Axisymmetric eddy current inspection of highly conducting thin layers via asymptotic models

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Axisymmetric eddy current inspection of highly conducting thin layers via asymptotic models

Houssem Haddar, Zixian Jiang

To cite this version:

Houssem Haddar, Zixian Jiang. Axisymmetric eddy current inspection of highly conducting thin layers

via asymptotic models. Inverse Problems, IOP Publishing, 2015. �hal-01214308�

onduting thin layers via asymptoti models

Houssem Haddar

∗

, ZixianJiang

†

Abstrat

Thin opper deposits overing the steam generator tubes an blind eddy urrent probes in non-

destrutivetestingsofproblematifaultsandarethereforeimportanttobeidentied.Existingmethods

basedon shape reonstrutionusing eddy urrent signalsenounter diulties ofhigh numerialosts

duetothelayer'ssmallthiknessandhighondutivity.Inthisartile,weapproximatetheaxisymmetri

eddyurrent problemwithsomeappropriateasymptotimodelsusingeetive transmissiononditions

representingthe thindeposits. Inthese models, the geometrial information related to the deposit is

transformedintoparameteroeientsonatitiousinterfae. Standarditerativeinversionalgorithmis

thenappliedtotheasymptotimodelstoreonstrutthethiknessofthethinopperlayers. Numerial

testsbothvalidatingtheasymptotimodelandshowingbenetoftheinversionproedureareprovided.

Keywords: axisymmetrieddyurrent inspetion,asymptotimodel, thiknessreonstrution ofthin

layers.

1 Introdution

Thinlayersofopperdepositsareobservedinnon-destrutiveeddyurrenttestingofsteamgeneratortubes

forthe safetyand failure-free operatingof nulearpowerplants. Coveringthe shell sideof thetube, these

deposits lead to a signiant signal feedbak in the eddy urrent inspetion due to the high ondutivity,

despite their extremely small thikness (see Table 1for a omparison with the tube). These deposits do

notdiretlyeet theprodutivityof steamnor thesafetyof struture. Howevertheirpresene may mask

otherkindsofproblematifaultssuhasloggingmagnetitedeposits, raksintubes,et. Thisis whyitis

importantto identifythem.

tubewall opperlayer

ondutivity(in

S · m

⁻¹⁾

σ t = 0.97 × 10 ⁶ σ c = 58.0 × 10 ⁶

thikness(in

mm

⁾

r t

2

− r t

1

= 1.27 0 ≤ f δ (z) ≤ 0.1

Table1: Condutivityandsaledierenesbetweentubewallandopperlayer.

Inpreviousstudies, we modeledthe axisymmetrieddy urrentproblem withan eientnite element

approximationwhih involvesartiial boundaryonditionsto ut o theomputational domainand thus

redue thenumerial ost(see [10℄). Basedon this model isdevelopedan inversionalgorithmusing shape

optimization methods to reonstrut the shape of some logging magnetite deposits [12℄. These methods

anbenaturally applied to thiknessreonstrution of thin opperdeposits. However, amajor numerial

hallenge to dealwith this aseis thehigh numerialost resulting from thefat that the omputational

domainshouldbedisretizedintoanemeshwiththesamesaletothelayerthiknessoftheopperdeposits.

Besidesadaptivemeshreningmethod(seeforexample[7℄),anotherwidelypratiedstrategytoredue

omputational ost onsists in modeling the thin deposit with eetive transmission onditions (see for

example[19℄, [21℄). [20℄ in partiular suppliesanalysis andomparison of dierent impedanetransmission

onditionsforthinsheetsintwodimensionaleddyurrentmodel. Thesetransmissiononditionsareobtained

bymathingtheasymptotiexpansionsofthesolutionwithrespetto asmallparameterharaterizingthe

∗

houssem.haddarinria.fr,DÉFIteam,INRIASalay,Palaiseau,Frane.

†

zixian.jiangpolytehnique.edu,DÉFIteam,INRIASalay,Palaiseau,Frane.

perspetive,anadditionalmajoradvantageofasymptotimodel istoavoidre-meshingat eahiteration. A

rih literature on asymptoti models has beendeveloped for dierent approahes and various appliations

(seeforexample[3℄,[2℄,[6℄,[5℄,[4℄, [18℄,[13℄,[14℄, [22℄andthereferenestherein).

