Leaky axisymmetric modes in infinite clad rods. I axisymmetric modes in infinite clad rods. I ... trix and many of the modes are leaky, transmitting energy ... between the modes created

Leaky axisymmetric modes in infinite clad rods. I John A. Simmons

Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899

E. Drescher-Krasicka

Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Mary/and 20899 and The Johns Hopkins University, Baltimore, Maryland 21218

H. N.G. Wadley a) Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899

(Received 1 November 1990; revised 24 March 1992; accepted 30 March 1992)

A detailed computational study is presented for the radial-axial modes-- both leaky and nonleaky--in an infinitely clad isotropic rods. The complex phase velocities of leaky modes are located using an application of the argument principle. Particle orbits are determined, and leaky modes are shown to have an asymptotic leakage angle away from the interface. By using the homotopic methods of varying densities and elastic constants, clad-rod modes are compared with those in a bare rod. The topology of the clad-rod mode dispersion diagram differs qualitatively from that of a bare rod, even when the cladding has negligible density, with no velocity cutoffs and with wave mode knitting. Comparison is also given with modes occurring in a cladding without a rod present (a tunnel) and for a planar interface. Most leaky modes can be correlated with rod modes; only a limited number of tunnel modes exist. Energy flow contours within modes are also calculated. The local energy velocity, which generalizes group velocity, can vary considerably in the radial direction for bare rod modes. For leaky modes the contours are quite complex due to the cylindrical geometry, giving rise to apparent shift in wave position across the interface.

PACS numbers: 43.20.Jr, 43.35.Cg, 43.40.Cw

LIST OF SYMBOLS

r,O,z

u(r,z)

subscript rn = C subscript rn = R pc,ac,bc

Io,I1,Ko,K1

A c ,A R ,B c ,B R

radial, polar, and axial coordinates in cylindrical coordinate system [Eq. (1)] displacement field in radial-axial co- tc ordinates [Eq. ( 1 ) ] ro potential functions [ Eq. (3) ] f frequency in radians [ Eq. (4) ] (complex) wave number [Eq. (4)] cladding [ Eq. (5) ] rod [Eq. (5) ] cladding density, longitudinal and shear wave speeds [Eq. (5) ] rod density, longitudinal and shear wave speeds [ Eq. (5) ] complex leaky mode velocity = co/k [Eq. (5) ] modified Bessel functions 0L [Eq. (5) ]

p,a c ,a n ,/3c,gb

qcr,qc•,qnr,qnz

[ E 1 (r,2) ,g 2 (r,2),

E3(r,z) ]

4- x/1 -- v2/a•, -i- •/1 -- v2/a• ,

4- x] l -- v:/b •, 4- 41 - v:/b • [Eq. (5) ] rok = 2•rrof /V [Eq. (5) ] radius of rod [ Eq. (6) ] frequency = w/2•r [ Eq. (6) ]

Pc ac a• bc rof [Eq. (7)] pR b• b• b• b•

complex moduli for particle orbits [Eq. (11)]

energy velocity field = (.(P1) (r,2) {E)(r,z)

(P2}(r, z) {P3}(r,z)) (E)(r,z)' (E)(r,z)' [Eq. (19)] asymptotic leakage angle [Eq. (22)]

INTRODUCTION

The topic of normal modes in clad rods has a substantial history much of which has been summarized by Thurston. 1 In most cases attention has been concentrated on the use of

a) Present address: Department of Mechanical Engineering, University of Virginia, Charlottesville, VA 22901.

such structures for waveguides. Relatively little effort has been expended on the study of other modes that could be used for interface characterization in fiber reinforced com-

posites. For such systems the fibers are stiffer than the matrix and many of the modes are leaky, transmitting energy into the surrounding medium. The existence of this leakage energy offers a potential means of monitoring, or even imag-

1061 J. Acoust. Soc. Am. 92 (2), Pt. 1, August 1992 0001-4966/92/081061-30500.80 @ 1992 Acoustical Society of America 1061

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

ing, the characteristics of the interface zone. Jen, et al. 2 have specifically addressed leaky modes in clad rods, but their emphasis has also been towards waveguide applications.

