Leaky axisymmetric modes in infinite clad rods. I
John A. Simmons
MetallurgyDivision,NationalInstituteofStandardsand Technology,
Gaithersburg,
Maryland 20899
E. Drescher-Krasicka
MetallurgyDivision,NationalInstituteof Standardsand Technology,
Gaithersburg,
Mary/and 20899 and
TheJohnsHopkinsUniversity,
Baltimore,Maryland 21218
H. N.G. Wadleya)
MetallurgyDivision,NationalInstituteofStandardsand Technology,
Gaithersburg,
Maryland 20899
(Received1 November1990;revised24 March 1992;accepted30 March 1992)
A detailedcomputationalstudyis presentedfor the radial-axial modes-- both leaky and
nonleaky--in an infinitelyclad isotropicrods.The complexphasevelocitiesof leaky modesare
locatedusingan applicationof the argumentprinciple.Particle orbitsare determined,and
leakymodesare shownto havean asymptoticleakageangleaway from the interface.By using
the homotopicmethodsof varyingdensitiesand elasticconstants,clad-rodmodesare
comparedwith thosein a barerod. The topologyof the clad-rodmodedispersiondiagram
differsqualitativelyfrom that of a bare rod, evenwhen the claddinghasnegligibledensity,with
no velocitycutoffsand with wave modeknitting. Comparisonis alsogivenwith modes
occurringin a claddingwithout a rod present(a tunnel) and for a planar interface.Most leaky
modescan be correlatedwith rod modes;only a limited number of tunnel modesexist. Energy
flow contourswithin modesare alsocalculated.The local energyvelocity,which generalizes
groupvelocity,can vary considerablyin the radial directionfor bare rod modes.For leaky
modesthe contoursare quite complexdue to the cylindricalgeometry,givingriseto apparent
shift in wavepositionacrossthe interface.
PACS numbers:43.20.Jr, 43.35.Cg, 43.40.Cw
LIST OF SYMBOLS
r,O,z
A c ,AR,B c ,BR
radial, polar, and axial coordinatesin
cylindrical coordinate system [Eq.
4-x]l -- v:/b•, 4-41- v:/b•
(1)]
displacement
fieldin radial-axialcoordinates
[Eq. ( 1) ]
u(r,z)
potential
functions
[Eq.(3)]
subscriptrn = C
subscriptrn = R
pc,ac,bc
Io,I1,Ko,K1
frequencyin radians [ Eq. (4) ]
(complex) wavenumber
[Eq. (4)]
cladding[ Eq. (5) ]
rod [Eq. (5) ]
cladding density, longitudinal and
shearwavespeeds[Eq. (5) ]
rod density, longitudinal and shear
wavespeeds[ Eq. (5) ]
complex leaky mode velocity= co/k
[Eq. (5) ]
modifiedBesselfunctions
[Eq. (5) ]
INTRODUCTION
The topicof normal modesin cladrodshasa substantial
historymuchof whichhasbeensummarized
by Thurston.1
In most cases attention
has been concentrated
on the use of
a)Presentaddress:
Departmentof Mechanical
Engineering,
Universityof
Virginia, Charlottesville,VA 22901.
1061
4-x/1-- v2/a•,-i-•/1 -- v2/a•,
tc
ro
f
p,a c,a n ,/3c,gb
qcr,qc•,qnr,qnz
[Eq. (5) ]
rok = 2•rrof /V [Eq. (5) ]
radiusof rod [ Eq. (6) ]
frequency= w/2•r [ Eq. (6) ]
Pc ac a• bcrof[Eq.
(7)]
pR
b•
b•
b•
b•
complex moduli for particle orbits
[Eq. (11)]
[E 1(r,2) ,g 2(r,2),
E3(r,z) ]
energy
velocity
field
=(.(P1)
(r,2)
{E)(r,z)
(P2}(r,
z){P3}(r,z))
(E)(r,z)'
(E)(r,z)'
[Eq.
(19)]
0L
asymptoticleakageangle
[Eq. (22)]
such structuresfor waveguides.Relatively little effort has
been expendedon the study of other modesthat could be
used for interface characterization
in fiber reinforced
com-
posites.For suchsystemsthe fibersare stifferthan the matrix and many of the modesare leaky, transmittingenergy
into the surroundingmedium.The existenceof this leakage
energyoffersa potentialmeansof monitoring,or evenimag-
J. Acoust.Soc.Am.92 (2), Pt. 1, August1992 0001-4966/92/081061-30500.80
@ 1992 AcousticalSocietyof America
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ing,thecharacteristics
oftheinterface
zone.Jen,etal.2have
specificallyaddressed
leaky modesin clad rods,but their
emphasis
hasalsobeentowardswaveguide
applications.
In the work reportedherewe studyin detailthe radialaxial modesin an infinitelycladisotropicrod. In spiteof the
simplicityof thegeometry,we showthat thetopologyof the
clad rod modesdiffersfrom that of a bare rod, evenwhen the
matrix densityis very small.
where/l andp are the Lam• elasticconstants.
In the fibergeometrycylindricalcoordinates
area more
convenientmeansof representation.Here, materialhomogeneitywith respectto axialtranslationandradialrotation
allowtheeigenfunctions
of thewaveoperatorto bewrittenin
the form F(r) exp(inO) exp(iwt- kz). A generaldynamic
isotropicelasticsolutioncanbeexpressed
in termsof potential functions in each material:
The topicof energyflow occupiesa significantpart of
this study.While the groupvelocity(dw/dk) hasbeenrou-
tinelycalculated
formodes
in thecylindrical
geometry,
3and
u = v• + v.T,
V.T = G,
(3)
train.
where G is an arbitrary function of r, 0, and z. It can be
shownthat only three of the four potential functionsin •
and T are independent,and that the r dependenceof these
can be expressedin terms of Besselfunctionsof, perhaps,
complexarguments.
In addition to (3) the boundaryconditionsdetermining
self-propagating
modesin a rod with finitelythick cladding
are continuityof normal tractionsand displacements
at the
rod/claddinginterfaceandvanishingsurfacetractionsat the
surfaceof the cladding.When the claddingisinfinitelythick,
thisleadsto a matrix equationof the form Mx = 0, whereM
is a 6 X 6 matrix dependingon phasevelocityand the poten-
I. THEORY
dition for solvingthis equationis that the determinantof M
A. Equation for axisymmetric modes
be zero.
whileSafaai-Jazi
et al.4 discuss
the overallPoyntingvector
field,it doesnot seemto havebeenrecognizedthat the cylindrical geometryintroducessignificantnonuniformityinto
theenergyflowfield.Evenin a bare-rodmode,whoseoverall
averageenergyvelocityis givenby the groupvelocity,the
localenergyvelocitycanvaryin theradialdirectionsomuch
as to changesign, delineatingregionsof reversedenergy
flow. In the caseof cladrods,thiscanleadto apparentshifts
in wave positionof severalwavelengths.Theseshiftsare
analogousto the Goos-H/inchenshiftsin optics,but they
will be derived herein without
resort to a Gaussian wave
tial functions, and x is a set of unknown constants.The con-
The equationgoverningthephasevelocityfor self-propagatinglinearelasticaxisymmetricmodesin an isotropicrod
with infinitecladdingfollowsfrom the more generalexpressionfor all modesin an isotropicrod with finitecladdingthat
appears
in severalplacesin theliterature.
1'3'•The dynamic
When there is no 0 dependence,i.e., when n = 0, and
when the claddingis infinite, i.e., when there are only four
boundaryconditions,oneof the components
of the potential
function becomes redundant, so that T reduces to (0, T,0)
andM becomesa 4 X 4 matrix. The potentialfunctionsthen
have the form:
elasticwaveoperator• maybewrittenin Cartesiancoordi-
d) = d)(r)e i(ø•'-•),
T = T(r)e •(•'- •).
nates:
,•/•(u)=.•-fx
• Cq•0x•--p8t----•-,
(1)
whereui are the displacement
components,
•5the massdensity, and the isotropicelasticconstantsare givenby
Cijk,
=Xaijakl
-3
L[.•(
ailajk
-3
Laikajl
),
,
(2t
The self-propagating
modesfrom (4) are the axisymmetric
modes.They are independentof 0 andhaveno torsional(or
transverseshearin the planar limit) component.The displacements
are,thus,onlyin the radialandaxialdirections.
The matrix equationtakesthe explicitform:
2psb•.4sll(.4sK)
2pcb•.4cKi(.• off)
2pc( 2b• -- v• )KI ( BcK)
pc{(• -- 2b•)Ko(.4cK
) - 2b•.4cKl(.4cK)/K
}
-- 2pcb•c{BcKo(Bc
K) •- Kl(BcK)/K}
(4)
ps(2b• -- 02)Ii(BsK)
a'
ps{(2b• -- o2)Io(AsK)
-- 2b•AsIi(AsK)/K} 2psb•{Bslo(BsK)
-- Ii(BRK')/K
'}
Ko(.4cK)
BcKo(BcK)
-- 4,(A•K)
.4cK I (.4cK)
Ki (BcK)
ARIi(ARK)
=0.
-Ii (BRK)
(5)
I
Here a', ]3', 7', •5'are constants--notall zero--to be discussed
in Sec.II B,Pm,am,andbmarethedensity,longitudinal, and shearwavespeedsof the rod (m = R ) and cladding
(m = C), respectively,v= w/k, is the complexvaluewhose
real part is the phasevelocityof the modeand whoseimaginary part describes
the growthor decayof the eigenwave,Io,
11,Ko,andK1aremodifiedBessel
functions
6'7and
z4m= -Jr( 1-- ve/a•)1/2
Bm= q- ( 1-- v:/b: ) ]/:
m
(6)
whererois the radiusof the rod andf is the frequency.This
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992
II B. Columns
1 and 2 refer to the contribution
in the clad-
ding, while columns3 and 4 refer to thosefrom the rod;
columns1 and 3 refer to longitudinalcontributionswhile
columns 2 and 4 refer to shear contributions; rows 1 and 2
refer to the balance of tractions while rows 3 and 4 refer to
the balanceof displacements;
and rows 1 and 3 refer to the
axialcomponentwhilerows2 and 4 referto the radial com-
•c= rok = roW/V= 2rrrof/V,
1062
equationis solvedby lookingfor valuesof v for which the
determinantof M vanishesas is explicitly discussedin Sec.
ponent.
Equation (5) is an equationfor v in termsof sevenparameters:
Pc, PR, ac, aR, bc, b•, androf The firsttwo parametershavedimensionsof densityand the last fiveparamSimmonseta/.' Clad rods. I
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etershavethe dimensionof velocity.By dividingthroughby
oneof the densitiesand oneof the velocitiessquared,Eq. (5)
can be reducedto a dependence
of a dimensionless
phase
velocityon five dimensionless
parameters.We shall frequentlyusesuchparameters
and,anticipating
thepredominant role of the rod modes,we shall designate:
P =Pc/P•,
tion exponentiallydecayingin the radial direction.Thus,for
such v's, the exponentiallydecaying (including Stoneleylike) wavesare type 3 modes;leaky modesarise from the
solutionsto any of the otherbranches.Becauseof the exponential increasein the radial direction, leaky modes have
infinite energy but if truncated outside of an energy flow
curve, as discussedbelow, such modes become wave beams
a• = a•/b•,
of finiteenergy,with boundeddisplacements
that areexponentially decayingaway from the flow curveand that almost
satisfythe waveequation.
/3c = bc/b•.
Other symmetriesof Eq. (5) could be exploitedto reWhen dealingeither with a bare rod or an infinite cladding
strict v to the first quadrant of the complexplane. If v is
with no rod present,Eq. (5) factors,and only the traction
replacedby •, its complexconjugate,the matrix M becomes
term need be retained. In that case,the dedimensionalizedr
M sincethe Besselfunctionshave this property. Furtheris a functionof only two dimensionless
parameters,e.g., v?b
more, v only occurssquaredexceptin combinationwith A m
is a function of rof/b and a/b (or Poisson'sratio). Even
or Bm (rn = C or R). Thus, if v is a solutionto the determiwhenp,theratioof thedensities,
isnearzeroor isverylarge,
nant equation,so is •; - v is alsoa solution,providedthat
the determinantof M essentiallyfactorsinto the productof
the modetype,s, is replacedby 3 - s. It is possibleto select,
two 2 X 2 subdeterminants.
Whenp is near0, thesetwo dethen, that solutionfor which both the real and imaginary
terminantsexpressthe vanishingof the tractionsat the surparts of v are non-negative.This givesrise to a (possibly)
face of the rod and the displacements
at the surfaceof the
attenuating(strictly, not growingin the z direction) mode
cladding,while for very largep, the two subdeterminants
with positivephasevelocity. However, as will be shown,
describethe vanishingof the tractionsat the claddinginterthere are certain "backward-leakingmodes"whoseenergy
face and the vanishingof the displacements
at the rod surflow in the rod isin the reversedirectionto the phasevelocity
ac = ac/b•,
qb= rof /b•,
(7)
face. In thesecases,either solutionsto the traction-free Neu-
mann problem or the dual displacement-freeDirichlet
problemwill yield acceptablevaluesof v. The closerelationship betweenthe infinitely many eigenvaluesfor the Neumannand Dirichlet problemsfor the rod and the very limited numberof solutionsfor the claddingcan be used,aswill
be seen,to qualitativelyexplain the closecorrespondence
between the modes created on bare and clad rods.
Becauseof the possibilityof usingeithera plusor minus
branchfor eachof the squarerootsin (6), Eq. (5) appearsto
be sixteendifferentequations.However,the modifiedBessel
functionsusedto representthe solidrod, Io and I•, are, respectively,evenand odd functionsof their argument.Since
I1 alwaysoccursmultiplyinganA mor Bin, oneseesthat the
plus or minus squareroots associatedto the rod have no
effecton thedeterminantequation.Thus,thereareonly four
distinctequations(leadingto fourbranches
) for axisymmetric modesin the rod with infinite cladding.Physicallythis
occurs because the solution must be finite at the center of the
rod, thuseliminatingthreetypesof divergentsolutionsthat
couldoccurif the rod werehollowor if oneweredealingwith
a planarinterface.
Sinceall the axisymmetricbrancheswill be frequently
used,the four branchessolvingEq. (5) will be labeledwith
an abbreviatedbinary notation, referring to a mode as an s
mode, wheres - 2p q- q goesfrom 0 to 3 and p and q are
either0 or 1accordingto the choiceof the negativeprincipal
squareroot value (0) or positiveprincipalsquareroot value
( 1) for the shear(p) or longitudinal(q) component,respectively. A 2 mode would then arise by choosing
at the interface (such modeshave also been discussedin the
contextof theopticsof planarinterfacesS).
For suchmodes
the phasevelocityis negativeeventhough the attenuation
occursin the positivez direction;the useof a positivephase
velocityand type 3-smodefor thesecasesreversesthe sense
of time, producingan "absorbing"rather than "leaking"
mode.Our convention,then, is to usea non-negativeimaginary componentfor v, precludingexponentialgrowthalong
the interfacein the positivez directionand, at largedistances
from the interface,to only permit energyto flow parallel to,
or away from, the interface.
B. Expressions for particle displacement
The determinantof the matrix M that occursin Eq. (5)
hasa rangein its orderof magnitudewhichis prohibitively
large. In order to solvethis equationover a wide rangeof
parameters,we make use of the asymptoticexpressions
(Reft 6, p. 203):
io(•.)•
q-ie
e
-_gg
e•
g+ie'
2 < argO<
2'
•-1/2
T/'
3rr
I1(•')•" •172
Km(•) • e- •1/2,
m • 0,].
(8)
We note that the exponentialbehavior of the I•'s can be
accommodated
by the useof cosh(•) and we divideeachof
the columnsof M by an appropriateexponentialfactorwhile
dividingtheentiredeterminant
by•/B •. Thelatterfactor
Bc = ( 1-- o2/b• )1/2andAc= -- ( 1-- o2/a•)1/2.Ascan
removesthe squareroot of z dependenceof the Besselfunc-
beseenfromthe asymptoticbehaviorof the Besselfunctions,
[ see(10) ] if o liesin the firstquadrantof the complexplane,
the negativeprincipalsquareroot value (0) givesriseto a
solution exponentiallyincreasingin the radial direction,
while a positiveprincipalsquareroot value ( 1) yieldsa solu-
tionswhiletheB • termremoves
a spurious
v -- B• root.