In the researh note [9℄ the authors built and ompared several asymptoti models using a family of

eetive transmission onditions

{Z ^m,n } m,n∈{0,1,2}

^{for the} axisymmetri eddy urrent inspetion of thin opper deposits. Roughly speaking, if the thikness of a thin deposit at vertial position

z

^{of the tube}

writes

f δ (z) = δd(z)

^{, then theindex}

m

^{of the}transmission ondition

Z ^m,n

^{denotes are-saling parameter}

of the opper ondutivity with regard to

δ

^:

σ c = σ m /(δ ^m )

^{, while the index}

n

^{stands for the order of}

asymptotiexpansionofthesolutionwith respetto

δ

^{. Numerialtests thereinonaredued1Dasewith}

onstant deposit thikness showthat theasymptoti modelswith

Z 1,n

⁽

n = 0, 1

^{)onditionsgivesatisfying}

approximationofthefullmodelandleadto easyonstrutionofinversionmethods.

Inthis artilewerst introduetheaxisymmetri modelforeddy urrenttesting andsomeusefulteh-

niquestodeveloptheasymptotimodels(Setion2). Usingthesamemethodpresentedin[9℄,wethenbuild

thevariationalasymptoti modelswith

Z 1,n

⁽

n = 0, 1

⁾transmissiononditionsfor generalaseswith vari- abledeposit thikness(Setion3). Byintroduing adjointstatesassoiatedtoderivativesoftheimpedane

measurements,weformulate theinverseproblem forthiknessreonstrutionastheminimizationof aleast

square ost funtional of layerthikness by gradient desent (Setions 4). Finally, numerialexamples of

modelvalidation,measurementapproximationandthiknessreonstrutionaregiveninSetion(5). Forthe

useofasymptotimodelsininverseproblemswemayite[8℄,[15℄and[16℄forvariousappliations.

2 Asymptoti approximation of axisymmetri eddy urrent model

This setion is devoted to a formal derivation of asymptoti models for eddy urrent problems with the

preseneofhighlyonduting thindeposits. Theobjetiveistogettheeetivetransmissiononditionson

theinterfaebetweenthethinlayerandthetubewithwhihthevariationalasymptotimodelhasnolonger

thevolumeintegralonthethinlayerdomain.

First let us briey introdue the axisymmetri eddy urrent problem. For more details readers may

refer to [10℄. In the ylindrial oordinates, a vetor eld

a

an be deomposed into the meridian part

a _m = a r e _r + a z e _z

and the azimuthal part

a _θ = a θ e _θ

.

a

is axisymmetri if

∂ θ a

vanishes. Under the assumption of axisymmetry and the low eletri permittivity / low frequeny regime (

ωǫ ≪ σ

^{), the 3-D}

time-harmoniMaxwell'sequationsfortheeletriandmagnetields

( E , H ) ( curl H + (iωǫ − σ) E = J

in

R ³ ,

curl E − iωµ H = 0

ⁱⁿ

R ³ ,

withadivergene-freeaxisymmetriappliedsoure

J

(

div J = 0

^{)anbereduedtoaseondorderequation}

ona2-Ddomain

R ² + := { (r, z) : r ≥ 0, z ∈ R}

^{fortheazimuthalpartoftheeletrield}

E θ

^{that wedenote}

inthesequelby

u = E θ

^:

− div 1

µr ∇ (ru)

− iωσu = iωJ θ = iωJ

ⁱⁿ

R ² + ,

⁽¹⁾

where

∇ := (∂ r , ∂ z ) ^t

^and

div := ∇·

^{are gradient and divergene operators in 2-D Cartesian} oordinates.