In the work reported here we study in detail the radial- axial modes in an infinitely clad isotropic rod. In spite of the simplicity of the geometry, we show that the topology of the clad rod modes differs from that of a bare rod, even when the matrix density is very small.

The topic of energy flow occupies a significant part of this study. While the group velocity (dw/dk) has been rou- tinely calculated for modes in the cylindrical geometry, 3 and while Safaai-Jazi et al. 4 discuss the overall Poynting vector field, it does not seem to have been recognized that the cylindrical geometry introduces significant nonuniformity into the energy flow field. Even in a bare-rod mode, whose overall average energy velocity is given by the group velocity, the local energy velocity can vary in the radial direction so much as to change sign, delineating regions of reversed energy flow. In the case of clad rods, this can lead to apparent shifts in wave position of several wavelengths. These shifts are analogous to the Goos-H/inchen shifts in optics, but they will be derived herein without resort to a Gaussian wave train.

I. THEORY

A. Equation for axisymmetric modes

The equation governing the phase velocity for self-propagating linear elastic axisymmetric modes in an isotropic rod with infinite cladding follows from the more general expression for all modes in an isotropic rod with finite cladding that appears in several places in the literature. 1'3'• The dynamic elastic wave operator • may be written in Cartesian coordinates:

,•/•(u)=.•-fx • Cq• 0x• --p 8t----•-, (1) where u i are the displacement components, •5 the mass density, and the isotropic elastic constants are given by

Cijk, = Xaijakl -3 L- [.•( ailajk -3 L- aikajl ), (2 t ,

2pc b •.4 cKi (.• off) 2pc ( 2 b • -- v • ) K I ( BcK )

pc{(• -- 2b•)Ko(.4cK ) - 2b•.4cKl(.4cK)/K } -- 2pcb•c{BcKo(Bc K) •- Kl(BcK)/K}

Ko(.4cK) BcKo( Bc K)

.4cK I (.4cK) Ki (BcK)

Here a', ]3', 7', •5' are constants--not all zero--to be discussed in Sec. II B, Pm ,am, and b m are the density, longitudinal, and shear wave speeds of the rod (m = R ) and cladding (m = C), respectively, v= w/k, is the complex value whose real part is the phase velocity of the mode and whose imaginary part describes the growth or decay of the eigenwave, Io, 11, Ko, and K1 are modified Bessel functions 6'7 and

z4 m = -Jr- ( 1 -- ve/a• )1/2 B m = q- ( 1 -- v:/b: ) ]/: (6) m

•c = rok = roW/V = 2rrrof /V,

where ro is the radius of the rod andf is the frequency. This

where/l and p are the Lam• elastic constants. In the fiber geometry cylindrical coordinates are a more

convenient means of representation. Here, material homo- geneity with respect to axial translation and radial rotation allow the eigenfunctions of the wave operator to be written in the form F(r) exp(inO) exp(iwt- kz). A general dynamic isotropic elastic solution can be expressed in terms of potential functions in each material:

u = v• + v.T, V.T = G, (3)

where G is an arbitrary function of r, 0, and z. It can be shown that only three of the four potential functions in • and T are independent, and that the r dependence of these can be expressed in terms of Bessel functions of, perhaps, complex arguments.

In addition to (3) the boundary conditions determining self-propagating modes in a rod with finitely thick cladding are continuity of normal tractions and displacements at the rod/cladding interface and vanishing surface tractions at the surface of the cladding. When the cladding is infinitely thick, this leads to a matrix equation of the form Mx = 0, where M is a 6 X 6 matrix depending on phase velocity and the potential functions, and x is a set of unknown constants. The condition for solving this equation is that the determinant of M be zero.