1063
J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992
These operationshave no effecton the phasevelocity
determinedfrom the vanishingof the determinant,but they
multiply eachof the coe•cientsa• - g• (which are obtained
as 3 • 3 minorsof M acrossany row of M) sothat:
Simmonsotal.: Clad rods. I
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at= Car'to
1/2exp(ActC),
/3= c/3'tc1/2exp(Bctc),
7/= c•'•d/2cosh
(A• •c),
6 = c6'•c
1/2cosh(Ba•c).
whose
ellipticalorbitshaveaxesalwaysoriented
paralleland
perpendicular
to thez axis,theellipseorientation
for leaky
(9)
modeschangesasa functionof r (but not z), andis inclined
at angles0m and 0m _ •r/2, where
Om=
----1
tan-1
Im( 2
The complexfactor c only affectsan arbitraryuniform
change
of magnitude
andstartingphase,
sothatat anyvalue
of r, theexpression
for thedisplacement
will bein theform:
Um,(r,z,t)= Re(qm,e
i(ø't-•'•)) ,
Um•
(r,z,t)= Re(qm•ei(ø't-•'•)),
+ il•
tc•/2exp(-- Acre)
(10)
,
td/2exp(-- Bctc)
qcz
= --flkBcKø(
Bctcr/rø
) -- ia kKø(
Actcr/rø
)
•c1/2exp(-- Bc•c)
2
(Re(q2•z
qmz
-[qmr
))'
+
q2m•)
(12)
The directionof rotationaboutan ellipticalorbit can
alsochange
withr. Thischange
ofdirectionissignaled
bythe
minororbitof theellipsedegenerating
to zero,i.e., inclined
rectilinearmotion.Suchchanges
arealsoa commonfeature
of ordinaryrod modesand Stoneleywaves.
whereRe(•) istherealpartof • andwhere
qc• = -- a
2
tc1/2exp( -- ActC)
Examinationof the determinantin Eq. (5) alsoshows
that the root v = b• existsfor everyvalueof rof. This root
comes from column 4 of the matrix and arises because of the
asymptoticexpansions
of Io (z) --. 1 and I1 (z) • z. The mode
determinedby this root is identically zero, so we shall removeit by dividingthe determinantby B•.
C. Limiting planar interface modes
(11)
Whentheinterface
isplanar,translational
symmetry
in
thetwoorthogonal
directions
parallelto theinterface
gives
riseto potentials:
The particleorbitsare thenellipticalwith Iqmrland
Iqa•I beingthe magnitudes
of the maximumexcursions
in
ther andz directions,
respectively.
Thenegative
phases
of
theq'sdetermine
thepositivephaseat whichthismaximum
q•= a exp( -- kAcx+ icot- ikz),
ß = • exp( - kBcx+ icot- ikz)
(13a)
in the claddingand
Occurs.
We haveobserved
in Eq. (5) thatv isa functionof rof
(or v/ba isa functionof • = rof/ba ). FromEq. ( 11) one
canseethatif rof isfixed(sothatbothtoandk varyas1/to),
thenroqmr
androqmz
arefunctions
of r/to.
Unlike ordinary modeswhosevelocitiesare real and
ß= 7/exp(
--kAax
+ icotikz),
ß = 6 exp( - kBax+ icot-- ikz)
(13b)
in the rod.
The boundaryconditionmatrixequationthenhasthe
form:
=0.
(14)
All sixteen
branches
ofthematrixequation
canproduce froma rod,whilebothmodes
correspond
tothelimitingcase
solutions
to theplanarinterfaceproblem.Consequently
we obtainedfromthe tunnel-shaped
claddingwith no rodinusea doubleindexnotationfor a planarinterfacemode,referringto s,tmodes,wheresandt rangefrom0 to 3 asin the
side.
cylindrical convention.The first index will refer to the cladding materialand the secondindexto the rod material.The
approaches
infinitywhilethe frequencyfremainsfixed,the
results
mightbeexpected
to converge
to thoseof theplanar
case.Thisis not so,however,
asthereare infinitelymany
modes
in thecylindrical
casefor whichtherearenoplanar
planarinterface
equation
hastheadditional
symmetry
thatif
v isa solutionfor ans,tmode,it isalsoa solutionfor a 3 - s,
3 - t mode.Onlythosemodesfor whichs or t equals3 can
belimitingcases
fromthecylindricalgeometry.
Thus,for example,althoughthebarerod solution,corresponding
tOpc = 0, hasonlyonebranchdueto theboundednatureof therod,thecorresponding
limitingplanarsolu-
Forthecylindrical
case,
whentheradiusofcurvature
ro
analog.
Thekeyto thisdifficultyliesin thefactthatIo andI•
havedifferentasymptotic
behaviors
asseenin (10). Exami-
nationofthecolumns
ofM in Eq. (5) shows
thatKoandKl
bothappear
in somecolumns
whileIo andI1 bothappear
in
tion has two branches--the 0 or 3 mode and the 1 or 2 mode.
othercolumns.The first-orderasymptotic
expansions
of
Only the 3 modecorresponds
to the limitingcaseobtained Ko(•) exp(•)/(•) 1/2 and K• (•) exp(•)/(•)1/2 are the
1064
J. Acoust.
Soc.Am.,Vol.92, No.2, Pt.1, August1992
Simmons
ota/.:Cladrods.I
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same, but the first-order asymptotic expansions of
Io(•)/(•) 1/2cosh(•) andI1(•)/(•)1/2 cosh(•) havethe
forms [ 1 + i exp( - 2•) ]/[ 1 q- exp( - 2•) ]. In orderfor
theseexpressions
to agree,then:
Re(•) >•1.
(15)
This conditionsuffices
to ensurethat a particularcylindrical
modeis convergingto a planarinterfacemode.
If Eq. (15) holds,a simplecorrespondence
canbe made
betweenthe a- 6 constantsin Eq. (8) and thosein Eq.
(13). One can seethat for large rof, the cosh(•) termsbecomeessentiallyexp(•) and the complexdisplacement
amplitudesqmrand qmzhavea local r dependencedominated
byexp[ cf(r -- ro)], wherecisa constant
depending
onlyon
v and, therefore,essentiallyindependentof rof Thus, for
planartypemodes,iffis heldfixedandrotendsto infinity,
theorbitshapedepends
onlyonthedistance
fromtheinterface,asin theplanarcase.However,for anymodeif rofis
heldfixedat somevalue,not necessarily
large,whilef and ro
arevariedreciprocally,
then,exceptfor an arbitraryamplitudeconstant,
theq'sdependonr/ro,i.e.,reciprocally
onfor
directly with "wavelength."
Planar interfacemodesexist whosephasevelocity exceedsbothaR and ac. The exponentialform of the displacementsthen requiresthat thesemodesshouldconsistof the
superposition
of a longitudinaland a shearplane wave in
each medium. Such modes,which we may call decomposablemodes,then consistof a specialplanewavereflection/
refractionsituation in which at least one input or output
planewaveis missing.
D. Energy velocity
of eigenfunctions
in termsofexp(icot-- kz), wherek maybe
complex for leaky waves, allows us to representall field
quantities(•ru,ui,P
i,E,etc.) in the formf (r,z,O), where
0 = cot- kRz and k• is the real part of k. Heref is periodic
in 0 with period2zr.We thendefinethe time-averagedquantity:
!Or)=
f ( r,z,O)dO.
(18)
Sincespatialdifferentiationisinterchangeable
with time
integration,Eq. (16) holdsfor the time-averagedPoynting
vector.
Thevectorwithcomponents
(P•)/(Po) defines
a velocityfieldE, calledthe energypropagationvelocity(or simply energyvelocity) field:
E•(r,z) =
(P•(r,z))
.
(Po(r,z))
(19)
For the clad rod, the cylindrical componentsof the
space-timeenergyflow vectorenergyvelocitymay be calculated:
=0,
(Pr) =pw exp[2zIm(k)] [a2Im(F:/;qr)
q-(a2-- 2b•)
X Re(kFtrqz)+b2Re(kqr7qz)
+b2Im(O•qz)],
(Pz)=pw exp[2zIm(k) ] [a2Re(k)lqzl'•
+ (a2- 2b2)
X Im(F:/;qz)
+ b2Re(k)lqrl
2q-b2Im(7q•qr)
],
(Po)= (Ekin).ql_
(Epo
t)
=pexp[2z
Im(k)](q
(Iqrl
2+Iqzl
•)
a2[q;.
[2d-Ikqz
b2 24-Ikqr
d-T(
I2) 4--(Iql
I2)
The elasticpower(EP) flowin andout of an arbitrary
volume Vis describedby the Poyntingvector:
+ (a2- 2b2) Im(k•;qz) + b2Im(k•qr )
•- d•
(Co•u,,•u•,•
+p•oi•,•
) dx
2b
(•q•.qr)
q'2b
r2Re
r2
(20)
=f•,(P6oii•
- Co•lU•,l•
)iti
dx
-l-fa•,(roit,
ds• Here, the qr and qz coefficientsare thosefor the appropriate
mediumasgivenin Eq. ( 11) but with theasymptoticcorrection terms removed.The symbol" "represents complex
-fa•oh•ds•fo
•ds•.
(16)conjugate,
q• = dqz/dr and q• = dqr/dr. If rofis held conv
v
Here, •r is the stressand c•V is the boundaryof V.
The firstexpression
of the secondequationof (16) con-
tains• (u) and,consequently,
vanishes.
Thecomponent
P•.
describesthe energyper unit area and unit time flowing
througha surfacewith normalin thex• direction.To the
threecomponents
P•.ofdimension
l - 2t- l, wecaninclude
the energydensityE asa fourth componentPo.This compo-
nenthasdimension
l - 3andcanbe thoughtof asthe time
stant,thenbothfro
q• andfro
q• arefunctions
of r?ro,sothat
r• P is a functionof r?ro.The energyvelocityis determined
by (20).
The conceptof energyvelocityhasbeenusedin electromagnetictheoryaswell asbeingintroducedfor planewaves
in elasticityand fluid mechanics,where it is seento agree
9
1
with the groupvelocity.- 2 However,whilethe energyve-
locityis constantand coincideswith the groupvelocityfor a
planewave,the energyvelocityfieldbecomes
variablewhen
the translationalsymmetryof the modeis limited. For axisymmetricleaky modes,for instance,the energyvelocity
field is a functionof r (the exponentialdecaysin the Pi's
Equation(16) thenstatesthat in a conservative
system cancelingout the z dependence).It hasbeenshownfor rod
P isdivergence-free,
P;j = 0.In thecase
ofamonochromaticmodesthat the groupvelocityin the axial directionis the
elasticmode,it becomessimplerand moreconvenientto use quotientof the total time averagedenergyflowoverthe rod
energyquantitieswhich havebeentime-averagedover one crosssectionwith the total time-averagedenergydensity
period.To thisend,we notethat thegeneralrepresentation overthe rod crosssection.13Thus, evenfor modesin a bare
componentof a space-timeenergyflow vectorP:
j=0,
P;= E,
tro/t,,
j = 1,2,3.
1065
J. Acoust.Soc.Am.,Vol.92, No.2, Pt. 1, August1992
(17)
Simmons
eta/.' Cladrods.I
1065
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rod, the more physicallyfundamentalenergyvelocityfield
displayslocalinformationnot containedin the groupvelocity. Indeed, the energyvelocityfield can vary with r by an
order of magnitudefor somerod modes.This resultscan
easilybe extendedto type 3 modesin a clad rod which have
real k, wherethe energyvelocityat largedistancesfrom the
cylinderaxisis asymptoticallyequalto the phasevelocityof
the mode.The situationfor leaky wavesis more complex,
andwhiletheenergyvelocityretainsitsphysicalvalidity,the
physicalinterpretationof groupvelocityis unclear.Yet we
areunawareof any work utilizingthe variableaspectsof the
energyvelocityfor monochromaticmodeswith fixedwave
number.
If onedrawsanarrowwithcomponents
E• throughout
a
medium,thesearrowsnot only point in the directionof maximum energyflow, but no energyflowsperpendicularto the
arrows (henceforthflowswill be interpretedas time-averagedovera period). Thusif onedrawsa smallcircle,or other
closedsurface,at time to, one can use the energyvelocity
fieldtogetherwith the outerperimeterof the circleto develop a tubularsurfaceinsideof whichenergyis conservedand
whosevelocityis governedby the energyvelocitywithin the
tube.An illustrationof thisisshownin Fig. 1 for the caseof a
wave mode in a rod enteringa clad half-space.Sincethe
techniques
we employhereare only valid for a clad rod system of infinite axial extent, it is assumedthat there is a small
reactionzonenearthe edgeof the claddingfollowingwhich
the rod mode is convertedinto a number of leaky modes,
only oneof which is shown.The energyflow curvesfor the
leaky modeare obtainedby integratingthe energyvelocity
(or Poynting vector) field. Translationalmaterial homogeneityrequiresthat all flow curvesshouldbe congruent
undershifting,althoughthe amountof energybetweenany
two equallyspacedcurves(actually surfacesof revolution
about the cylinder axis) decreasesexponentiallywith increasingz. Sincethe boundaryconditionsrequirethat Pn,
the normalcomponentof P, be continuousacrossthe interface,the discontinuityin directionat the interfaceisdictated
by the discontinuityin the tangentialcomponentof P. When
Pis moreparallelto theinterface,the flowandenergydensitiesare increasedby the factor 1/Pn. In the caseof nonleaking modes,all energyflow vectorsare parallelto the interface, and this area correction is unnecessary.For leaky
modes,however,theenergyflowcurvesbendawayfrom the
axis and becomeasymptoticallyinclinedat some"leakage
angle,"aL to thecylinderaxis.(Uniquenessandshiftinvariance of the flow curvesprecludea changein sign of slope
alongthe flowcurve.) This angleandthe associated
limiting
energyvelocitycan be found from Eq. (20) by usingthe
dominantasymptoticform of the q'sat largevaluesof r. The
two asymptoticgrowth exponentsof the q's are given by
Re(kAc ) andRe(kBc). Thus,if Re(kAc ) < Re(kBc),
the substitutions
qr •-'
q;
( _
c ),
qz'"'-'
--i,
(21a)
q; • ikAc,
maybe used,whileif Re(kAc ) > Re(kBc ), the substitutions
qr""--i,
q;'"-'-- ikBc,
qz'•--Bc,
(2lb)
q;.-..k ( 1 -- v•/b• )
are appropriate.
If Re(kAc ) = Re(kBc ), or if the dominant growth exponentis zero and v is real, the asymptotic
form is ambiguous.
It is usefulto simplifythe expression
for the asymptotic
leakageangle.After somealgebraicmanipulationit can be
shownthat if the conditionsin (21a) are applicable,
tan 0L = Im(k)/Re(kAc),
(22a)
while if thosein (21b) are applicable,
tan 0• = Im (k)/Re ( kBc),
(22b)
where0• is the asymptoticleakageangle. [The asymptotic
expressions(22) can alsobe found by a differentargument:
The asymptoticamplitudeof leaky wavesexponentiallyincreasesawayfrom the cylinderaxiswith an exponentlinearly dependenton r, while exponentiallydecayingwith an expon.entz Im (k) alongthe cylinderaxis.The limitingleakage
anglea• isthat at whichthesetwo exponentialtrendsexact-
ly canceleach other.] In the specialcasethat I Ira(k) I
'•l Re(k)[ [or [ Im(v)],•[ Re(v)l], wehave
0•0, forRe(v)<a
c
tan OL•[Ac[,
when Re(v)>
(23a)
for Re(v) >ac,
=0 and conditions (21a) are applicable,
while
FIG. 1.Energyflowcurves(a, b, c, c', andd ) for a leakywavestartingat the
axis-perpendicular
edgeof thecladding.Curvesa, b, c, andd arerotationally equivalent.Curve c', which showsthe energyflow startingnearerthe
edgeof therod,istranslationallycongruentto flowcurvec. This isindicated
by the shiftedinnercircle.The self-parallelfeatureand the propagation
of
energytubesare shown.
1066
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
0•0,
for Re(v)<bc
(23a)
tan 0• -• IBc[, for Re(v) > bc,
when(22b)applies.
{Strictlyspeaking
theexpression
asderived for Re(v)<ac
(or bc) requires that [ Im(v)/
Simmonseta/.: Clad rods.I
1066
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Re(v)],•(1 -- [ac/Re(v)]2} or {1 -- [bc/Re(v)]2},
but thislatter inequalityis rather forgiving,especiallywhen
Re(v) is near ac or bc in which casethe leakageangleis
flow lines---oscillate
with r, indicatingthe interactionof the
individualplanewaves.
smallanyway.