Assume that

J ∈ L ² (R ² ₊ )

^{has ompatsupport, and that}

µ

^and

σ

^arein

L

^∞

(R ² ₊ )

^{suh that}

µ ≥ µ v > 0

^,

σ ≥ 0

^{and that}

µ = µ v

^,

σ = 0

^for

r ≥ r 0

^{suiently large. Then problem (1) with a deay ondition at}

innity(

u → 0

^as

r ² + z ² → 0

^{)hasauniquesolutionin}

H (R ² ₊ )

^whereforany

Ω ⊂ R ² +

^wedenote

H(Ω) := n

v : r

¹

^/

²

(1 + r ² )

^−λ

^/

²

v ∈ L ² (Ω), r

⁻¹

^/

²

∇ (rv) ∈ L ² (Ω) o

with

λ

^anyreal

> 1

^{(see[10℄). Letusindiatethatif}

Ω

^{isboundedinthe}

r

^{-diretionthen}

H (Ω)

^isequivalent

tothefollowingspaeforwhihweshallusethesamenotation

H(Ω) := n

v : r

¹

^/

²

v ∈ L ² (Ω), r

⁻¹

^/

²

∇ (rv) ∈ L ² (Ω) o

.

a(u, v) :=

Z

Ω

1

µr ∇ (ru) · ∇ (r¯ v) − iωσrw¯ v

dr dz = Z

Ω

iωJ ¯ vr dr dz ∀ v ∈ H (Ω).

⁽²⁾

Fornumerial reasons, the omputational domain will be trunated in radialdiretion at

r = r

∗ ^with

r

∗

> 0

^{suientlylarge,andwesetaNeumannboundaryonditionon}

r = r

∗^{. Thesolutiontothetrunated}

problemshouldsatisfy(1)on

Ω = B r

∗

:= { (r, z) ∈ R ² : 0 ≤ r ≤ r

∗

}

^{. Thisiswhyweshalluseinthesequelthe}

generinotationforthevariationalspae

H (Ω)

^with

Ω

^denoting

R ² ₊

^or

B r

∗^{. Wealsoreallthattheproblem}

on

Ω = B r

∗ ^anbeequivalentlytrunated toaboundeddomain

B r

∗

,z

∗

= { (r, z) ∈ R ² : 0 ≤ r ≤ r

∗

, | z | < z

∗

}

byintroduingappropriateDirihlet-to-Neumannoperatorsontheboundaries

Γ

±

:= { (r, z) : 0 ≤ r ≤ r

∗

, z =

± z

∗

}

^{. This would be onvenient for aelerating numerial evaluation of the solution. The analysis and}

examplesofdomaintrunationusingNeumannandDirihlet-to-Neumannboundaryonditionsanbefound

in[10℄. Weremarkthatthederivationoftheasymptotimodelsandtheinversionmethodsinthesequelare

independentof the domain trunation strategy. This is whyweuse the variationalformulation(2) as our

startingpointand thenweapplydomaintrunationinthenumerialexamples.

Finally letusspeifytheproblem settingsof eddy urrenttestingfor thin layerdeposit. Weonsider a

thinopperdepositoveringaxisymmetriallytheshell sideofthetube(seeFigure1foraradialutofthe

setting in theylindrial oordinate system, the

z

^{-axis is the tube axis, the thin deposit shown in blue is}

exaggeratedin thikness). The eddy urrent probeis omposed oftwooaxialoils, whose radialutsare

representedbytwosmallretanglesinFigure 1,that movein the

z

^{-diretionduring asan.}

Onthedomainofproblem

Ω

^,wedenote

Ω

±

:= { (r, z) ∈ Ω : r ≷ r t

2

}

^{. Theappliedurrent}

J

^issupported

bytheoils,therefore

suppJ ⊂ Ω

−^{. Thedepositlayerisdepitedby}

Ω ^δ _c ⊂ Ω +

^{withavariablethikness}

f δ (z)

(exaggeratedinthikness,showninblue). Wedenoteby

u ^δ

_±^{theeletrieldsoutsidethedepositlayer,with}

u ^δ

₋ ^on

Ω

− ^and

u ^δ ₊

^on

Ω + \ Ω ^δ _c

^{, andby}

u ^δ _c

^{thein-layereletrield, i.e. on}

Ω ^δ _c

^.

Tube Deposit

z

r

Coils

uδ− uδc uδ+

fδ(z) rt1 rt2

Γt₂ Γc Γt₁

Ωδ c

Ω− Ω+

Figure1: Representationof theeddyurrenttestingofathinlayerdeposit.

