(t;z)=y(t;~ z)+(t 1)g(z);
and theresultingequations fory are
y c
2
(z)y 00
+c 2
(z)(t 1)g 00
(z)=0
y(0;z) =0 y(t;1) =0
_
y(0;z) =h(z)+g(z) y(t;_ 0) c
1 y
0
(t;0) =0
y(t;z
1
)=y(t;z
1
+) c
2
(z
1 )y
0
(t;z
1 )=c
2
(z
1 +)y
0
(t;z
1 +)
y(t;z
2
)=y(t;z
2
+) c
2
(z
2 )y
0
(t;z
2 )=c
2
(z
2 +)y
0
(t;z
2 +)
where c(z), g(z), and h(z) are as dened previously, and we use that fact that g 0
(0) = g(1) =
g(0)=0 dueto thelocationof thepressureimpulseentirelywithinthematerial at t=t
1 .
Usingsimilarnotationand techniquesasbefore,we ndthattheweakform of ourequationis
R
T
0
<y;_ > _
dt + R
T
0 <c
2
(z)y 0
; 0
> dt
R
T
0
(t 1) dt<c 2
(z)g 0
; 0
> <h+g;> (0) + R
T
0 c
1 _
y(;0)(0) dt=0
(22)
forall 2H 1
R
(0;1) and forall 2H 1
R
(0;T) whereT =T
F t
1
withy(0;z)=0 and y(t;1)=0.
Thus,we seeksolutions y2H 1
L
(0;T;V), whereV H 1
R
(0;1),that satisfy(22).
Giventheweakform (22)of ourequation,wecan again approximatey byalinear combination of
y(t;z)y MN
(t;z)= X
i=1 X
j=0
ij i (t)
j (z);
where
i 2H
1
L
(0;T) and
j 2H
1
R
(0;1) arepiecewise linearspline functions. We note that inour
computationsM;N neednotbethesameasthosefortheapproximationofpontheinterval[0;t
1 ],
although we use the same notation here for ease of exposition. Followingthe same procedure as
before,we substitutethe approximation into theweak form ofourequation, ndingthat
P
M
i=1 P
N 1
j=0
ij n
<
j ;
l >
R
T
0 _
i _
m
dt + <c 2
(z) 0
j ;
0
l >
R
T
0 i m
dt+c
1
j (0)
l (0)
R
T
0 _
i m dt
o
= R
T
0
(t 1)
m dt<c
2
(z)g 0
; 0
l
>+ <h+g;
l >
m (0)
holds for the piecewise linear spline functions
m 2 H
1
R
(0;T); m = 0;:::;M 1; and
l 2
H 1
R
(0;1); l=0;:::;N 1:Then,asbefore,we can writethese equationsasa system
G=K
where containsthecoeÆcients
ij
,Gisasdenedpreviously,andKisanalogoustothepreviously
denedH . Thisequationcan be solvedforthecoeÆcientvector ,andthecoeÆcientscan inturn
beused to determineapproximationsfory and y.~
Summarizing,we use
p(t;z) p M
p N
p
(t;z)= M
p
X
i=1 N
p 1
X
j=0 a
ij i (t)
j
(z) forz2[0;1]; t2[0;t
1 ]
y(t;z) y MyNy
(t;z)= My
X
i=1 Ny 1
X
j=0
ij i (t)
j
(z) forz2[0;1]; t2[0;T
F t
1 ]
and
~ p
MpNp
(t;z) = p MpNp
(t;z)+zf(t) forz2[0;1]; t2[0;t
1 ]
~ y
MyNy
(t;z) = y MyNy
(t;z) (t 1)g(z) forz2[0;1]; t2[0;T
F t
1 ]
~ p
MpNp
(t;z) = y~ MyNy
(t t
1
;z) forz2[0;1]; t2[t
1 ;T
F ]
to dene an appropriate approximation to the behavior of the wave on the spatial interval [0;1]
and theentiregiven timeinterval[0;T
F ].