) Theseexpressions
pointtowards
thedistinc-
E. Locating leaky mode branches and calculating
tion betweenleakingcylindricalwavesat high valuesof •
[definedin Eq. (?)] havingphasevelocitieswith vanishingly smallimaginarypart and possibleplanarinterfacelim-
dispersion curves
its where v would be real. This distinction will be discussed
below. They also show the need in computationto avoid
underflowingthe imaginarypart of v to zero.
If one draws an energy flow curve starting from the
junctionof the rod andthe half-space,therecanbeno energy
detectedbetweenthe edgeof the half-spaceand this limiting
curveexceptfor a fringefield associated
to elasticnonequilibrium alongthe limiting curve (as couldalsobe seenfrom
the nontime-averaged
energyflow). The lacunathusformed
is a characteristicof leaky modesand offersone experimen-
tal meansfor their detection.
TM
For valuesof z beyondthe
limiting flow curve,the waveamplitudedecreases
exponentially for any fixed r. (Since there may be a "mixing zone"
nearthejunction,thepeakenergymaybeshiftedslightlyin a
way that cannot be predictedfrom this theory. The major
effectof thisshiftwill beon the apparentasymptoticleakage
angle.If required,it canbe removedby usingpeakmeasurements at two different valuesof r with the sametransmitter. )
Becauseof interferenceeffectsbetweenshearand longitudinal components,the energyflow curveis not a straight
line and fluctuates in some manner near the interface. If one
wereto draw a straightline inclinedat crLfrom the intersection of the rod with the half-spaceto any distancer, this line
wouldbeshiftedsomedistancefrom the limitingenergyflow
curve.If rofis fixed,the dependence
of (P) on r/ro shows
that the magnitudeof this shift will vary inverselywith f.
Sucha shift alsooccurswith leaky modesat planar interfaces.It is particularlyilluminatingto think of a mode
whereone of the type indicesis 3 and the other is 0, 1, or 2,
sincethesemodeshave leaky cylindrical analogs.In such
cases,thereis alsoa limiting "approach"anglein the type 3
medium with an associatedasymptotic approach wave.
Generallythiswaveis of mixedtype,but if the energyvelocity is closeto eithera• or b•, the modewill be essentiallya
planewaveof longitudinalor sheartype. Near the interface
inhomogeneous
interfacewavesare generatedon eitherside
of the interface, and the interference of thesewaves with the
asymptotic incoming and outgoing waves producesthe
bendingof the energyflowlines,with a characteristic
shiftof
the expectedpositionof the asymptoticoutput wave. This
shift,whichis inversefrequencydependent,is analogousto
the Goos-Hanchen shift in opticsor the Schocheffect at
liquid-solidinterfaces.
8'is(When severalleakymodesare
simultaneouslypresent, they will interfere and produce
changesin the energy flow curves,which will affect the
amountof the shift.However,for many casesof experimental interestthe exponentialdecaywith z and the relativeinsensitivityof the flow curveto smalldisturbancesallow usto
treat the flow curvesof the leaky modesindividually.)
In the caseof decomposable
modesrepresenting
a plane
wavescatteringreactionat a planarinterface,the vectorP•
whichrepresentsthe total energyflow•as well asthe energy
1067
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
Individualmodesarefoundassolutions
to theequatio•
IM(v) I =0
(24)
from Eq. (5) as modifiedby the asymptoticexpression
in
(9). The modifiedBesselfunctionsoccurringthereinhavea
branchcut alongthe negativereal axis and the coshfunc-
tionshavezerosalongtheimaginary
axis.Thus,IM (v) [ will
be analyticinsidethe firstquadrantof thecomplexv plane.
Complexrootsmay thenbe locatedby applyingthe "argumentprinciple":•6
1f f.(,)
# zeros
inside
F,
f(v) ao=
2rci
(25)
whereF is anyclosedcurvein the firstquadrant.Real roots
and pure imaginaryrootsof (24) can be foundby ordinary
root searchtechniquesfor functionsof a real variable.
Newton'smethodwasusedfor computingeitherreal or
complexrootsto (24) oncethe approximatezerowaslocated.Root accuracydependson theboundsof Newton'smethod, but wasalwaysgreaterthan eightplaces.
At chosenfixedvaluesof• = rof/b•, an automatedimplementation
of theargumentprinciplewasemployed
to detect roots where
0.01<•<10.
(26)
Similar bounds were used for root searches on the real and
imaginaryaxes.
Oncean individualroot wasfound,the dispersioncurve
for the mode branch containingthat root was obtainedby
varyingrofover a desiredrangeand employinga Newton's
method algorithm with forward interpolation,which was
doublenested
to allowthreeordersofmagnitude
in 1(rof) to
squeezethroughdifficultspotsin the dispersioncurve.This
method was sufficientlyrobust that one could follow the
dispersioncurveof a rod modenear the cutoffto a phase
velocityof 1000timesthat of the longitudinalwave.
The roots associatedto numerousvaluesof rof were
found, the dispersioncurves traced and the redundant
curveseliminated.Nonetheless,
thereis alwaysthe possibility that individual anomalousmodeswere overlooked.
Oncethe root v is found,the real part of v directlygives
the mode'sphasevelocityalongthe interface.In dimensionlessform, this velocityis v = Re(v/b• ). The attenuation
alongtheinterfaceisdetermined
fromtheimaginarypartof
the wavenumberk and can be expressed
in dimensionless
form in dB by
attenuation= log10(e) 40•r Im (v)
-•40•rlOglo(e)b•Im(v)/[vl2.
(27)
The units of b• are in distancetimes frequency,and the
dimensionalform of the attenuation(without the b• ) expresses
the fact that the attenuationdependslinearlyon frequencyas well as on distancetraveled down the interface.
This attenuation
could also be dedimensionalized
to mea-
Simmonset al.: Clad rods. I
1067
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suredistancealongthe interfacein termsof unitsof %, but
sucha form wouldnot be extendibleto the planarinterface,
so the units will be left in thoselaboratoryunits usedto
expressbe.
In orderto assess
the effectsof materialparameterson
leakymodes,a nestedNewton'smethodwasusedto vary an
individualelasticconstantor densityor groupsof suchparameters.By meansof theseparameterhomotopics,for instance,it was possibleto connectleaky modeswith rod
modesaswell as with modesin the claddingwithout a rod
isnot zero,mostleakymodesare leakyrod modes.Figures2
and 3 showthe dispersionrelationsfor the caseof a silicon
carbiderod in an aluminum cladding.The plots in these
figureshave been givenin dimensionedform. The dimen-
sionless
parametersfor thisexamplearethenp -- 0.866,ac
-- 3.21,•c -- 1.68,as -- 1.92. All bare, real, or partly real
SiC rod modeswith cutoff value of •b- rof/b• <6 are
shownasdottedlines.The associatedleaky modesare plotted as solidor dashedlines.The dispersioncurvesin these
figureshave been arbitrarily truncatedat their maximum
present.
valuesfor clarity of graphicalpresentation;they actually
continueto a valueof •b--0.
This codealsotestsfor changeof modetype.The comThe type0 modesarethemostabundantandareshown
plex velocityv hasbeenchosensothat Im(v)>•0, implying
that thewavecannotexponentially
growalongtheinterface, in Fig. 2. As canbeseen,thereisonesuchmodefor eachrod
nor can energyflow into the interface.However, at certain
mode.In Fig. 3 areshownbothtype1andtype2 modes.The
values•b,the characterof a modecanchange.This can only
numberof type1 andtype2 modescombinedin thisfigureis
happenwhenA • or B • arereal,andwhenthe displace- onemorethan the numberof rod modes.Theserelationships
will be discussed below.
ments do not jump discontinuously.In particular, the
The modesthat exist in a bare rod have beenstudiedby
asymptoticdisplacements
cannotundergoa changeof sign
Onoeet al.'7 and reviewedby Thurston.'In the studyby
in the real part of the exponent,whichmustthenbe zeroat a
Onoe,the frequency(dimensionless)
wasappropriatelycalbranchchange.Sincetheseexponents
areproportionalto the
culated asa function of wave number (dimensionless),rathrealpartsofA c/v andBc/v, andsinceBc < Ac, wehavethat
frequencyas
1/v2mustberealandthat2/v2< 1/A • isa necessary
condi- er than the velocityversusradius-compensated
is
done
herein.
From
the
v
vs
•b
point
of
view,
rod modes
tion for a changeof mode.Dispersioncurvessatisfyingthis
show
a
cutoff
at
k
-0
(infinite
velocity)
or
df/dk
- 0. Beconditionarecheckedfor modechangesat apparentcutoffs.
yondthe k -- 0 cutoffthe velocitybecomespureimaginary
asdoesk. The df/dk - 0 cutoffrepresents
a nodein k space
II. RESULTS
at whicha complexbranchcontactseithera pureimaginary
A. Dispersion curves
or purerealbranchat a localminimum.Only therealminima havebeenstudiedherein, and in that casethe complex
Sincea clad rod consistsof the conjunctionof a barerod
branchhasbeenconnected
with that part of thew vsk curve
with an infinite cladding,it seemsnatural to seekthe relathat continuesdown to where k becomeszero, since the distionshipbetweenleaky modesand the modesthat existin a
continuityin slopeislessandthatisthepathfollowedby the
bare rod or in an infinite material with a cylindrical hole
(tunnel modes).
Newton'smethodalgorithm.In Figs.2 and3 therealpartof
thelowestsuchcomplexrodbranchhasbeenincluded,since
Qualitativelyspeaking,aslongasthe densityof the rod
20.0
o
:
:
...
o
0.0
4.0
8.0
12.0
16.0
20.0
2ti. 0
28.0
32.0
36.0
•0.0
Radius X Frequency (mm/•s)
FIG. 2. Dispersioncurvesfor the modalphasevelocityfor type0 leakymodesand rod modesas a functionof rod radiustimesfrequency(tof) in the
aluminum-silicon
carbidesystem(aluminum:
density- 2.77g/cm3, ac --6.323 mm//•s,bc -- 3.1 mm//•s.Siliconcarbide:
density
-- 3.2 g/½m
3, aR
-- 9.649mm//•s,bR ----5.193mm//•s). The leakymodeshavebeentruncatedat theirpeakvaluefor clarity.The extra-widespaceddottedrod modeis the
complexpart of the secondrod mode.
1068
J. Acoust.Soc.Am.,Vol. 92, No. 2, Pt. 1, August1992
Simmonseta/.: Clad rods I
1068
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25.0
20.0
• •5.0
•
•0 0
5.0
0.0
0
0
I•. 0
8.0
12.0
!6.0
20.0
2U,. 0
28.0
32.0
36.0
U,0
Radius X Frequency(mm/•s)
FIG. 3. Dispersioncurvesfor themodelphasevelocityfor types1and2 leakymodesaswell asrod modesasa functionof rodradiustimesfrequency(raf) in
the aluminum-silicon
carbidesystem(aluminum:density= 2.77g/½m3,ac --6.323 mm//•s,bc = 3.1 mm//•s,siliconcarbide:density= 3.2 g/cm, ac
= 9.649mm//•s, bc = 5.193mm//•s). The leakymodeshavebeentruncatedat their peakvaluefor clarity.The extra-widespaceddottedrod modeis the
complexpart of the secondrod mode.
it seemsto berelatedto the lowesttype0 leakymode.For the
siliconcarbiderod, only two df/dk minima were found.
Rod modes have been classifiedinto two kinds by the
function equations,are approximatelyequally spaced;and
near a Poissonratio of 1/3, there are about twice as many
axial modesas radial modes.Using a homotopictechnique
particlemotionat thecutofffrequency.'
In theonekindof
on densitiesto be discussedbelow, one can show that, with
modethe particlemotion is radial, while in the secondit is
axial. (While this particlemotion existsin the limit, it is not
very usefulasa practicalmodesignature,evenin thesebare
rod modes,sinceto obtain elliptical trajectorieswith an aspectratio morethan,say,10-1 in the radialor axialdirection
for all valuesof r oftenrequiresa phasevelocityof morethan
100bR.) For eachof the two kinds of rod modes,the cutoff
valuesof •b,which are solutionsof transcendentalBessel
the exceptionof one type 1 leaky mode, in the rangesof raf
somewhatabove the cutoff value for the appropriaterod
mode,the type 1 leaky modesin Fig. 3 correspondto radial
modes and the type 2 leaky modes correspondto axial
modes.
Figures4 and 5 showthe type 0 leaky modedispersion
curveswith the low-frequencysegmentsincluded.Thosein
Fig. 4 correspondto the axial modesand thosein Fig. 5 to the
30.0.
2•.0
K
L
18.0
E
F
12.0
8.0
0.0
0
0
•1.0
8.0
6.0
12.0
20.0
2ti.0
28.0
32.0
36.0
U,0
Radius X Frequency(mm/•s)
FIG. 4. Dispersioncurvesfor themodalphasevelocityfor thetwo, axialrod moderelatedsubfamilies
of type0 leakymodesasa functionof rod radiustimes
frequency
(raf) in thealuminum-silicon
carbidesystem(aluminum:
density
= 2.77g/½m
3,ac = 6.323mm//•s,bc = 3.1 mm//•s;siliconcarbide:
density= 3.2g/cm3,ac = 9.649mm//•s,bc = 5.193mm//•s).Thelow-frequency
portions
ofeachcurveareshownhereandeachcurveisidentified
bya letter
for future reference.
1069
J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
Simmonseta/.' Clad rods. I
1069
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•.0
õ 20.0
P
._
>,
12.0
u,.o
0.0
U,. 0
8.0
12.0
16.0
20.0
2U,. 0
28.0
32.0
36.0
U,0
Radius X Frequency(mm/t.t,s)
FIG. 5. Dispersion
curvesforthemodalphasevelocityfortheradialrelatedsubfamily
oftype0 leakymodesasa functionofrodradiustimesfrequency
(rof)
in thealuminum-silicon
carbide
system
(aluminum:
density
= 2.77g/cm3,ac = 6.323mm//•s,bc -- 3.1mm//•s;siliconcarbide:
density
= 3.2g/cm,ac
-- 9.649mm//•s,bc = 5.193mm//•s). The low-frequency
portionsof eachcurveareshownhereandeachcurveisidentifiedby a letterforfuturereference.
radial modes. For future reference each of the modes in these
figureshasbeenassigneda letter. Basedon their peakbehavior, the axial modescan be divided into two alternate families
of modes,which we designateasthe lesserand greateraxialrelatedmodes,while the other family of type0 modeswill be
called the radial-related
modes. This third class of modes is
the only one observedto exhibit negativephasevelocities,
signalingbackward-leakingmodes.In this case,stoppingat
rof= 30, there are 6 modesin eachfamily. Figures6 and 7
showthe equivalentdispersioncurvesfor type 1 and type 2
leakymodes,againwith a designationletter assignedto each
mode.There are seventype 1 modesshownin Fig. 6, and all
but the lowesttype 1mode,labeledA in Fig. 6, correspondto
radial-relatedtype 0 modes.The extra modein family 1 can
be seenin Fig. 17(a)--although the elasticconstantsused
theredifferslightly--to correspond
not to a rodmode,but to
a mode arisingin a hollow aluminum claddingwith rigid
boundary.This correspondence
holdsfor valuesof rofbetweenabout6.5 and 22.5 mm//•s. Outsidethis rangedifferent correspondences
hold. Two familiesof type 2 modes,
shownin Fig. 7, correspond
to the equivalentaxiallyrelated
type0 althoughthe differencein peakbehavioris muchless
pronouncedthan for type 0 modes.There are six modes
shownin eachfamily.Therearenotypethreemodesfor this
materialcombination.[Thoughnot strictlyobeyedat every
valueof ½,,we statea general"rule of thumb" that the numberof 0 modesplusthe numberof 3 modesequalsthe number of 1 modesplusthe numberof 2 modes.This arisesheur-
30.0
F
2LI.0
G
E
D
18.0
C
12.0
6.0
0.0
0
ti. 0
8.0
12.0
16.0
20.0
2U,. 0
28.0
32.0
36.0
El0
RadiusX Frequency(rnrn/t.t,
rn)
FIG. 6. Dispersioncurvesfor the modalphasefor the subfamilyof type 1 leaky modesasa functionof rod radiustimesfrequency(rof) in the aluminum-
siliconcarbide
system
(aluminum:
density-- 2.77g/½m3,ac = 6.323mm//•s,bc = 3.1mm//•s;siliconcarbide:
density= 3.2g/cm,ac = 9.649mm//•s,bc
-- 5.193 mm//•s). The low-frequencyportionsof eachcurveare shownhereand eachcurveis identifiedby a letter for futurereference.