2.1 Resaled in-layer eddy urrent equation

Therststep isto rewritethein-layereddyurrentequationby resalingtheoordinatein thetransverse

diretion and theondutivity with respet to thelayerthikness. Theanalytialsolution of this resaled

equation allowsto get arelation between the boundary valuesof Dirihlet andNeumann type onthe two

longitudinalboundariesofthethinlayer.

Weassume thatthe thikness

f δ (z)

^writes

f δ (z) = δd(z)

^{, with}

δ > 0

^{asmall parameterand}

d(z) > 0

^a

dimensionlessquantity. Asweareonernedonlywiththe

Z ^1,n

⁽

n = 0, 1

⁾transmissiononditions(see[9℄), weresalethedepositondutivityby

σ c = σ 1 /δ

^{,wheretheresaledondutivity}

σ 1

^isonsiderablysmaller than

σ c

^{. Weresalealsotheoordinateinthetransversediretionofthedepositsbythehangeofvariable}

ρ = (r − r t

2

)/δ

^with

ρ ∈ [0, d(z)]

^{,and wedenote by}

u(ρ, z) := ˜ u ^δ _c (r t

2

+ δρ, z)

^{the resaledin-layersolution.}

Let

k 1

^besuhthat

(k 1 ) ² = iωµ c σ 1

^{. From(1) andtheresalingsettings,weobtain}

∂ _ρ ² u ˜ = − δ B 1 ¹ u ˜ − δ ² B 1 ² u ˜ − δ ³ B ³ 1 u ˜ − δ ⁴ B 1 ⁴ u, ˜

⁽³⁾

B ¹ 1 = 2ρ r t

2

∂ _ρ ² + 1 r t

2

∂ ρ + k ₁ ² , B 1 ² = ρ ²

r ² _t

₂

∂ _ρ ² + ρ

r ² _t

₂

∂ ρ − 1

r ² _t

₂

+ ∂ _z ² + 2ρ r t

2

k ² ₁ , B ³ 1 = 2ρ

r t

2

∂ _z ² + ρ ²

r _t ²

₂

k ² ₁ , B 1 ⁴ = ρ ² r ² _t

₂

∂ _z ² .

2.2 Transmission onditions between in-layer eld and elds outside the layer

Thetransmissiononditionsbetweentheeldinside thetube

u ^δ

₋ ^{andthein-layereld}

u ^δ _c

^are

u ^δ

₋

= u ^δ _c

^and

µ

⁻¹

_t ∂ r (ru ^δ

₋

) = µ

⁻¹

_c ∂ r (ru ^δ _c )

^on

Γ t

2

.

⁽⁴⁾

Thetransmissiononditionsbetween

u ^δ _c

^{andtheeldoutsidethedeposit layer}

u ^δ ₊

^write

u ^δ ₊ = u ^δ _c

^and

µ

⁻¹

_v ∂ n (ru ^δ ₊ ) = µ

⁻¹

_c ∂ n (ru ^δ _c )

^on

Γ c .

⁽⁵⁾

Withtheunit normalandtangentialvetorson

Γ c

^{at thepoint}

(r t

2

+ δd(z), z) n = (1, − δd

^′

(z))

p 1 + (δd

^′

(z)) ²

and

τ = (δd

^′

(z), 1) p 1 + (δd

^′

(z)) ² ,

werewrite(5) usingvetordeompositiononthesediretions

( n , τ ) u ^δ _c = u ^δ ₊

^and

∂ r (ru ^δ ) = µ c /µ v + (δd

^′

) ²

1 + (δd

^′

) ² ∂ r (ru ^δ ₊ ) + (1 − µ c /µ v ) δd

^′

1 + (δd

^′

) ² ∂ z (ru ^δ ₊ )

^on

Γ c .

⁽⁶⁾

Togetherwiththepartialdierentialequationforthein-layereld(3), thesetransmissiononditionson

bothsidesofthethin layerasboundaryonditionswillservetoestablishCauhyproblems whosesolutions

linktheeldsonbothsidesofthethin layer,thatisarelationshipbetween

u ^δ

₋ ^on

Γ t

2 ^and

u ^δ ₊

^on

Γ c

^{. Toget}

theeetivetransmissiononditionson

Γ t

2^{, wefurtherapplyaTaylorexpansionof}

u ^δ ₊

^.