Priorto presentingsome solutionsobtained fromthisapproximation,wediscussbrieythe
out on a Sun Sparc Ultra 10 workstation. The coeÆcient vectors and were computed with
MATLAB's slash command, which nds solutions by Gaussian elimination. We recall that the
elementsinKare
Z
T
0
(t 1)
m dt<c
2
(z)g 0
; 0
l
>+ <h+g;
l >
m (0):
Here,g representsp~ MpNp
(t;)at a speciedtime t
1
. Asa result,weonlyhave accessto valuesof g
at thenodal points, z
k
. Furthermore,h(z) represents _
~ p
MpNp
(t;z) at t
1
, butthese values must be
approximatedat the appropriatespatialnodes. So,in order to compute theterms in K, we must
rst numericallyapproximateg_ (which is thesame as h) and g 0
from the known data points and
then calculate theinner products. We use a centered dierencemethod to approximateg 0
at the
nodal points and a backward dierence method to approximate g_ at the same points. Then we
use linear interpolation via the MATLAB command interp1 to obtain values for g(z);g 0
(z); and
_
g(z) atintermediatevaluesz betweenthenodes. Withthese \enhanced"datasets,we canusethe
trapezoidalmethod,via MATLAB'strapz command,to computetheinnerproducts.
Finally,weshowplotstoillustratethebehaviorofthewaveasitpassesthroughthelayeredmedium.
EachoftheplotsinFigure3isasnapshotintimeofthepressureinthemediumfortheparameters
giveninTable1 withM =N =256 basiselements. Lookingatthesnapshots sequentially,we can
seethepressurewavemovethroughthelayers. Thenoiseinfrontofandbehindthewaveisaresult
ofapproximationerrorandshouldnothavesignicantimpactontheelectromagneticinterrogation
process when usedintheproblemdescribed inSection1.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=0 − Initial Condition
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=0.375
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=0.625
Figure 3(c) Pressurevsdepth {t=0.5 Figure 3(d)Pressurevsdepth {t=0.625
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=0.75
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=0.875
Figure 3(e)Pressurevs Depth{ t=0.75 Figure 3(f)Pressurevs depth{ t=0.875
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=1.125
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=1.25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=1.375
Figure 3(i) Pressurevs depth{ t=1.25 Figure 3(j) Pressurevs depth{ t=1.375
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
Pressure vs Depth − t=1.5
Figure 3(k)Pressurevs depth{ t=1.5
Parameter Value
c
1
1.5
c
2
1.485
z
1
0.5605
z
2
0.7012
0.25
t
1
0.5
T
F
1.5
Table 1 Parametervaluesforcomputations inFigure 3
To address approximation error, we compare solutions as thenumberof basis elements increases.
In Figure 4,we seethe solutions of pressureversusdepth fora xed time computed withvarying
appear to converge. Figure 5 gives the corresponding plots for solutions of pressure versus time
at a xed depth. Again, we see apparent convergence as the number of basis elements increases.
Thissuggeststhatanyerrorintheapproximatesolutionisduetoapproximationerror. Thevalues
of parameters used inthese computations are given inTable 2. Moreover, Table 3 illustratesthe
convergenceinnormweseeasweincreasethenumberofbasiselements. Thenormsusedtoobtain
theresultsgiven inthetable are denedasfollows
jf gj
l
1 = sup
k sup
l jf(t
k ;z
l ) g(t
k ;z
l )j
jf gj
l 2 =
1
N
k N
l P
k P
l (f(t
k ;z
l ) g(t
k ;z
l ))
2
1
2
where(t
k ;z
l
)arethenodalpointsof thepiecewiselinearelementsf
i g;f
j
goftheapproximation
(20) andN
k ;N
l
arethenumberofnodalpoints.