1070
J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992
Simmonseta/.' Clad rods. I
1070
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2u• O.
L
v
E •8.o i
G
E
D
C
6.0
•---•
---/ ..............
0.0
0
0
•.0
8.0
•2.0
20.0
2U,.0
28.0
32.0
36.0
L10.0
Radius X Frequency(mm/l•m)
FIG. 7. Dispersioncurvesfor the modalphasevelocityfor the subfamilyof type 2 leakymodesasa functionof rod radiustimesfrequency(tof) in the
aluminum-silicon
carbidesystem(aluminum:density= 2.77 g/cm3, ac = 6.323mm//ts,bc = 3.1 mm//ts;siliconcarbide:density= 3.2 g/cm3, as
= 9.649mm//ts;bR = 5.193mm//ts). Thelow-frequency
portionsof eachcurveareshownhereandeachcurveisidentifiedby a letterfor futurereference.
istically from the two type wave terms (longitudinal and
shear)cancelingfor pair of branchesand combiningfor the
other pair.]
It is apparentfrom thesefiguresthat for this clad rod
systemk (or v) is a well-definedfunctionof the frequency-therearenopointsfor whichk = 0 or d.f/dk = 0•and there
is no interconnection
between modes as occurs for the bare
rod.'7It appears
thatforvanishing
Pc theconfiguration
oc-
in the range27<rof<28. The intermediatedensitychosenis
just below the crossoverdensity.As a consequence,there
seemsto exist a subclassof modes in infinitely clad rods
whosedispersioncurveshave, as their asymptoticlimit, the
longitudinalvelocityof the rod, eventhoughno longitudinal
velocity mode existsin the limiting planar interface case.
This featureisquitedistinctfrom the caseof barerodswhere
only the shearor Rayleigh velocity is an asymptoticlimit.
curring in the bare rod is topologicallyuniqueand that the
leakymodesseparateat pointswhered.f/dk = 0 in the bare
rod diagram.Indeed, one might expectthat modesleaking
into an acousticor elasticcladding (where there are four
distincttypesof leaky modesversusonly one type for the
rod) wouldbettermodelrod modeexperiments(where the
environmentof the rod is not a perfectvacuum).
Another distinctionoccursin the mid-frequencyrange
of the dispersioncurves,whererod modecurvesexhibitterracing.Eachrod modeisknownto reacha plateaujust above
the longitudinalvelocityof the rod beforecontinuingdown
For low claddingdensity,p•0, the numberof modesthat
existat a particularvalueof•bincreasedwith increasingdensity.The samesituationobtainsfor low rod density,p•
with the number of modesincreasingwith increasingrod
to theasymptotic
shearvelocity.'For thesystems
wehave
on any one mode.
studied,only type 0 leaky modesexhibit the terracingfeature; but, there, an important additional feature occurs,
whichwe callmodeknitting (seeFig. 8 ). If oneincreases
the
densityof the cladding,for instance,startingfrom somevalue "near" zero,onecansee,startingat "very high" valuesof
•b,the endof the terraceof a certainmodeapproachingthe
cornerof the next higheradjacentmode.At a "critical" density thesemodesactually touch, the complexvelocitiesbecomingequalat a particularvalueof •b.This is shownin Fig.
8 (a) for the 11 and 12 modesfrom Fig. 2 (o) (or modesH
and/in Fig. 4) at a valueof rof•27.520 218 076 87 mm//•s
(•b= 5.299483 55) over a range of densitiescloseto the
critical density.Above that densitythe modal lines interchange;the terraceof the highermode,togetherwith those
of previouslyknit modes,are transferredto the lower mode.
Figure 8 (b), for instance,showsa blowupof thephasevelocity curvesfor the sametwo modesat threedifferentdensities
At high valuesof •bsomeaspectsof the clad rod solutionsapproachthoseof the planarinterface.Of course,there
always exist infinitely many clad rod solutionswhile there
are only a finite number of planar interface solutions,and
planeinterfacemodesare oftenapproximatedasthe limiting
caseof a givencladrod mode;but this doesnot alwaysoccur.
Whenevera leaky planar interfacewave or a Stoneleywave
of typem,3 (m = 0 -- 3 ) exists,it isthe limit of a similartype
m cylindricalmode;but this isnot the casefor decomposable
planarinterfacemodes.For instance,in thecaseof an aluminum matrix encasinga steel rod, there exist two limiting
decomposable
planarinterfacemodeswith velocitiessignificantlyhigherthan the longitudinalvelocityof eithermaterial (7.41 mm//•s for a type 1,3 modeand 6.78 mm//•s for a
type 2,3 modeasopposedto VL= 6.32 mm//•s in A1 and VL
= 5.92mm//•s in steel). (SeeFig. 32 captionfor the material constantsof the aluminum-steelsystem.) (There is alsoa
1071
J. Acoust.Soc.Am.,Vol. 92, No. 2, Pt. 1, August1992
density.There, aswill be discussed,
the limiting caseconsists
of Dirichlet boundaryconditionsfor the rod with no displacementsat the outer surface.Thus, there is someintermedi-
atevalueofp wheremodeknittingis maximized.Note that,
however, since we have not identified which particular
modesstay at the higher longitudinal velocity, we cannot
rigorouslyconcludethat thesehigh velocity"tails" remain
Simmonset aL: Clad rods.I
1071
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10.970
0.620
•
REAL
PART
OFMODE
H
....
IMAGINARY PART OF MODE H
REAL PART OF MODE I
•--
IMAGINARY PART OF MODE I/
/
10.968
0.619
/
iI
iI
!
E
0.618
10. 966
._•
>
0.617
10.96q
\,
whoseasymptoticlimit is 7.41mm/fts. Rather,for veryhigh
modesnumbers,asthe velocityof an individualmodepasses
from its peak value near the associatedrod mode cutoff
downto the asymptoticvalueof 5.92 mm/fts (in this case),
it takeson the characterof a decomposable
modefor values
of ½wherev• 7.41 mm/ps.
To completethe discussionof dispersion,we wish to
describethe rod modessatisfyingthe Dirichlet--zero displacement--boundaryconditions as well as the tunnel
modesthat existin an infinite claddingwithout a rod present. The real rod mode structures for both the Dirichlet
•,,•
10.962,
10
0.616
0.615
2.85662
2.8566q
2.85666
2.85668
2 85670
Density (gm/cma)
{a!
11.5'.
,,
11.3•
•.E 11.1'
•
10.9
10.7
10.5'
27
{b)
0
27.2
27.q
27.6
27.8
28.0
Radius X Frequency (mm/txs)
FIG. 8. (a) The real and imaginarypartsof modesHand IofFig. 4 plotted
asa functionof densitynearthe criticalcrossoverdensity---2.856 645 108 7
g/cm3 at a valueof rof= 27.520218076 87 mm/ps.Thisisjust pastthe
criticalvalueof rof, andherethe realpart ofv hasalreadycrossedoverwhile
the imaginarypart has not. (b) Blowupof the dispersioncurvesshowing
modeknitting for modesH and I of Fig. 4 plottedas a functionof rof at
threedifferent
densities:
onebelowthecriticaldensity(2.77g/cm3whichis
thatof aluminum),onejustbelowthecriticaldensity(2.856g/cm3) and
oneabovethecriticaldensity(2.97g/cm3).
2,1 typedecomposable
modethat doesnothavea cylindrical
analog.)
The first of thesedecomposable
modesis made up of
threeinputplanarwaves(oneshearwavein A1at -- 65ø,one
(weak) shearwavein steelat 61ø,and onelongitudinalwave
in steelat 29ø) and only oneoutputwave (one longitudinal
wavein A1at 31ø).(The reversewavewith a singlelongitudinal inputandthreeoutputsis of 2,0 typeanddoesnot havea
cylindricalanalog.Generally speaking,0 type waveshave
both planar modesgoingaway from the interface,1 type
waveshavethe longitudinalwavegoingawayand the shear
planewavegoingtowardtheinterface,2 typewavesarejust
the reverse,and 3 type waveshaveboth planewavesgoing
toward the interface.) However, there is no type 1 mode
1072
Neumannboundaryconditionsareshownin Fig. 9. The Dirichlet rod mode structure, while somewhat distorted from
that of the Newmann (traction-free) rod mode structure,
96o
2.85660
and
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992
can still be connected to that structure for modes in the inter-
mediateto high valuesof • exceptfor the traction-freesurface modes, which cannot exist with Dirichlet boundary
conditions.(Note that neither of theselimiting rod mode
structuresdependson density.) However, for thosemodes
that arein their low • range--nearor belowtheir peak--the
correspondence
betweenNeumannand Dirichlet boundary
condition modescannot be simply made, but must be inferredfrom the part of the dispersion
curvefor higher•. The
displacementfor all Dirichlet modesappearsto beaxial near
the cutofffrequency.
In the traction-freeNeumann case,thereexistonly four
puretunnelmodes--usually
oneof eachtypeformostvalues
ofac/l•c = at/be; thisisillustratedin Fig. 10forthecaseof
the aluminum cladding used in the previous examples,
whereac/l•c • 2.04. In general,type 0 and 1 modesappear
to existat all valuesof ac/l•c and•c = rof/bc. (Valuesof
•c lessthan0.01 andvaluesof ac/lgc greaterthan 10 were
notinvestigated.
) The type2 and3 modesexistfor all values
of •c > •maC/•C, rn = 2,3. Justbelowthese"switching"
frequencies,
the type 2 modechangesto a type 1 modeand
the type 3 modechangesto a type 2 mode.In the type2 to
type 1 mode transition,the exponentialterm controlling
asymptoticgrowthswitchesfrom the shearto longitudinal
type.Justabovethevalueof •2, thephasevelocityisrealand
exceeds
2bc,theasymptoticgrowthexponentiszeroandthe
asymptoticenergy velocityis bc, while just below the
switchingfrequencythe asymptoticgroupvelocitychanges
to ac. The "low-frequency"type 1 mode alsohas a lower
cutoffvalueOf•c that occursfor a valueof oslightlylessthan
ac, andceases
to existfor valuesOfac/lgc somewhere
above
2.35, wherethe type 2 modemerely exhibitsa lower cutoff
when v=a•
(e.g., in the case when ac/]?c = 10,
•: •0.0636). The switchingfrequencyfor the type 3 to type
2 transitionoccursat that frequencywherethe exponentfor
exponentialdecayof the type 3 modebecomeszeroandthen
turns positiveproducinga type 2 mode. At the transition
pointboththephaseandasymptoticenergyvelocitiesarebc.
The variation of the cutoff frequencywith ac/]•c is quite
small [e.g.,•b3(1.1) = 0.224 and ½3(10.0) = 0.378]. Below
the transitionpoint, the type 2 modeis a leaky modewith
nonzeroimaginarypart.
The zero displacement
Dirichlet boundaryconditions
appearto giveriseto onlyonetunnelmode(of type1) in the
infinitecladding.This mode,alsoshownin Fig. 10, exists
Simmonset al.' Clad rods.I
1072
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30.0
2L•.0
E
E ]8.oi
._
•
]2.0.
8.0
..................................................................................................................................
•......
'
i
Ill
,
I
'
r
'1.........
iii
.
.
..'
.
.
.
0.0
0
0
L•. 0
8.0
12.0
!8.0
20.0
2L•. 0
28. 0
32.0
38. 0
L•0. 0
L•L•. 0
L•8. 0
52. 0
58. 0
80
Radius X Frequency (mm/l•S)
FIG. 9. Dispersioncurvesfor the modalphasevelocityfor both ordinaryrod modes(zero tractionsat the surface-- Neumannmodes)and Dirichlet rod
modes(zerodisplacements
at thesurface)asa functionof radiustimesfrequency
(rof) in a siliconcarbiderod (siliconcarbide:density-- 3.2 g/cm3,ac
-- 9.649mm/lts, bc -- 5.193mm/lts). The extra-widespaceddottedrod modeis the complexpart of the secondNeumannrod mode.
over the whole range of •bc valuesinvestigatedwhen ac/
bc <--4.2, but has a forbidden range of •bc values for
ac/bc > 4.2. The most critical value of ro/bc at which the
forbiddenrangebeginsis about 0.41. The existenceof this
singlemode,aswell asanomaliesoccurringat variousvalues
of •bnear modecutoffs,are exampleswherethe numberof
type 0 plus 3 modesis not the sameas the numberof type 1
plus2 modes.
We are now in a position to attempt a catalog of all
axisymmetricleaky modes.As mentionedearlier, when the
densityof eitherthe rod or the claddinggoesto zero (p goes
to infinity or 0), the determinantof the matrix in Eq. (5)
essentiallyfactorsinto the productof two subdeterminants,
oneassociatedto the tractionsand the other to the displacements at the interface. The condition for a free rod or clad-
ding is the vanishingof the traction subdeterminant,while
the vanishingof the displacementsubdeterminantdescribes
a rod or claddingbondedto an infinitelyrigid material.
There are alwaysinfinitely many modesfor any clad rod
aslongasthe densityof the claddingisnonzero.Thus, in any
cladrod withp near0 or infinity,the setof cladrod modes
can be identifiedwith all rod modesof one boundarycondition typetogetherwith all tunnelmodesof the otherboundary conditiontype.As we havealreadyseen,nearsmallvaluesof p the networkof complexand imaginaryrod modes
becomesgreatlysimplifiedwhencladdingmaterialon non-
8.0
zerodensityis present,
leadingto a singlefamilyof mod.
es
which can be identified with the real rod modes both above
_
•.6
0.0
0.0
•.0
8.0
12.0
16.0
20
and below their cutofffrequency.This samesituationappearsto occurwith Dirichletrodmodesfor largevaluesofp.
In suchcircumstances
eventhoughthe rod has very small
density,it hassufficientstiffness
to balancethe motionof the
cladding. The rod displacements,then, are very large
vis-h-visthoseof the claddingand the tractionsat the interface are non-negligible.
The questionthen occurs:Can onemap all leaky modes
for p near zero, say, onto all leaky rod modeswith large
valuesofp andthuscompletelyencompass
all leakymodes,
or aretheremodesthat occurat intermediatevaluesofp that
FIG. 10.Dispersion
curves
forthemodalphase
velocity
alongthefiberaxis
cannot be linked to any extremal modes?While we cannot
answerthisquestionin general,whichrequiresthe variation
of four otherparameters,we shalladdressspecificaspectsof
for both Neumann and Dirichlet modes as a function of rod radius times
it.
Radius X Frequency(mm/l•S)
frequency
(rof) in analuminum
tunnel(an infinitespace-filling
solidof
aluminumwith a cylindricalholeboredout) (aluminum:density-- 2.77
g/cm3,ac -- 6.323mm/lts,bc = 3.1mm/lts).Thecrossover
fromtype2
to typeI modeoccurs
at rof-•0.833665mm/lts,andthetype3 cutoffoccursat about1.067202 4 mm/s, wherethephasevelocitybecomes
bc.
1073
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
Calculationswerecarriedout for a cladrod consistingof
a pair of materialswith the sameelasticconstants
as the
aluminumand siliconcarbideusedherein,but with p rang-
ingfrom0.001to 1000while•b-- rof/bR washeldat a fixed
Simmonseta/.: Clad rods.I
1073
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valueof 5.78 (tof= 30 mm//•s). At the valueofp = 1.16,
correspondingto the A1/SiC exampleused,there were 37
modes,asseenin Figs.2 and 3 showingsignificantrealphase
velocitysomewhere
below• = 5.78. Betweenp = 1.16and
0.001, as discussedearlier, two clad rod modescould be cor-
The appropriatecladrod dispersion
curvesatp = 1000
werecomparedwith thosefor rodmodesobtainedusingDirichlet boundary conditions.They were seento almost coincidedownto nearthe appropriaterod modecutoffwherethe
clad rod mode velocity acquiresa large imaginarycompo-
related with each of the 17 real SiC rod modes with a cutoff
nent. Of course, unlike the rod modes, there are an infinite
below • = 5.78 as well as with the lowestrod mode. This
gaveriseto 36 modes.The 37thmode,of type2, wasfoundto
crossoverto a type 1 modeat a valueofp•0.176, andthis
numberof leakymodesat any fixedvalueof •; almostall of
mode continued
down to the Dirichlet
tunnel mode. This
situationdiffersfrom that obtainedat lower valuesof •,
wherethe lowesttype 1 modewasfound to transform,with
changingvaluesof p, to the Dirichlet tunnelmode.At this
valueof •, that lowestmodecrosses
overat a smallvalueofp
to a type2 modeandthencontinuesdownto a rod mode.At
still higher valuesof • the lowest type 1 mode smoothly
these modes, however, are associatedto the low end tails of
higherrod modes.Theseother modeshave smallphasevelocity and tend to complexDirichlet rod modes.