2.3 Extension of

u ^δ ₊

^{using Taylor's expansions}

Togeteetivetransmissiononditionslinkingup

u ^δ

_±^{ontheinterfae}

Γ t

2 ^{betweentubeandthindeposit,we}

shouldextendtheeld

u ^δ ₊

^{,originallydened onlyon}

Ω + \ Ω c

^{,tothewholedomain}

Ω +

^{. Anaturalapproah}

would be to assumethat the extendedeld, still denoted by

u ^δ ₊

^{, satises the sameeddy urrentequation}

(1)as

u ^δ ₊

^on

Ω + \ Ω c

^{, i.e. with oeientsof thevauum}

µ = µ v

^,

σ = σ v = 0

^{. We remark that theright}

hand side of equation(1) vanishesin

Ω +

^sine

J

^{hassupport onlyin}

Ω

−^{. Usingthe variable} substitution

ν = r − r t

2^{,onerewrites(1)in}

Ω +

^as

X 4 j=0

ν ^j A j (ν∂ ν , ∂ z ) u ^δ ₊ = 0,

⁽⁷⁾

where

A 0 (ν∂ ν , ∂ z ) = (ν∂ ν ) ² − ν∂ ν , A 1 (ν∂ ν , ∂ z ) = 2 r t

2

(ν∂ ν ) ² − 1 r t

2

ν∂ ν , A ² (ν∂ ν , ∂ z ) = 1

r ² _t

₂

(ν∂ ν ) ² − 1

r _t ²

₂

+ ∂ _z ² , A ³ (ν∂ ν , ∂ z ) = 2 r t

2

∂ _z ² , A ⁴ (ν∂ ν , ∂ z ) = 1 r ² _t

₂

∂ _z ² .

Theasymptotiexpansionof

u ^δ ₊

^withrespetto

δ

^isintheform

u ^δ ₊ (r, z) = P

∞

n=0 δ ⁿ u ⁺ _n (r, z)

^{. Obviouslyeah}

term

u ⁺ _n (r, z)

^{veriesthesameequation(7). With Taylorseriesexpansion,onehas}

u ⁺ _n (r t

2

+ ν, z) =

X

∞

k=0

ν ^k u ⁺ _n,k (z)

^where

u ⁺ _n,k (z) = 1

k! ∂ _ν ^k u ⁺ _n

(r t

2

, z).

Sine

ν∂ ν (ν ^k u ⁺ _n,k (z)) = k(ν ^k u ⁺ _n,k (z))

^{, we an indeed write}

A ⁱ (ν∂ ν , ∂ z )

^as

A ⁱ (k, ∂ z )

^{while it is applied to}

(ν ^k u ⁺ _n,k (z))

^{. Thus,from(7)onehas}

P 4

j=0

P

∞

k=0 A j (k, ∂ z )(ν ^k+j u ⁺ _n,k ) = 0.

^{Theequalityatorder}

O (ν ^k )

^gives

A ⁰ (k, ∂ z ) u ⁺ _n,k = −

X 4 j=1

A ^j (k − j, ∂ z ) u ⁺ _n,k−j ,

with

u ⁺ _n,−1 = u ⁺ _n,−2 = u ⁺ _n,−3 = u ⁺ _n,−4 = 0

^{. Nowweonsider}

A 0 (k, ∂ z ) = k ² − k

^{. For}

k ≥ 2

^,

A 0 (k, ∂ z ) 6 = 0

^,

thus invertiblewithitsinverse

A

⁻¹

0 (k, ∂ z ) = _k

2

¹

−k^{. Sowehave}

u ⁺ _n,k = −A

⁻¹

0 (k, ∂ z )



 X 4 j=1

A ^j (k − j, ∂ z ) u ⁺ _n,k−j



 , k ≥ 2.