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Pressure vs Depth
depth
0.55
0.56
0.57
0.58
0.59
0.6
0.61
0.62
0.63
0.64
0.65
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Pressure vs Depth
depth
Figure 4(a)Convergenceof elementsindepth Figure 4(b)A close-up ofFigure 4(a)
0.25
0.3
0.35
0.4
0.45
0.5
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Pressure vs Time
time
0.35
0.4
0.45
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
Pressure vs Time
time
c
1
1.5
c
2
1.485
z
1
0.5605
z
2
0.7480
0.25
t
1
0.5
T
F
0.5
Table 2 Parameter values forcomputationsinFigures 4and 5
Norm Value Norm Value
j~p 32;32
~ p
16;16
j
l
1 0.119126 j~p 32;32
~ p
16;16
j
l
2 0.048853
j~p 64;64
~ p
32;32
j
l 1
0.080170 j~p 64;64
~ p
32;32
j
l
2 0.032970
j~p 128;128
~ p
64;64
j
l
1 0.069472 j~p 128;128
~ p
64;64
j
l 2
0.025776
j~p 256;256
~ p
128;128
j
l 1
0.048119 j~p 256;256
~ p
128;128
j
l
2 0.021199
Table 3 Convergenceinnorm
4 Conclusions and future work
Wehavederivedandtestednumericallyanadequateapproximationmethodforthebehaviorofthe
pressurewave. Thenexttaskistocoupletheacousticsystemwiththeappropriateelectromagnetic
system to develop theory and computation forthe interrogation technique describedin Section1.
Thiswillinvolvederivingamodelsimilartothose outlinedinSection1fortheinteractionbetween
thetwo waves.
Inthispaper,wehave developedan approximation schemefortheacoustic waveequations.
Ques-tions of the well-posedness of the decoupled pressure system remain to be addressed. Since the
variationalform oftheequation isweak inbothtimeand space,thisisanon-trivial issueand will
be treated in a forthcoming paper. Moreover, a theoretical and computational treatment of the
This research was supported in part by the Air Force OÆce of Scientic Research under grants
AFOSR-F49620-98-1-018 0, AFOSR-F49620-01-1-0026 , and AFOSR-F49620-98-1-043 0 and inpart
throughaDepartment ofEducation GAANNFellowshipto J.K. Raye underGrant P200A70707.
References
[1] H.T.BanksandT.Lin.Determiningthestructureofabiologicalmediumusingacousto-optic
probes. J. Inv. Ill-Posed Problems, 7:61{82, 1999.
[2] H. T. Banks and J. K. Raye. Pressure-dependent polarizationmodelsin thecontext of
elec-tromagnetic interrogation. in preparation.
[3] R.S.Beezleyand R.J.Krueger. Anelectromagneticinverseproblemfordispersivemedia. J.
Math.Physics, 26:317{325, 1985.
[4] R.W.Boyd. Nonlinear Optics. Academic Press,San Diego, 1992.
[5] H.T.BanksM.W.BuksasandT.Lin.ElectromagneticMaterial Interrogation Using
Conduc-tive Interfaces and Acoustic Wavefronts. SIAM Frontiers in Applied Mathematics,
Philadel-phia,2000.
[6] R. S. Elliot. Electromagnetics: History, Theory, and Applications. IEEE Press, New York,
1993.
[7] C.A. J.Fletcher. Computational Galerkin Methods. Springer-Verlag,Berlin,1984.
[8] J.D. Jackson. Classical Electrodynamics. JohnWiley,New York,2nd edition,1975.
[9] J. Lundand K. L.Bowers. Sinc Methods for Quadrature and Dierential Equations. SIAM,
Philadelphia,1992.
[10] P.M. Morse and K. U.Ingard. Theoretical Acoustics. McGraw-Hill, NewYork, 1968.
[11] R.A.AlbaneseJ.PennandR.Medina. Short-rise-timemicrowave pulsepropagation through