Thus, in this case, while no unusual modes were found
to existonly for intermediatevaluesof p, the completehomotopiccorrespondence
amongmodesfromp = zeroto infinity has to be qualifiedat any value of • by the possible
oversightof cladrod modesneartheir peakvalueaswell asa
type zero mode correspondingto the lowestorder traction
transforms into a rod mode and the Dirichlet tunnel mode is
free rod mode. Further, the correspondence
betweenclad
found to correspondto a type 2 mode after a crossover. rod modesand rod or tunnelmodesmay changewith different valuesof •.
Theseresultsshowthat homotopiccorrespondences
are by
A secondspecificaspectof the homotopicevolutionof
no meansunique,or equivalently,that homotopiesin • (disclad rod modeswith densitywasinvestigated.Using orderpersioncurves)andp do not commute.
ing relationsamongthe elasticvelocitiesin a mannersimilar
Corresponding
to highvaluesofp wenotethatthereare
to that suggested
by Thurston,
• a setof 12 examples
was
16Dirichlet rod modeswith cutoffsbelow• = 5.78 and one
chosenand the evolutionof the tunnelmodeswith changing
Neumann tunnel mode of type 2 to which there shouldbe
correspondents.
[ Examinationof Fig. 15(a) showsthat the
p wascalculated.Table I givesthe characteristics
of the 12
other three Neumann modeswill have no correspondent orderingtypes.The propertiesof the claddingwerekept conmodeat valuesof p as low as 1.16. In investigating
these stantthroughall examples,with the ratio ac/bc fixedat 2, a
valuecloselyapproximatingthat of aluminum,andthe value
threetunnelmodesfor largevaluesofp, boththetype0 and 1
modeswere found to cut off near zero phasevelocityat valof rofwas fixed at 15.5 ( = 5be). The 12 order typeswere
uesofp• 2.8 at which the shearmodulusof the rod is apbrokeninto sizeclassesinvolvingrelationshipsamongthe
proximatelyequal to that of the cladding,and the type 3
longitudinaland shearvelocitieswith two subclasses
in each
typedeterminedby the relativemagnitudeof an appropriate
tunnelmodecut off at p-• 7.5, sincesuchsurfacemodesdo
not existovermostof the rangeofp when/3c< 1.] In going surfacewave and bulk velocity. Figures 11-16 show the
fromp = 1.16to p = 1000,everyreal valuedDirichlet rod
changewithp, forp greaterthan0.2, in thefivetunnelrelatmodewasagainthe limit of two leakymodes,oneof type0
ed modes(four of Neumann and one of Dirichlet type) for
and the other of type 1 or type 2; and the type 2 Neumann
eachof thesecases.The aluminum/siliconcarbideexample
modewasthe limit of the lowesttype 2 leakymode.Three of
usedthroughoutmuch of this studyis similarto case5a. In
the type 1 or type 2 modescrossedoverto betype 2 or type 1
the first three classes,where bR • be,the modebehaviorappearsto break into a tunnel dominatedand rod dominated
modesat valuesofp exceeding
3.35;andthelowesttypezero
modeaswell asthe highestmodesof types0-2 all tendedto
regionwith a boundaryoccurringapproximatelyat the valcomplexDirichlet rod modes.
ueofpR wherethe shearimpedance(lob)of thetwo media
TABLE I. Characteristicsof the 12 orderingtypes.
Class
la:a n <sc<b c
lb: sc < an < bc
2a:bn <sc<bc<an <ac
2b: sc < bn < bc < an < ac
3a: bn < sc < bc < ac < an
3b:sc < bn < bc < ac < an
4a: ac < sn < bn
4b: sn < ac < bn
5a: bc < sn < bn < ac < an
5b:sn < bc < bn < ac < an
6a: bc<sR <b n <a n <ac
6b:st < bc < bn < an < ac
1074
sn
bn
an
Sc
bc
ac
Examplesystem
1.4
1.56
2.61
2.80
2.56
2.8
7.46
5.97
4.85
2.98
3.17
3.03
1.5
1.67
2.8
3.0
2.75
3.0
8.0
6.4
5.2
3.2
3.4
3.25
2.7
3.0
5.04
5.4
6.6
7.2
14.4
11.52
9.36
6.4
6.12
5.85
2.89
2.89
2.89
2.89
2.89
2.89
2.89
2.89
2.89
2.89
2.89
2.89
3.1
3.1
3.1
3.1
3.1
3.1
3.1
3.1
3.1
3.1
3.1
3.1
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.2
6.2
AI-Th (p = 4.2)
Al-graphite (p..•0.54)
AI-W (p--•6.9)
J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992
AI-B (p--•0.85)
AI-SiC (p--• 1.2)
Al-steel (p--•2.8)
Simmonseta/.: Clad rods. I
1074
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10.0
10.0
LEGEND
CASE
CASE
CASE
CASE
CASE
8.0
1A:TYPE
1A:TYPE
1A:TYPE
1A:TYPE
1A:TYPE
0
1
2
3
2
NEUMANN
NEUMANN
NEUMANN
NEUMANN
DIRICHLET
8.0
/
/
8.0
8.0
LEGEND
........
CASE
CASE
CASE
CASE
CASE
2A:TYPE
2A:TYPE
2A:TYPE
2A:TYPE
2A:TYPE
0
1
2
3
1
NEUMANN
NEUMANN
NEUMANN
NEUMANN
DIRICHLET
I
2.0
2.0
0.0
0.0
0
p-1 (RodDensity
/ Cladding
Density)
p-1 (RodDensity/Cladding
Density)
(8)
10.0
10.0
CASE
CASE
LEGEND
lB:TYPE
0 NEUMANN
lB:TYPE
CASE
CASE
CASE
CASE
1 NEUMANN
•SE
NEUMANN
SE I•:•YPE
TPE • NEUMANN
....... CRSE lB TYPE 1 DIBICHLET
8.0
8.0
LEGEND
2B:TYPE
0
2B:TYPE
1
2B:TYPE
2
2B:TYPE
3
NEUMANN
NEUMANN
BOTH KINDS
NEUMANN
...............................................................
:•.....
.•.Z...........................................................................
8.0
•'"-'-- ........
•'
8.0
i //
\/,!
/'\
2.0
0.0
0.0
0.0
{b)
1.0
2.0
3.0
k•.O
5
0.0
1.0
2.0
3.0
Lt.0
5.0
p-1 (Flod
Density
/ Cladding
Density)
JCI-1
(Flod
Density
/ Cladding
Density)
FIG. 11. Dispersioncurvesfor the modalphasevelocityalongthe fiberaxis
for both Neumannand Dirichlet modesas a functionof rod density/clad-
FIG. 12.Dispersioncurvesfor the modalphasevelocityalongthe fiberaxis
for both Neumann and Dirichlet modesas a function of rod density/clad-
dingdensity(p - 1) forthecladrodofclassla fromTableI. TheDirichlet
dingdensity(p - •) forthecladrodofclass
2afromTableI. (b) Dispersion
modehascrossed
overto type2 in thisregionofp ratherfar removedfrom
the Dirichletregion(p--•0). (b) Dispersioncurvesfor the modalphase
velocityalongthe fiber axisfor both Neumann and Dirichlet modesas a
curvesfor the modalphasevelocityalongthe fiberaxisfor both Neumann
functionofroddensity/cladding
density(p - ' ) forthecladrodofClasslb
andDirichletmodes
asa function
of roddensity/cladding
density(p- ')
for the cladrod of Class2b from Table I. The Dirichlet andtype2 Neumann
modescoincidefor thisrangeofp.
from Table I.
areequal.In the lastthreeclasses,
whereb• > bc, thetype0
and 1 modesexhibita cutoffat a valueofp• wheretheshear
moduli of the claddingand rod are equal and the type 3
modescutoffwhenv = bc. Only the type2 Neumannrelated
and the Dirichlet
related modes exist for all classes and all
valuesofp > 0.2. In four of the casesthesemodescoincide,
givingriseto a totally tunnelrelatedmode;in the othereight
casesthesemodestendto rodmodesat theotherendof thep
scale.Boththe type 1 and type2 relatedmodescanexhibit
1075
crossovers.
More detailedstudiesare requiredto determine
whether the orderingclassesusedhere give an exhaustive
breakdownof wavemodetypes.
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
B. Attenuation
Figures17-20 displaythe attenuationin dB/mm MHz
for the A1-SiC systemwhosedispersionrelationsweregiven
previouslyin Figs. 4-7. Thesefiguresusethe dimensional
Simmonseta/.' Clad rods. I
1075
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.143.68.142 On: Wed, 30 Oct 2013 18:53:49
10.0
10.0
CRSE
C•SE
C•SE
CRSE
8.0
LEGEND
3R:TTPE
0
3R:TTPE
!
3R:TTPE
2
3•:TTPE
3
NEUMANN
NEUMANN
BOTH KINDS
NEUMANN
8.0
2.0
0.0
I)-1 (RodDensity
/ Cladding
Density)
10.0
0.0
(8)
l.O
2.0
3.0
•.0
I3-1 (RodDensity
/ Cladding
Density)
5
10.0
LEGEND
CRSE 3B:TTPE
0 NEUMRNN
CRSE 3B:TTPE
CRSE 3B:TTPE
CRSE 3B:TTPE
8.0
! BOTH KINDS
2 NEUMRNN
3 NEUMRNN
8.0
8.0
8.0
2.0
o
CRSE
CRSE
CRSE
CRSE
CRSE
CRSE
2.0
o
o.o
i.o
•.o
•.o
,'0 ....... .•
D-1 (RodDensity
/ Cladding
Density)
LEGEND
NB:TTPE
NB:TTPE
NB:TTPE
NB:TTPE
N:TTPE
NB:TTPE
0 NEUMRNN
! NEUMRNN
2 NEUMRNN
3 NEUMRNN
2 DIRICHLET
! DIRICHLET
0.0
0
13-1 (RodDensity
/ Cladding
Density)
FIG. 13.(a) Dispersion
curves
forthemodal
phase
velocity
along
thefiber FIG. 14.(a) Dispersion
curves
forthemodal
phase
velocity
along
thefiber
axisforbothNeumann
andDirichlet
modes
asa function
ofroddensity/ axisforbothNeumann
andDirichlet
modes
asa function
ofroddensity/
claddingdensity(p - ' ) for thecladrodof class3afromTableI. The Dir-
ichlet
andtype2 Neumann
modes
coincide
forthisrange
ofp.Thetype1
modeforthiscaserequired
a somewhat
largererrorboundonthevalueof
thedeterminant
arising
from(5) duetoa section
ofapparently
realroots
lyingbetween
--•0.0859
and--•0.0862.
(b) Dispersion
curves
forthemodal
phasevelocityalongthefiberaxisfor bothNeumannandDirichletmodes
asafunction
ofroddensity/cladding
density
(p - ' ) forthecladrodofClass
3bfromTableI. TheDirichletandtype2 Neumann
modes
coincide
forthis
case.
cladding
density(p - ' ) forthecladrodofclass4afromTableI. Thecutoffs
forthetype0 and1modes
occur
at --•0.1501
andthecutoff
forthetype3
modeoccurs
at --•0.0499.
TheDirichletandtype2 Neumann
modes
coin-
cideforthisrange
ofp.(b) Dispersion
curves
forthemodal
phase
velocity
alongthe fiberaxisfor bothNeumannandDirichletmodesasa functionof
roddensity/cladding
density(p - ]) forthecladrodof Class4bfromTable
I. Thecutoffsforthetype0 and1modesoccurat --•0.2346andthecutofffor
thetype 3 modeoccursat --•0.0806.The crossover
of the Dirichletmode
fromtype1totype2 (withdecreasing
roddensity)
occurs
at --•3.3936.
formof theattenuation
thatdoesnothavethefactorbR C. Energyflowanddisplacementplots
shown
in Eq.(27). Corresponding
curves
fromtheappropriatepairsof figurearelabeledwith thesameletter.To find
therelative
attenuation
indBbetween
twopoints,
(r,zl) and
(r,z2), locatedat equalradialdistances
fromthecenterof the
rod,onemustmultiplythevaluegivenbythefrequency
of
themodeandtheaxialdistance
z2-zl.
1076
J.Acoust.
Soc.
Am.,
Vol.92,No.2,Pt.1,August
1992
Evenconsidering
therelativesimplicityof radialaxial
modes
in aninfinitelycladrodwithperfectly
bonded
interface,a largevarietyofwavephenomena
canoccurasthefive
parameters
describing
a givenmodearechanged.
We show
hereonlysome
ofthese
phenomena.
Theplotsasgiven
show
Simmons
ota/.'Cladrods.
I
1076
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10.0
20.0
18.0
E
....
LEGEND
œRSE 5R:TYPE
ϥSE 5R:TYPE
œ•SE 5•:TYPE
ϥSE 5R:TYPE
ϥSE 5R:TYPE
0 NEUMRNN
1 NEUMRNN
2 NEUMRNN
3 NEUMANN
1DIBIœHLET
8.0
6.0
12.0
._
O
'
>
8.0,
LEGEND
g
œRSE
œRSE
œRSE
œRSE
œRSE
œRSE
2.0
o,
0.0
o.o
o
0
].0
2.0
3.0
u,.0
0
5
0....
1.0
2.0
GR:TTPE
6R:TTPE
6R:TTPE
6R:TTPE
6R:TTPE
6R:TTPE
0
I
2
3
2
I
3.0
NEUMRNN
NEUMRNN
NEUMRNN
NEUMRNN
DIBIœHLET
DIRIœHLET
I-1.0
5
lb-1(RodDensity
/ Cladding
Density)
p-1 (RodDensity
/ Cladding
Density)
10.0
]0.0
LEGEND
8.0
8.0
6.0
6.0
.......
CRSE
CRSE
CRSE
CRSE
CRSE
6B:TTPE
6B:TTPE
6B=TTPE
6B•TTPE
6B:TTPE
0 NEUMRNN
1NEUMRNN
2 NEUMRNN
3 NEUMRNN
! DIRIœHLET
ti.0
LEGEND
2.0
.......
œRSE
œRSE
œRSE
CRSE
CRSE
5B:TTPE
5B:TTPE
5B:TYPE
5B:TYPE
5B:TTPE
0 NEUMRNN
1 NEUMRNN
2 NEUMRNN
3 NEUMRNN
1 DIRIœHLET
2.0
0.0
0.0
0
0
1.0
2.0
3.0
q.O
0
5
Lt.0
5
D-1 (RodDensity
/ Cladding
Density)
•-l(RodDensity/CladdingDensity)
FIG. 15. (a) Dispersion
curvesfor themodalphasevelocityalongthefiber
FIG. 16.(a) Dispersion
curves
forthemodalphase
velocityalongthefiber
axis for both Neumann and Dirichlet modes as a function of rod den-
axis for both Neumann and Dirichlet modes as a function of rod den-
sity/cladding
density(p - •) forthecladrodofclass5afromTableI. The
cutoffsfor thetype0 and 1 modesoccurat .•0.3554 andthecutofffor the
type3 modeoccursat -0.1213. TheDirichletandtype1Neumannmodes
coincidefor thiscase.(b) Dispersion
curvesfor the modalphasevelocity
alongthefiberaxisforbothNeumannandDirichletmodesasa functionof
sity/cladding
density(p - •) forthecladrodofclass6afromTableI. The
cutoffs
forthetype0 and1 modesoccurat --•0.8313andthecutofffor the
type3 modeoccurs
at --•0.4070.
Thecrossover
oftheDirichletmodefrom
type1totype2 (withdecreasing
roddensity)occurs
at •0.0535. (b) Dispersion
curves
for themodalphasevelocityalongthefiberaxisfor both
roddensity/cladding
density(p - •) forthecladrodofClass5bfromTable
Neumann and Dirichlet modesas a function of rod density/claddingden-
I. Thecutoffs
forthetype0 and1modesoccurat --•0.9385andthecutofffor
the type 3 modeoccursat --•0.5308.The Dirichletand type 1 Neumann
sity(p - •) forthecladrodofclass
6bfromTableI. Thecutoffs
forthetype
modes coincide for this case.