⁽⁸⁾

Nowweindutivelydenetwofamiliesofoperators

{S k ⁰ (∂ z ) , S k ¹ (∂ z ) }

^:

S 0 ⁰ := Id, S 0 ¹ := 0, S 1 ⁰ := 0, S 1 ¹ := Id,

k ≥ 2

 

 

 

 

 

 

 



S k ⁰ := −A

⁻¹

0 (k, ∂ z )



 X 4 j=1

A ^j (k − j, ∂ z ) S k−j ⁰ (∂ z )



 ,

S k ¹ := −A

⁻¹

0 (k, ∂ z )



 X 4 j=1

A ^j (k − j, ∂ z ) S k−j ¹ (∂ z )



 .

(9)

From the indution (8) one gets

u ⁺ _n,k (z) = S k ⁰ (∂ z ) u ⁺ _n (r t

2

, z) + S k ¹ (∂ z ) ∂ r u ⁺ _n (r t

2

, z)

^{. Therefore we havethe}

followingexpansion

 

 

 

 

 



u ⁺ _n (r t

2

+ ν, z) = X

∞

k=0

ν ^k

S k ⁰ (∂ z ) u ⁺ _n + S k ¹ (∂ z ) ∂ r u ⁺ _n

(r t

2

, z),

∂ r u ⁺ _n (r t

2

+ ν, z) = X

∞

k=0

ν ^k (k + 1)

S k+1 ⁰ (∂ z ) u ⁺ _n + S k+1 ¹ (∂ z ) ∂ r u ⁺ _n

(r t

2

, z).

Wealsodene theoperators

S e k ⁰ := S k ⁰ − r ¹

t

2

S k ¹

^and

S e k ¹ := _r ¹

_t

2

S k ¹

^{. ThentheaboveTaylorexpansionswrite}

 

 

 

 

 



u ⁺ _n (r t

2

+ ν, z) = X

∞

k=0

ν ^k

S e k ⁰ (∂ z ) u ⁺ _n + S e k ¹ ∂ r (ru ⁺ _n )

(r t

2

, z),

∂ r (ru ⁺ _n )(r t

2

+ ν, z) = X

∞

k=0

ν ^k (k + 1)

(r t

2

S e k+1 ⁰ + S e k ⁰ )u ⁺ _n + (r t

2

S e k+1 ¹ + S e k ¹ )∂ r (ru ⁺ _n )

(r t

2

, z).

(10)

Althoughonly asymptotiproblems oforder

0

^and

1

^{aredisussed in thesequel, theabove}expressionsfor anyordersouldfailitatefurtherexploitationsbeyondthesopeoftheurrentwork.

3 Asymptoti models for deposits with variable layer thikness

Inthissetion,webuild theasymptotimodelsoforder

0

^and

1

^{forthin deposits ofvariable thikness,that}

is to establishthe transmissiononditions

Z 1,n

^for

n = 0, 1

^{basedon the materialof the previoussetion.}

ThegeneralproedureonsistsinsolvinganalytiallyaCauhyproblemoftheresaledin-layereddyurrent

equationin itsasymptoti expansionform(3)withboundaryonditionson

Γ t

2 ^{(4). The resultingsolution}

yieldstheDirihletandNeumannboundaryvalueson

Γ c

^{whihshould mathwith} transmissiononditions (6). FinallywiththeTaylorexpansionsof

u ^δ ₊

^{(10) ,onegetstheeetive}transmissiononditionson

Γ t

2^.

Tosimplifythepresentation,espeiallytheomplexitiesintroduedbythetransmissiononditions(6)on

theurvedboundary

Γ c

^{,weassumethatthemagneti}permeabilityofthedepositsequalstothatofvauum

µ c = µ v

^{. Thisassumptionmathesthepratialappliation wehaveinmindsinetherelative}permeability ofopperis

1

^.

Weintroduethenotationforjumpandmeanvalueson

Γ t

2

[v] := v + | ^r

^t2

− v

−

| ^r

^t2

, [µ

⁻¹

∂ r (rv)] = µ

⁻¹

_v ∂ r (rv + ) | ^r

^t2

− µ

⁻¹

_t ∂ r (rv

−

) | ^r

^t2

, h v i := 1

2 v + | ^r

^t2

+ v

−

| ^r

^t2

h µ

⁻¹

∂ r (rv) i = 1 2

µ

⁻¹

v ∂ r (rv + ) | ^r

^t2

+ µ

⁻¹

t ∂ r (rv

−

) | ^r

^t2

.