..•0.4905.
the elliptical particle trajectoriesat a variety of points
throughoutthe cladrod. Sinceonly the relativesizeof the
ellipses
isimportantin a givenplot,thescalevariesfromplot
to plotfor easeof graphicalpresentation.
A largearrowhead
is shownon each trajectory to indicateparticle location
throughout
thecladrodat onemomentoftime,thusindicatingpositions
of equalphase.The directionof thearrowhead
indicatesthe senseof motion of the particle.Two smaller
arrowheadsvisibleon someof the larger ellipsesshowthe
positionsat two otherequallyspacedperiodsof time asthe
particleproceeds
in its orbit. If the sizeof the ellipseis too
largeor too smallto permit convenientgraphing,it is not
plotted.From the centerof eachellipseextendsan arrow
which givesthe energyvelocityat that point.
1077
J. Acoust.Soc.Am.,Vol.92, No.2, Pt. 1, August1992
0 and 1 modesoccurat • 0.9098andthe cutofffor the type 3 modeoccursat
Simmons
eta/.' Cladrods.I
1077
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20.0
16.0
E 12.0
•
8.0
•t.0
0.0
0
0
q. 0
8.0
12.0
16.0
20.0
2q. 0
28.0
32.0
36.0
gO. 0
RadiusX Frequency(mm/•s)
FIG.17.Attenuation
indB/(mmMHz)along
thefiberaxisforthetwo,axialrodmode
related
subfamilies
oftype0leaky
modes
asafunction
ofrodradius
times
frequency
(rof) inthealuminum-silicon
carbide
system.
These
attenuation
curves
correspond
tothephase
velocity
curves
ofFig.4.
20.0
16.0'
12.0
•.0
•t.0
0.0
0
tt. 0
8.0
12.0
16.0
20.0
2tt. 0
28.0
32.0
36.0
RadiusX Frequency(mm/l•s)
FIG.18.Attenuation
indB/(mm
MHz)along
thefiber
axis
fortheradial
related
subfamily
oftype
0leaky
modes
asafunction
ofrodradius
times
frequency
(rof)inthealuminum-silicon
carbide
system.
These
attenuation
curves
correspond
tothephase
velocity
curves
ofFig.5.
20.0
16.0
N
X
E 12.0
•
8.0
it.0
0.0
0 0
tI.0
8.0
12.0
16.0
20.0
2tI.0
28.0
32.0
36.0
LI0 0
Radius X Frequency(mm/ixs)
FI6. 19.Attenuation
indB/(ram
MHz)along
thefiber
axis
forthefamily
oftype1leaky
modes
asafunction
ofrodradius
times
frequency
(tof ) inthe
aluminum-silicon
carbide
system.
These
attenuation
curves
correspond
tothephase
velocity
curves
ofFig.6.
1078
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
Simmonseta/.: Clad rods.I
1078
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.143.68.142 On: Wed, 30 Oct 2013 18:53:49
20.0,
]6.0
EE]2.O
•
8.0
g.0
0.0
0 ..... ,'0 ..... 8'0
•2 o
•
o
2o o
•, o
•
o
3• o
a• o ",0.o
Radius X Frequency (mm/lxs)
FIG. 20.Attenuation
indB/(mmMHz) along
thefiberaxisforthefamilyoftype2 leakymodes
asa function
ofrodradius
times
frequency
(rof) inthe
aluminum-silicon
carbide
system.
Theseattenuation
curves
correspond
tothephase
velocity
curves
ofFig.7.
Startingfrom a convenient
locationwithinthe rod, an
integratedcurveshowsthe flowof energyproceeding
from
that point.Sinceall energyflowcurvesareparallelwith respectto a shiftin theaxialdirection,all otherenergytraveling througha tubeof a givencross-sectional
areawill decreaseexponentiallywith increasing
z. To betterindicate
graphically
thenatureofthelacunathatliesto theleftofthe
"leading"energyflowcurve,the plottingof particletrajectories to the left of the energyflow curve has been suppressed.
Usuallythiscurveis notthe leadingcurve,but representsthat curveshiftedto the right to givean indicationof
energyflowwithinthe rod. Rigorously,
the actualposition
oftheleadingenergyflowcurveisunknown.If oneconsiders
a rod in a claddingof infiniteradius,but only half-infinite
axially (suchas >•0), thenthe modesdescribed
in thisstudy
modesof types0-2 for the aluminum-clad
siliconcarbide
rod.The modesin Figs.21 and22 areoneswhichhavebeen
generated
fromwaves
sentdown
abarerodintoacladding.
19
Theyarecalculated
at rof= 9 mm/ps,whichisjustbeyond
the peakof thesetwo modes.The displacement
andenergy
plotsoftherodmodeassociated
withtheleakymodeof Fig.
21isshownin Fig.28,alsofor rof= 9 mm/ps.At thisvalue
s.ø
1
q
O•
/'/
represent
onlyanapproximation
totheactualmodes
ofsuch
a configuration,
althoughwecanexpectthattheapproxima-
I
/
I
3.0
tion becomesbetter the further one movesaway from the
edgeof the cladding.Evenif onewereto acceptthat the
infiniterod modeswerean accurateapproximationto those
of a rod with a truncatedinfinite cladding,especiallyfar
from the edgeof the cladding,the presenceof multiple
modesnearthe edgewouldproducemodeinterference
with
a resultingcomplicatedenergyflow curve.
Only in regionswhereone modepredominates
could
oneexpectthe curvesshownto givean accuraterepresentation of theenergyflow.In particular,onemightexpectthat
evenwithonemodepredominating
in regionsawayfromthe
faceof thecladding,thepresence
of manyhighlyattenuated
(.//
/ .................
/ ..............
/ .............
...............
.....................
................
................
1.0
0.0
0
0
1.0
2.0
3.0
•.0
•.0
Z / •adius
num and with SiC rods clad with aluminum, seven leaky
FIG. 21.Displacement
orbitsandenergyvelocity
diagrams
formodeD of
Fig. 4 at a valueof rof-- 9 mm//ts.For thismode,v = 18.41+ il 1.09
mm//ts,theasymptotic
leakage
angle-- 75.58øandtheasymptotic
energy
velocity
= 6.25mm//ts.Thetipofthelargest
arrowhead
oneachelliptical
trajectory
marksa synchronous
position(equalphase).Thedimensioned
insetarrowscales
theenergyvelocities
whichrun fromthecenterof each
orbit(unless
suppressed
for graphical
reasons).
Thoseenergyvelocityar-
modeshavebeendetected(of types0 and 2), but the above
with solidheadsto flowsin thecladding.Because
of thetangentialdiscon-
modeswith high leakageanglecouldcausea shift of the
leadingedgeof the energycurveof the predominantmode
awayfrom thejunctionpointof the rod andthe cladding.In
experiments
thusfarconducted
in steelrodscladwithalumi-
mentioned
possible
shiftwasnotobserved.
•4'•8
Figures21-27 showa seriesof displacement
plotsfor
1079
J.Acoust.
Soc.Am.,Vol.92,No.2, Pt.1,August
1992
rows with hollow heads are associatedto flows inside the fiber and those
tinuityinenergy
flowattheinterface,
twoenergy
velocity
arrowsappear
for
orbitscenteredalongthe interface.
Simmons
eta/.:Cladrods.I
1079
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5.0
'" I
/
q.0
3.0
•.o .:........••
o.o
/
/
/
/
/
/
.
•'0 .....
•'0 ..... •'0 .....
\
/
\
\
.........................
.....................
......................
................
...........
............
•
..............
• ..................
• ...................
•.o
/
0.0
•.o
-1.0
.......
•
................
0.0
•
1.0
Z / Radius
2.0
,
...........
3.0
•.0
Z / •adius
FIG. 22.Displacement
orbitsandenergyvelocitydiagrams
formodeN of
Fig. 5 at a valueof tof= 9 mm//•s. For this mode,v = 12.05-4-i3.21
FIG. 24. Displacement
orbitsandenergyvelocitydiagrams
formodeD of
Fig. 4 at a valueof tof-- 4.5 mm//•s.For thismode,u- -- 1.18-F i 6.4
mm//•s,theasymptotic
leakage
angle= 61.01
øandtheasymptotic
energy mm//•s,theasymptotic
leakage
angle- 94.46
øandtheasymptotic
energy
velocity
- 6.25mm//•s.Thetipofthelargest
arrowhead
oneachelliptical velocity
-- 2.81mm//•s.Thetip ofthelargest
arrowhead
oneachelliptical
trajectory
marksa synchronous
position(equalphase).Thedimensioned trajectorymarksa synchronous
position(equalphase).Thedimensioned
insetarrowscalesthe energyvelocities
whichrun fromthe centerof each
insetarrowscalesthe energyvelocitieswhichrun from the centerof each
orbit(unless
suppressed
forgraphical
reasons).
Thoseenergy
velocity
arorbit(unless
suppressed
forgraphical
reasons).
Thoseenergy
velocity
arrows with hollow headsare associatedto flows insidethe fiber and those
rows with hollow headsare associatedto flows inside the fiber and those
withsolidheads
toflowsin thecladding.
Because
ofthetangential
discontinuityinenergy
flowattheinterface,
twoenergy
velocity
arrows
appear
for
withsolidheadsto flowsin thecladding.
Because
ofthetangential
discontinuityinenergy
flowattheinterface,
twoenergy
velocity
arrows
appear
for
orbitscenteredalongthe interface.
orbitscenteredalongthe interface.
of rof the phase velocity for this mode -•29.36
mm/Fs-• 5.65bR.Sincethereisnodissipation
forthismode,
occursat abouttwo-thirdsof therodradius;andtheoverall
theorbitsareunchanged
alongthez axis,andonlyoneset
wasplotted.As canbe seen,thereareregions
in the rod
wheretheenergy
flowisreversed
indicating
thatnoenergy
will be conducted
throughthat regionof the rod for that
mode.Through
theremainder
oftherod,theenergy
velocity
ranges
fromzerotoabout1.27bR.Thepeakenergy
velocity
seenfrom the figure.
groupvelocity-•orbital
rotationchanges
signare clearly
Figure23 shows
thesecond
lowesttype.0 mode,alsoat
rof-- 9 mm/Fs.Thisisa relatively
largevalueof•bforthis
mode,andalthough
theenergyvelocityisbeginning
to approachtheb•, theorbitsfarfromtheinterface
remainellipticallyshaped
andoriented
in thedirection
of energyglow.
q.0
3.2
1.6
0.8
........
::-'1::'-.'::.
. ........ .......... ..... _ ............
.............
: ............
:...
o.8 0
0.8
•.6
2.q
3.2
q.O
q.8
5.6
6.q
7.2
8.0
8.8
9.6
•0.•
•.2
•2.0
Z / Radius
FIG.23.Displacement
orbits
andenergy
velocity
diagrams
formode
AofFig.4atavalue
oftof= 9mm/Fs.
Forthismode,
v-- 5.66-4-/0.19
mm//•s,
the
asymptotic
leakage
angle
-- 75.58
øandtheasymptotic
energy
velocity
-- 5.65mm//•s.
Thetipofthelargest
arrowhead
oneach
elliptical
trajectory
marks
a
synchronous
position
(equal
phase).
Thedimensioned
inset
arrow
scales
theenergy
velocities
which
runfrom
thecenter
ofeach
orbit
(unless
suppressed
for
graphical
reasons).
Those
energy
velocity
arrows
with
hollow
heads
areassociated
toflows
inside
thefiber
and
those
with
solid
heads
toflows
inthecladding.
Because
ofthetangential
discontinuity
inenergy
flowattheinterface,
twoenergy
velocity
arrows
appear
fororbits
centered
along
theinterface.
1080
J.Acoust.
Soc.
Am.,
Vol.
92,No.2,Pt.1,August
1992
Simmons
eta/.'Clad
rods.
I
1080
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8.0
5.0
:'ddd
0.0'
O.
Radius
FIG. 25. Displacementorbitsand energyvelocitydiagramsfor modeC of
Fig. 4 at a valueof tof= 40 mm//zs. For this mode,v -- 9.689 + t0.0066
mm//zs,the asymptoticleakageangle-- 49.26øand the asymptoticenergy
velocity-- 6.323mm//zs.The tip of thelargestarrowheadoneachelliptical
trajectorymarksa synchronous
position(equalphase).The dimensioned
inset arrow scalesthe energyvelocitieswhich run from the centerof each
orbit (unlesssuppressed
for graphicalreasons).Thoseenergyvelocityar-
FIG. 27. Displacementorbitsand energyvelocitydiagramsfor modeC of
Fig. 7 at a value of rof= 9 mm//zs. For this mode, v- 7.44 + •0.059
mm//zs,the asymptoticleakageangle-- 65.55øandthe asymptoticenergy
velocity-- 3.098mm//zs.The tip of thelargestarrowheadoneachelliptical
trajectorymarksa synchronous
position(equal phase).The dimensioned
insetarrow scalesthe energyvelocitieswhichrun from the centerof each
orbit (unlesssuppressed
for graphicalreasons).Thoseenergyvelocityar-
rows with hollow heads are associated to flows inside the fiber and those
rows with hollow heads are associated to flows inside the fiber and those
with solidheadsto flowsin the cladding.Because
of the tangentialdiscontinuity in energyflowat the interface,two energyvelocityarrowsappearfor
orbitscenteredalongthe interface.(In thiscasethe largersizeof the orbits
insidethe rod prohibittheir inclusionon the figure.)
with solidheadsto flowsin the cladding.Becauseof the tangentialdiscontinuity in energyflow at the interface,two energyvelocityarrowsappearfor
orbitscenteredalongthe interface.
5.0
082
3.0
-%L%•.
[
0.62
2.0
0.•12
1.0
0.22
ß
0.0
0
0
1.0
2.0
3.0
LI.0
5
Z / Radius
O. 02! .............
-0.5
FIG. 26. Displacementorbitsand energyvelocitydiagramsfor modeB of
Fig. 6 at a value of rof= 9 mm//zs. For this mode, v- 10.29-t-•O.35
mm//zs,the asymptoticleakageangle-- 52.16øandthe asymptoticenergy
velocity-- 6.32mm//zs.Thetip of thelargestarrowhead
oneachelliptical
trajectorymarksa synchronous
position(equalphase).The dimensioned
insetarrow scalesthe energyvelocitieswhich run from the centerof each
orbit (unlesssuppressed
for graphicalreasons).Thoseenergyvelocityarrows with hollow heads are associated to flows inside the fiber and those
with solidheadsto flowsin thecladding.Because
of thetangentialdiscontinuityin energyflowat theinterface,twoenergyvelocityarrowsappearfor
orbitscenteredalongthe interface.
1081
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
-0.3
:--..•. --O.l
0.1
0.3
0
Z / Radius
FIG. 28. Displacementorbitsand energyvelocityvectorsfor the rod mode
at a valueof rof= 9 mm//zsassociated
to modeD of Fig.4. Thisisthefifth
rodmodecountingfromthetop left in Fig. 2. The tip of thelargestarrowheadon eachellipticaltrajectorymarksa synchronous
position(equal
phase).The dimensioned
insetarrow scalesthe energyvelocitiesthat run
fromthecenterof eachorbit (unlesssuppressed
for graphicalreasons).The
phasevelocityis29.357for thiscaseand,sincethereisno attenuation,orbits
and energyvectorsare unchangedundertranslation.
Simmonseta/.' Clad rods. I
1081
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This feature is retained even up to very high valuesof •,
where the imaginary componentof the phasevelocityand
leakageanglesare very small,the energyvelocityis almost
exactly bR, and the asymptoticorbits have a major axis/
minor axisratio • 1.75.Figure24 showsthe modeof Fig. 22
at tof-- 4.5 mm/lts, whereit is backwardleaking;and Fig.
25 showsthe third lowesttype 0 modeat tof= 40 mm/lts
that lieson one of the almostlongitudinalvelocitytails arising after modeknitting.