3.1 Formal derivation of

Z ^{1 , 0}

transmission onditions We reall the asymptoti expansion

u ^δ ₊ = P

∞

n=0 δ ⁿ u ⁺ _n

^{and expand}

u(ρ, z) = ˜ P

∞

n=0 δ ⁿ u n (ρ, z)

^and

u ^δ

₋

= P

∞

n=0 δ ⁿ u

⁻

_n

^{. Theeddyurrentequation (3)andthe}transmissiononditions(4)on

Γ t

2 ^{fortherstterm}

u 0

yieldtheCauhyproblem

( ∂ _ρ ² u 0 = 0 ρ ∈ [0, d(z)], u 0 | ^ρ=0 = u

⁻

₀ | ^r

^t2

, ∂ ρ u 0 | ρ=0 = 0.

This problem hasaonstantsolution

u 0 (ρ, z) = u

⁻

₀ | r

t2

for

ρ ∈ [0, d(z)]

^. Consideringitsvalue at

ρ = d(z)

andthersttransmissiononditionof (6)for

u 0

^on

Γ c

^,weget

u

⁻

₀ | r

t2

= u ⁺ ₀ | r

t2

.

⁽¹¹⁾

TheCauhyproblemfor

u 1

^{withinitialvaluesgivenby(4)is}

 

 



 

 

∂ _ρ ² u 1 = −B 1 ¹ u 0 = − k ² ₁ u 0 | r

t2

ρ ∈ [0, d(z)], u 1 | ^ρ=0 = u

⁻

₁ | ^r

^t2

,

∂ ρ u 1 | ρ=0 = − 1 r t

2

u

⁻

₀ | ^r

^t2

+ 1 r t

2

µ c

µ t ∂ r (ru

⁻

₀ ) | ^r

^t2

.

Itfollowsthat

∂ ρ u 1 = − 1 r t

2

u

⁻

₀ | r

t2

− µ c

µ t

∂ r (ru

⁻

₀ ) | r

t2

− ρk ² ₁ u

⁻

₀ | r

t2

,

⁽¹²⁾

u 1 = u

⁻

₁ | ^r

^t2

− ρ

r t

2

u

⁻

₀ | ^r

^t2

− µ c

µ t

∂ r (ru

⁻

₀ ) | ^r

^t2

− ρ ²

2 k ² ₁ u

⁻

₀ | ^r

^t2

.

⁽¹³⁾

Theseond transmissiononditionof (6)for

u 1

^on

Γ c

^implies

∂ ρ u 1 | ρ=d(z) = − 1 r t

2

u ⁺ ₀ | ^r

^t2

− µ c

µ v ∂ r (ru ⁺ ₀ ) | ^r

^t2

.

⁽¹⁴⁾

Mathing(14)and(12)at

ρ = d(z)

^gives

1

r t

2

µ c

µ t

∂ r (ru

⁻

₀ ) | r

t2

− k ² ₁ d(z)u

⁻

₀ | r

t2

= 1 r t

2

µ c

µ v

∂ r (ru ⁺ ₀ ) | r

t2

.

⁽¹⁵⁾

Equalities(11),(15)andthefat that

k ² ₁ = iωσ 1 µ c

^implythat

[u 0 ] = 0

^and

[µ

⁻¹

∂ r (ru 0 )] = − iγ 1 h u 0 i ,

⁽¹⁶⁾

where

γ 1 = ωσ 1 d(z)r t

2

= ωσ c f δ (z)r t

2

.

3.2 Asymptoti model of order

0

If

u ^δ := u ^δ

_± ⁱⁿ

Ω

± ^{denote an}approximationoftheexatsolutionupto

O(δ)

^{error,thenfrom (16),possible}

transmissiononditionsanset as

[u ^δ ] = 0

^and

[µ

⁻¹

∂ r (ru ^δ )] = − iγ 1 h u ^δ i

⁽¹⁷⁾