The energyflow curvesfor thesemodesexhibitsomeof
the more common characteristicsencounteredfor leaky
modes.The waveformsin eachof the rod and claddingare
madeup of the sum of two components,one arisingfrom a
shearpotentialandonefrom a longitudinalpotential.In the
cladding,in particular, if only one of thesecomponentsis
present,the waveformdescribedappearsto haveits angleof
leakage and energy velocity almost independentof r, dependingonly on v. When both componentsare presentat
approximatelyequal magnitudes,the waveform produced
hasan energyflow curve (actually a conicalsurfacein three
dimensions)in the overalldirectiondeterminedby the predominant componentbut with overlyingoscillationsproduced by interactionwith the secondcomponent.When
dealingwith wavesof very low exponentialgrowth,this osciliatorycharacterin the energyflow curvecancontinuefor
a substantialdistancefrom the interface,as in Figs. 25 and
32, beforethe exponentiallypredominantterm takesoverto
producean essentiallystraightline. In other casesthe relative decayof oneof the components
may be of the orderof a
wavelengthproducinga flowcurvethatjust appearsto bend
asin Fig. 23, or oneof the componentsmay predominatein
regionsvery closeto the interfaceto producean effectively
straightenergyflow curve.An extremecaseof the curvature
of the energyflow curveis discussed
belowfor an example
drawn from the aluminum-steelsystem.
Figures26 and 27 showexamplesof type 1 and type 2
modes (here modeB in Fig. 6 and mode C in Fig. 7) also
takenat valuesof rof-- 9 mm/lts. The predominantlylongitudinal characterof the type 1 modeand the predominantly
sheartype characterof the type2 modesare seenfrom comparing the energyflow directionwith the orientationof the
particleorbits.
Type 3 modes,whichare not leakymodes,do not exist
in the A1-SiCsystem.We showinsteadthe displacements
and energyvelocityfor two type 3 modesfor the system
listedas2ain TableI usingtof= 15.5mm//•sanda density
3.0
! •
p- 1 = 6.9,whichliesslightly
outside
thatfigure.
These
figuresusethe customaryplotsof the relativemaximum dis-
placements
in theradialandaxialdirections.
Thosepoints
wherether andz displacements
areof thesamesignindicate
a clockwise
particlemotionandthosewheretheyareof oppositesignindicatea counterclockwise
motion.Changes
in
1082
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
AXIAL
DISPLACEMENT
INTERFRC V•LOCITY
.... ENERGY
.........
3.0
2.8
2.0
2.•
E
--
2.L;
1.0
0.0
•\'•.L
Tungsten
0.0
i
o
•
2.2
Aluminum
0.8
0.•
1.2
1.6
2.0
0
2
R / Radius
FIG. 29.Displacement
andenergyvelocityplotsfortheStoneley-like
mode
forthesystem
listedas2ain TableI. Thedensity
wastakenas19.32g/cm3
and rof-- 15.5 mm//ts. That makesthis systemsomewhatsimilarto an
aluminumcladtungsten
rod.The phasevelocityandoverallgroupvelocitiesforthismodeare2.71mm//ts.Whentheradialandaxialdisplacements
areof thesamesign,theorbitalmotionisclockwise;
otherwiseit iscounterclockwise.
the directionof rotationare usuallyseparatedby a point of
rectilinear
motion.
Examination
of several of these modes
showsthat, as the phasevelocityincreases,the numberof
changesin rotational direction increase,and the relative
minimaof the energyvelocitycurvestend to lie near points
of rectilinearmotion. Positionsof equalphaseare not indi-
3.0
-•
'3.0
.
•.o
._>
•
1.o
2•
,• -o.o
._•
E
i
/.•..i•
'
"\
J'
'
\
of 19.32g/½m
3.Thisis similarto thealuminum-tungsten .IE:m - 1ß 0'•• / o•õ•L•C•E•T
\•J
cladrodwithp- 1-• 6.9.Fortype3modes
allelliptical
trajec-
torieshavetheirmajorandminoraxeseitherperpendicular
to or parallelto thecylinderaxis.In Fig.29isshowna plotof
thelowesttype3 mode,whichtendsto theStoneley
modeas
rof tendsto infinity.Figure30 showsthedisplacements
for
that type3 modeshownin Fig. 12(a), exceptat a valueof
RADIAL DISPLACEMENT
....
•
'
RRDIRLDISPLACEMENT
•'i-"•
-x L
.... ENERGY
VELOC
I TT
-2.0
0.0
O.L$
0.8
1.2
1.6
2
2
2
•
2
R / Radius
FIG. 30. Displacementand energyvelocityplotsfor the type 3 Neumann
tunnel related modefor the systemlistedas 2a in Table I. The densitywas
takenas 19.32g/cm3androf-- 15.5mm//ts.Thismodeisshownin Fig.
12(a)forvalues
ofp- • upto5,whileherethevalueofp- • -- 6.9tobetter
matchthe aluminum-tungsten
system.But thecurveisalmostflat overthe
intervening
values
ofp- •. Thephasevelocity
andoverallgroupvelocities
for thismodeare2.88mm//ts. Whenthe radialandaxialdisplacements
are
of the samesign,the orbital motion is clockwise;otherwiseit is counterclockwise.
Simmonseta/.' Clad rods.I
1082
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cated in this figure;doing so would require an additional
curve.
We haveindicatedthat it is possiblefor type 3 modesto
makea transitionto type2 modeswhenthe decayexponent
becomeszero. If the phasevelocityof the resultanttype 2
modebecomesgreaterthan bc, then, in the cylindricalgeometry it alwaysseemsto have an imaginarycomponent,
and,by formulae(23) mustbea leakywave.However,in the
planarinterfacegeometryit maybe possibleto havea purely
real phasevelocity.Sucha modewouldbea combinationof a
planeshearwaveand an exponentiallydecayingwave,and,
thus,be partially decomposable
on the appropriatesideof
theinterface.On that sideof theplanarinterface,the asymptotic energyflow is governedby the energyflow of the plane
waveand is not governedby the asymptoticleakageangle
formulae(22), whichare ambiguousin this case.If, on the
otherhand,the phasevelocityof the resultanttype 2 mode
becomeslessthan bc and its imaginarypart is very small,
then,in eitherthe cylindricalor planarinterfacegeometries,
it has exponentialgrowth away from the interface,but the
leakageangle is essentiallyzero. Strictly speaking,such
wavesdo not leak energy,and we refer to them insteadas
divergentwaves.In additionto divergentwavesof type 2,
divergentwavesof types0 or 1 exist wheneverthe phase
velocityis lessthan ac. Sincethe energyflow for suchmodes
is parallelto the cylinderaxis,the only way to propagatea
beamOffiniteradiusof divergenttypeis to bringthe energy
in from the edgeof the cladding.We are not aware of the
experimentaldetectionof suchmodes.When type 3 modes
of almostthe samephasevelocityas a divergenttype 1 or
type 2 modeexist, then all 3 typesof divergentwaves,althoughpossibleare not likely to be physicallypresent.Divergentwaveshavebeendiscussed
elsewherefor the planar
interface. :ø
There are ten nonredundantplanar interfacemodesfor
the aluminum-silicon carbide system.Four of thesemodes
are decomposable.
The energyflow patternsfor the two type
2,3 modesare shownin Fig. 31(a) and (b). The shift at the
interfaceis apparentin the nondecomposable
mode of Fig.
31(a) and has a magnitudeof about 1.1 mm for a 2-MHz
wave. This is about 70% of the shortest of the four wave-
lengthsinvolvedin this system.In this modethe waveforms
areessentiallyof sheartypeonboth sidesof the interface,but
in the types0,3 and 1,3 they changefrom shearto longitudinal type producinga rather dramatic hump in the energy
flow curves.The type 1,3 mode,for instance,has a 2.6-mm
shift for a 2-MHz
wave that is about 165% of the shear
wavelengthin aluminumand equalto the shearwavelength
in siliconcarbide.Althoughtherearefour planewavesmaking up the decomposable
modein Fig. 32(b), the longitudinal componentin the aluminum is very small giving rise to
an almostpure shearmode in the aluminum.
Five of the 10 modes(thosewith the secondtype index
equalto 3) canbe approximatedby modesin the aluminumclad siliconcarbiderod. Two of the five are decomposable
modesand, therefore,not the limit of a singleleaky mode
branch.The otherthreeleakymodesare high • limits of an
associated
branchof eachof the threeleaky modetypes(02). For the type 1 and 2 modes,this mode has the lowest
limiting phasevelocity, but the type 0 mode has a phase
velocitybetweenbR and ac. Table II liststheseten modes.
We concludethis sectionwith an exampleof possible
technicalinterestin its own fight drawn from the alumi-
num-steelsystem(steel:density= 7.9 g/cm3, aR = 5.92
mm//zs, bR = 3.25 mm//zs). We have notedbeforein discussingchangesof densitythat type2 modescanshiftoverto
type 1 modes.In somesystems,suchasaluminum-steel,this
transitioncanoccurwith changing• whileat a fixeddensity.
TABLE II. The ten planarinterfacemodesfor the aluminum-siliconcarbidesystem.
Nondecomposable
modes
Type
0,3
1,3
2,3
1,1
3,1
3,0
v= Re(v) + iIm(v)
5.56 +
8.64 +
4.73 +
4.62 +
10.48 +
5.56 +
1.1 li
2.46i
0.23i
3.42i
2.45i
1.1 li
Asymptotic
leakage
angle(A1)
in degrees
Asymptotic
energy
velocity(A1)
in mm//zs
Asymptotic
leakage
angle (SIC)
in degrees
Asymptotic
energy
velocity(SIC)
in mm//zs
21.03
5.40
31.82
49.01
6.13
56.89
4.92
5.10
49.21
3.097
6.60
4.71
48.29
4.75
-- 41.62
5.34
-- 73.76
3.09
-- 34.76
9.08
-- 57.88
3.08
-- 13.82
5.62
Decomposablemodes
Longitudinal
planewave
v = Re(v)
1083
(A1)
Shear
planewave
(A1)
Longitudinal
planewave
(SIC)
Shear
planewave
(SIC)
Type
in mm//zs
in degrees
in degrees
in degrees
in degrees
0,3
2,3
3,2
2,1
10.03
10.08
11.53
16.38
50.92
- 51.14
-- 56.75
-- 67.29
72.00
72.08
-- 74.40
79.09
15.84
16.75
33.20
-- 53.90
51.84
58.98
--63.23
71.51
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
Simmonseta/.' Clad rods. I
1083
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7.0
,/
5.0
/'
3.0
./
,/
1.0
-1.0
•
-3.0
2.0
0.0
•
q.0
•
•---•-
6.0
•
8.0
•
10.0
•
12.0
•
,
lq.0
Distancein Tangential Direction (mm)
5.0
FIG. 31. (a) Displacement
orbitsandenergyvelocitydiagramsfor thetype
0,3 nondecomposable
planarinterfacemodelistedin TableII. The frequency usedfor the calculationwas2 MHz, and the apparentshiftalongthe
interface
forthismodeisabout1.1mm.Thetip of thelargestarrowhead
on
eachellipticaltrajectorymarksa synchronous
position(equalphase).The
dimensionedinset arrow scalesthe energyvelocitieswhich run from the
centerof eachorbit (unlesssuppressed
for graphicalreasons).Thoseener-
3.0
•
1.o
/
gy velocityarrowswith hollow headsare associated
to flowsinsidethe fiber
andthosewith solidheadsto flowsin thecladding.Because
ofthetangential
discontinuityin energyflow at the interface,two energyvelocityarrows
appearfor orbitscenteredalongthe interface.(b) Displacement
orbitsand
energyvelocitydiagramsfor the type 0,3 decomposable
planarinterface
modelistedin Table II. The frequencyusedfor the calculationwas6 MHz.
The tip of the largestarrowheadon eachellipticaltrajectorymarksa synchronousposition(equal phase).The dimensionedinsetarrow scalesthe
energyvelocities
whichrunfromthecenterof eachorbit (unlesssuppressed
for graphicalreasons).Thoseenergyvelocityarrowswith hollowheadsare
._
z
-l.O
i:5 -3 0
associated to flows inside the fiber and those with solid heads to flows in the
-5.0
0
{b)
": ..............
0
2.0
q.O
6.0
8.0
10.0
Distance
inTangential
Direction
(mm)
In this system,for instance,one such transition occursat
rof--• 10.177 mm//•s, where type 2 mode transformsto a
type 1 mode at a velocity --•6.32358 mm//•s. The type 1
mode continues for only a short interval down to
rof--• 10.1348whereit hasa cutoffat the longitudinalvelocity of the aluminum cladding.The important thing to observein thiscaseisthat the leakageangleof the type2 mode,
which obeys(23b), is about60øright downto the crossover
point while that of the type 1 mode,which obeys(23a), is
lessthan 1ø.Justabovethe crossover
pointthe phasevelocity
and the two leakageexponentstend to zero. As one moves
abovethe crossoverpoint,the discrepancy
betweenthe leakageexponentsincreases,but becauseit's still small,and because the mode near the rod should look like an almost non-
attenuatinglongitudinalwave,onecanexpectthat in a range
1084
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
cladding.Because
of thetangentialdiscontinuity
in energyflowat theinterface,two energyvelocityarrowsappearfor orbitscenteredalongtheinterface.
of rofabove 10.177mm//•s, there existsa modewhich is
essentiallyconfinedto the rod for somedistance,which is a
functionof rof, and then turns rather sharply--due to the
takeoverof thetype2 exponent--toleakawayat about60øto
the interface.This distancewill be a very sensitivefunction
of rof and, in principle,can producea wave shift of any
magnitude.Figure 32 showsthe displacementand energy
velocityprofile for such a wave at a value of rof-- 10.5
mm//•s, wherethe magnitudeof the apparentshiftis about
14.6 times the radius of the rod.
D. Asymptotic leakage angle
The distributionof asymptoticleakageanglesfor the
A1-SiC exampleagaincorresponding
to Figs.4-7 is presentSimmonseta/.' Clad rods.I
1084
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20.0
16.0
,
/
ß
12.0.
//
8.0,
/
LJ O'
0.0
0
0 .....
,'0 .....
8'0 ....
i2 'ø ....
i6'0 .... 20'o
Z / Radius
FIG. 32.Displacement
orbitsandenergy
velocity
diagrams
fortype2 modeinthealuminum-steel
system
atrof= 10.5,whichliesnearacrossover
toa type1
mode(aluminum:
density
= 2.77g/cm3,ac = 6.323mm//•s,bc= 3.1mm//•s;steel:
density
= 7.9g/cm3,an = 5.92mm//•s,bR= 3.25mm//•s).Forthis
mode,v = 8.33+ i0.02mm//•s,theasymptotic
leakageangle= 60.66*andtheasymptotic
energyvelocity= 3.09999mm//•s.Thetip of thelargestarrowheadoneachellipticaltrajectorymarksa synchronous
position(equalphase).Thedimensioned
insetarrowscales
theenergyvelocities
whichrunfromthe
centerof eachorbit (unlesssuppressed
for graphicalreasons).Thoseenergyvelocityarrowswith hollowheadsareassociated
to flowsinsidethe fiberand
thosewithsolidheadstoflowsin thecladding.
Because
ofthetangential
discontinuity
in energy
flowat theinterface,
twoenergy
velocityarrowsappearfor
orbitscenteredalongthe interface.The apparentshiftalongthe interfacefor thismodeis almost15 timesthe rod radius.
ed in Figs. 33-36. As with the dispersioncurves,a uniform
trendisapparent:at valuesof•bwill belowthephasevelocity
peak,the leakageangleis large,near90ø, and,in the caseof
the type 0 radial modes,the leakageangleactuallyexceeds
90øfor the valuesof •bwherethe modeis backwardleaking.
As •bapproaches
the positionof the maximumin phasevelocity,theleakageanglestartsto drop,andcontinuessountil
reachingan asymptoticlimit. Dependingon the relativevaluesofac, aR, bc, and bR,theremay be a modewith leakage
angletendingto that of an associated
planarinterfacemode,
but all other rod-relatedmodes,whosephasevelocity possessesa small--but usually nonzero--imaginary component, have a leakageanglegivenby (23a) or (23b). Note
that, sincethelimitingphasevelocitiesareindependent
ofp,
soaretheselimitingleakageangles(althoughtherangeofp
wherethe limiting valuescanbe usedmay dependon •b).
It is possibleto analyzethe leakageanglefor thistypeof
mode,with relativelysmallimaginarycomponentin v, rather completely.If we set c = ac or bc and C = A c or -Bc,
respectively,
then, whenRe(v) > c, the leakyexponenttakes
the form
exponent
• d- Im (k) / tan( IC I),
(28a)
the plussignoccurringfor types0 or 1 whenc = ac and for
types0 or 2 whenc = bc. Similarly,whenRe(v) < c,
exponent•---F_
Re(k) [ 1 -- Re(l•)2/c2]1/2
(28b)
(23b) determinesthe leakageanglefor type 2 modes,thus
giving rise to a possiblediscontinuityin leakage angle at
crossoversfrom type 1 to type 2 modesas noted above.
At any fixedvalueof •b,then,thereare infinitelymany
modeswith leakageangle near 90ø, sinceanglesnear 90ø,
togetherwith very low phasevelocitiesis characteristicof
high-ordermodeswell belowtheir peakvalueof •b.At higher
valuesof •bthere are alsomany modeswith leakageangle
near their limiting value. In addition, one can expecta few
modeswith an intermediateleakageangle. Except for low
valuesof •b,where there are no other modeswith a given
limiting leakageangle,it is theseintermediatemodesthat
best offer the possibilityfor experimentaldetectionin the
cladding.Usually, thosemodeswith a higher leakageangle
have a greaterattenuation,so that their amplitude will be
relativelyweak at points (r,z) in the claddingwhere r/z is
the tangent of the mode with intermediateleakageangle.
(Here, z has been measuredfrom the leading edge of the
leakageandwehaveassumedthat the initial amplitude--the
"injection coefficient"--of the intermediate mode is sufficiently large comparedto modeswith higher leakageangles.) The intermediatemode will then appear in almost
pureform overa rangeofz (at fixedr) up to the leadingedge
of the next leaky mode. Even when pairs of intermediate
modescoexist,one may have a sufficientlylarger injection
coefficientto make it the predominantmode.
and when Re (v) • c
exponent• d- (2•rf/c)[ Im(v)/c].
(28c)
From Eqs. (28) it canbe seenthat for type0 or 1 modes,for
instance,and for almost all instanceswhere a c/bc is not
near one,Eq. (23a) determinesthe leakageangle.Similarly
1085
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
E. Limiting energy velocity
The limiting energyvelocityrepresentsthe magnitude
of the energyflow at large distancesfrom the rod/cladding
Simmonseta/.' Clad rods.I
1085
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100.0
i
60.0
•iO.
2o.
0.0
0 0
•.0
8.0
12.0
16.0
20.0
2•.0
28.0
32.0
36.0
N0.0
RadiusX Frequency(mm/l•S)
FIG. 33.Asymptotic
leakage
angle
indegrees
forthetwo,axialrodmoderelated
subfamilies
oftype0 leakymodes
asafunction
ofrodradius
timesfrequency
(rof) in thealuminum-silicon
carbidesystem.
Thesecurves
correspond
to thephasevelocitycurves
of Fig.4
100'O
i
80.0
60.0
q0.0
20.0
0.0
0
N. 0
8.0
12.0
16.0
20.0
2N. 0
28.0
32.0
36.0
gO
Radius X Frequency(mm/•s)
FIG. 34.Asymptotic
leakage
anglein degrees
for theradialrelatedsubfamily
of type0 leakymodesasa functionofrodradiustimesfrequency
(rof) in the
aluminum-siliconcarbidesystem.Thesecurvescorrespond
to the phasevelocitycurvesof Fig. 5.
•
•
80 0
-
,-
60
•
•0.0
0
._o
E
• 20.01
0.0
0
0
4.0
8.0
! 2.0
16.0
20.0
2N. 0
28.0
32.0
36.0
NO. 0
RadiusX Frequency(mm/ixs)
FIG. 35. Asymptoticleakageanglein degreesfor the family of type 1 leakymodesasa functionof rod radiustimesfrequency(rof) in the aluminum-silicon
carbidesystem.Thesecurvescorrespondto the phasevelocitycurvesof Fig. 6.
1086
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
Simmonseta/.' Clad rods.I
1086
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FIG. 36.Asymptoticleakageanglein degrees
for thefamilyof type2 leakymodesasa functionof rodradiustimesA frequency(rof) in thealuminum-silicon
carbidesystem.Thesecurvescorrespondto the phasevelocitycurvesof Fig. 7.
interface.At thesedistancesthe directionof energyflow is
givenby the asymptoticleakageangle.The limiting energy
velocityisthenderivedusingwhicheverof the regimes(21a)
or (2lb) is valid. For the examplesshownherein, this is
usually(21b) for type 2 modesand (21a) for types0 and 1
modesexceptfor backwardleakingmodes,whereit becomes
(2lb).
Figures37-40 showthe limiting energyvelocityfor the
A1-SiC example.Exceptfor the tunnelrelatedtype 1 mode,
the types 1 and 2 curveslimit at high valuesof • near the
longitudinaland shearvelocitiesof the cladding,respectively, whilethe type0 behavioris morecomplex.The asymptotic energyvelocitycurvesfor the type 0 modesin Fig. 37
peakat a valuejust belowac at valuesof • slightlybeyond
the phasevelocitypeaksin Fig. 4. [The type0 modelabeled
"B," which is relatedto type 0 Neumanntunnel modeis an
exceptionto this. This mode, which tendsat high • to a
planarmode,hasa largecomplexcomponentto its velocity,
and doesnot follow (23a). ] In this region the asymptotic
particlemotionis describedby eccentricellipsesorientedin
the directionof leakage,andthe valuesof Im (v) / Re( v) are
alsorathersmall,sothat Eq. (23a) isapproximatelycorrect.
This equationcan, there,be interpretedas givingthe phase
velocityalongthez axisin termsof a projectionof the natural wave motion in the direction of asymptoticenergymotion, where the phaseand energyvelocitiesare equal, onto
thez axis.As the phasevelocityapproaches
that of the cladding,theasymptoticenergyvelocityalsodrops,andboththe
energyand phasevelocitieseventuallyapproachthe shear
velocityof the rod. The asymptoticparticlemotionhereremains elliptical and parallel to the interface. For those
modeswhosephasevelocityremainsnear as, suchas that
whosedisplacements
near the interfaceare.indicatedin Fig.
25, the asymptoticenergyvelocityremainsnearac. At large
]0.
2.0
0.0
................................................................
O. 0
•l. 0
8.0
12.0
16.0
20.0
2•. 0
Radius X Frequency(mm/i•s)
28.0
32.0
36.0
•to
o
FIG. 37. Asymptotic
energyvelocityin mm//zsfor the two,axialrodmoderelatedsubfamilies
of type0 leakymodesasa functionof rodradiustimes
frequency
(rof) in thealuminum-silicon
carbidesystem.
Thesecurvescorrespond
to thephasevelocitycurvesof Fig.4.
1087
J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992
Simmons et al.' Clad rods. I
1087
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I0.
0.0
0
0
0,. 0
8.0
12.0
16.0
20.0
2•i. 0
28.0
32.0
Radius X Frequency
FIG. 38.Asymptotic
energyvelocity
in mm//•sfortheradialrelatedsubfamily
oftype0 leakymodes
asa function
ofrodradiustimesfrequency
(rof) in the
aluminum-siliconcarbidesystem.Thesecurvescorrespond
to the phasevelocitycurvesof Fig. 5.
o
o
O. 0
0,. 0
8.0
12.0
! 6.0
20.0
2•i. 0
28.0
32.0
36.0
NO
0
Radius X Frequency (mm/l•s)
FIG. 39.Asymptotic
energy
velocity
inmm//•sforthefamilyoftype1leakymodes
asafunction
ofrodradius
timesfrequency
(rof) inthealuminum-silicon
carbidesystem.Thesecurvescorrespond
to the phasevelocitycurvesof Fig. 6.
10.0
.-:I.
8.0
E
E
o
6 0
c
g.O
__o
ß
A
• 2.0
0.0
0
Radius X Frequency (mm/ixs)
FIG. 40.Asymptotic
energyvelocityin mm//•sforthefamilyoftype2 leakymodesasa functionofrodradiustimesfrequency
(rof) in thealuminum-silicon
carbidesystem.Thesecurvescorrespond
to thephasevelocitycurvesof Fig. 7.
1088
J. Acoust.Soc. Am., Vol. 92, No. 2, Pt. 1, August1992
Simmonset al.: Clad rods.I
1088
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distancesfrom the interface,wherethe asymptoticformula
holds,the particleorbitsare very eccentricand essentially
that of a longitudinalwave in the claddingmovingin the
directionof asymptoticleakage.
The curvesin Fig. 38 showsimilarbehaviorexceptfor
the discontinuities
occurringwhenthe modesbecomebackwardleaking.The discontinuityin asymptoticenergyvelocity that occursupon transitionfrom forward to backward
leakingmode (or vice versa) is due to the changein the
dominantasymptotictermsfrom (21a) to (21b). Justbelow
the value of •b marking this transition,both asymptotic
growthexponents
arealmostzero,andthedistanceto reach
the asymptoticlimit is very large.The asymptoticshapeof
theparticletrajectories
tendsto beelongated
in thedirection
of leakageperpendicularto the interface.Just abovethe
transitionvalue of •b, the growth exponents,which have
changedsign,remaincloseto zero,but nowthe decayof the
shearrelatedexponentis lessthan that of the longitudinal
relatedexponent.The consequentasymptoticshapeof the
particletrajectoriestendsto be elongatedperpendicularto
the directionof leakage,or parallelto the interface.
III. CONCLUSION
The classof nonattenuatingguidedmodesin a clad rod
systemis restrictedto thosematerialswherethe shearvelocity of the rod is lessthan that of the cladding.In other systemsall modeswill leak energyfrom the rod into the cladding, and even in this restrictedclassof materials,modes
leakingenergyfrom the rod into the claddingexist.
Over a largerangeof elasticparameters,thewavemodes
in a clad rod system,which break down into four families-three of which are leaky or divergent--can be correlated
with the modesof the bareelasticrod or the claddingwithout a rod (tunnel), sometimeswith traction-free boundary
and sometimeswith zero-displacementboundary conditions. There are infinitely many rod-relatedmodesand five
tunnel-related
modes.
There are five dimensionless
parametersneededto describean axisymmetricmodein a cladrod with infinitecladding. This presentsa variety of wave behaviorthat is too
greatto presentin detail. Instead,the variationof modebehavior with changein rod radiusor frequencyis given in
detailfor oneexampleof a siliconcarbiderod embeddedin
an aluminum
matrix.
We may summarizetheseresultsin the A1-SiC system
by observingthat, in additionto the correlationwith rod and
tunnelmodes,overmuchof the rangeof•bwherethe attenuationis not excessive,
mostleakymodesin the claddingmay
be thoughtof approximatelyaslongitudinalor shearwaves
in the claddingwhich are modifiedlocallyin the neighborhoodof the interfaceby interactionwith a waveof the other
type.The principalwavetype is longitudinalfor types0 and
1 and shearfor type 2. The asymptoticorbit shapesare eccentricellipsesorientedin directions,respectively,parallel
(perpendicular)to the directionof energytravel for modes
of type0 or 1 (2). Far from the interfacethesewaveshavea
naturaltraveldirectionalonga conethat is tracedout by the
energyvelocityfield, and they exponentiallydecayin a di1089
J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992
rection perpendicularto this direction.The phasevelocity
alongthez axisis essentiallyderivedfor thesewavesby projection from the natural directionof motiononto the z axis.
For a variety of systemsother than A1-SiC, the behavior
of the tunnel-relatedmodes under changeof density was
studied.Thesesystemswere chosenwith differentordering
regimesin the wave velocities.In those systemswhere
bR < bc, the modebehaviorbreaksinto a tunnel dominated
and rod dominatedregion with a boundaryoccurringapproximatelyat the valueofpR wherethe shearimpedance
(pb) of the two mediaare approximatelyequal.In those
systemswhere b• > bc, the type 0 and 1 modesexhibiteda
cutoffat a valueofp• wheretheshearmoduliof thecladding
and rod are equal.
The pattern of energyflow was analyzedusingthe detailedenergyvelocityfieldin a geometricallydispersivesituation. This type of analysisdoesnot seemto havebeencarried out before.It wasshownthat a lacunacan be expected
after the start up of a leaky wave, and that the generatorof
this conelikelacuna, after tracing a curved path near the
interfacedue to the interferenceof the two componentsof
the leaky wave,becomesa straightline inclinedat someangleto the rod axis.Evenat highfrequencies,
whenthe imaginary part of the highermodesin a family is quite small,the
leakageanglefor a modecan be finite and largeif the wave
speedsin the rod are largerthan thosein the cladding.
Becauseof the interferencepattern near the interface
betweenlongitudinaland sheartypecomponents,the energy
flow curvesand, therefore,the edgeof the lacunaof a leaky
modemay shiftfrom the positionestimatedfrom the asymptotic leakageangleof themode.This shift,analogousto similar shiftsfound in opticsand in acoustics,can be obtained
directlyby followingthe energyflow pattern.In someleaky
waves,the energypattern actually reversesitself and flows
backwards;if this occursasymptoticallyin the cladding,the
leaky wavesare backwardleaking. In other casesthe shift
canbearbitrarilylargeandadjustedasa sensitivefunctionof
frequency,perhapsallowingunusualwaveguidesto be constructed.
In addition to the complex patterns found in leaky
waves,the energyvelocityfield displaysthe interferencebetween planeand/or inhomogeneous
wavesnear a plane interface.This pattern differsfrom the averagedenergyflow,
givenby the groupvelocity,evenin bare rods,and for clad
rod systemspossessing
guidedmodes,the energyvelocityat
largedistancesfrom the interfaceis equalto the phasevelocity of the cladding,not, as might be expected,the group
velocity of the mode.
Leaky modeswereconnectedwith similarmodesoccurring at planarinterfaces.For the cylindricalgeometrythere
are an infinitenumberof suchmodes,but only a finite number in the planar interfacegeometry.It was seenthat one
clad rod systemmode, not always that with lowest phase
velocity,tendsto that of the planarinterfacesystem,if sucha
modeexistsandhasnonzeroasymptoticleakageangle.Otherwise,sucha modeis a clusterpoint for rod-systemmodes,
but is not the limit of any one mode. Decomposableand
partially decomposable
planar interfacemodeswere describedas was high-frequency
phenomenonof mode knitSimmonseta/.: Clad rods. I
1089
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ting in clad rod modes.Indeed,it wasshownthat the topologicalcharacteristics
of the modal structurein clad rods
seemsto differ substantiallyfrom that of a bare rod; that is,
we have found that there is a rich structure of leaky, or
evanescent,
modesin a clad rod systemand that, depending
on the particularmodeand frequency,the energyleakedby
suchmodesconveysinformationaboutboth the rod and the
cladding.
If the structurenearthe interfacezoneis morecomplex,
i.e., if the elasticconstantsvary with r andwithf, mostof the
methodsdevelopedherein can be extendedto describethe
leakymodes
in thatstructure.
19Amongthemodes
ofsucha
clad rod systemwith interfacezone, one can expectto find
some that are sensitive to the elastic structure of the interface
zone.Torsionalsensitivitycanbe includedby extendingthe
aboveanalysisto 0 dependentmodeswhere n • 0. In such
casesthe particleorbitswill no longerbe planarellipsesbut
will follow slightly more complicatedthree-dimensional
paths.Anisotropyin thedirectionof therod axiscanalsobe
includedfor the cladrod system,but morecomplexanisotropiescanonlybetreatedin thecontextof theplanarinterface
usingthe type of formulationdescribedin this study.
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14E.Drescher-Krasicka,
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StrahlsbeiU1ACKNOWLEDGMENTS
traschallwellen," Acustica 18-19, 2 (1952).
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Analysis,
editedbyG. Springer
andE. H. Spen-
We are particularlygratefulfor the early supportfrom
the Office of Naval Researchand the continuingfinancial
supportfrom the National Instituteof StandardsandTechnology'sOffice of NondestructiveEvaluation.In addition
we would like to acknowledgevaluablediscussions
with A.
H. Kahn and W. L. Johnson.
ier (McGraw-Hill,
New York, 1979), 3rd ed.
•?M.Onoe,H. D. McNiven,andR. D. Mindlin, "Dispersion
of Axially
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J. A. Simmons,and H. N. G. Wadley,"Guided
InterfaceWaves,"in Reviewof Progress
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19H.N. G. Wadley,J. A. Simmons,
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IR. N. Thurston,"ElasticWavesin RodsandCladRods"(reviewpaper)
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Simmonseta/.: Clad rods.I
1090
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.143.68.142 On: Wed, 30 Oct 2013 18:53:49