When there is no 0 dependence, i.e., when n = 0, and when the cladding is infinite, i.e., when there are only four boundary conditions, one of the components of the potential function becomes redundant, so that T reduces to (0, T,0) and M becomes a 4 X 4 matrix. The potential functions then have the form:

d) = d)( r)e i(ø•'- •), T = T(r)e •(•'- •). (4)

The self-propagating modes from (4) are the axisymmetric modes. They are independent of 0 and have no torsional (or transverse shear in the planar limit) component. The displacements are, thus, only in the radial and axial directions. The matrix equation takes the explicit form:

2psb•.4sll(.4sK) ps (2b• -- 02)Ii(BsK)

ps{(2b• -- o2)Io(AsK) -- 2b•AsIi(AsK)/K} 2psb•{Bslo(BsK) -- Ii(BRK')/K '} -- 4,(A•K) --

ARIi(ARK) Ii (BRK)

a'

=0.

(5)

I equation is solved by looking for values of v for which the determinant of M vanishes as is explicitly discussed in Sec. II B. Columns 1 and 2 refer to the contribution in the clad-

ding, while columns 3 and 4 refer to those from the rod; columns 1 and 3 refer to longitudinal contributions while 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 balance of displacements; and rows 1 and 3 refer to the axial component while rows 2 and 4 refer to the radial component.

Equation (5) is an equation for v in terms of seven parameters: Pc, PR, ac, aR, bc, b•, and rof The first two parameters have dimensions of density and the last five param-

1062 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons eta/.' Clad rods. I 1062


eters have the dimension of velocity. By dividing through by one of the densities and one of the velocities squared, Eq. (5) can be reduced to a dependence of a dimensionless phase velocity on five dimensionless parameters. We shall frequently use such parameters and, anticipating the predominant role of the rod modes, we shall designate:

P =Pc/P•, a• = a•/b•,

ac = ac/b•, qb = rof /b•, (7)

/3c = bc/b•.

When dealing either with a bare rod or an infinite cladding with no rod present, Eq. (5) factors, and only the traction term need be retained. In that case, the dedimensionalized r is a function of only two dimensionless parameters, e.g., v?b is a function of rof/b and a/b (or Poisson's ratio). Even whenp, the ratio of the densities, is near zero or is very large, the determinant of M essentially factors into the product of two 2 X 2 subdeterminants. When p is near 0, these two de- terminants express the vanishing of the tractions at the surface of the rod and the displacements at the surface of the cladding, while for very large p, the two subdeterminants describe the vanishing of the tractions at the cladding interface and the vanishing of the displacements at the rod surface. In these cases, either solutions to the traction-free Neu- mann problem or the dual displacement-free Dirichlet problem will yield acceptable values of v. The close relation- ship between the infinitely many eigenvalues for the Neu- mann and Dirichlet problems for the rod and the very limited number of solutions for the cladding can be used, as will be seen, to qualitatively explain the close correspondence between the modes created on bare and clad rods.

Because of the possibility of using either a plus or minus branch for each of the square roots in (6), Eq. (5) appears to be sixteen different equations. However, the modified Bessel functions used to represent the solid rod, Io and I•, are, respectively, even and odd functions of their argument. Since I1 always occurs multiplying an A m or Bin, one sees that the plus or minus square roots associated to the rod have no effect on the determinant equation. Thus, there are only four distinct equations (leading to four branches ) for axisymmetric modes in the rod with infinite cladding. Physically this occurs because the solution must be finite at the center of the

rod, thus eliminating three types of divergent solutions that could occur if the rod were hollow or if one were dealing with a planar interface.

Since all the axisymmetric branches will be frequently used, the four branches solving Eq. (5) will be labeled with an abbreviated binary notation, referring to a mode as an s mode, where s - 2p q- q goes from 0 to 3 and p and q are either 0 or 1 according to the choice of the negative principal square root value (0) or positive principal square root value ( 1 ) for the shear (p) or longitudinal (q) component, respectively. A 2 mode would then arise by choosing Bc = ( 1 -- o2/b • )1/2 andAc = -- ( 1 -- o2/a• )1/2. As can be seen from the asymptotic behavior of the Bessel functions, [ see (10) ] if o lies in the first quadrant of the complex plane, the negative principal square root value (0) gives rise to a solution exponentially increasing in the radial direction, while a positive principal square root value ( 1 ) yields a solu-

tion exponentially decaying in the radial direction. Thus, for such v's, the exponentially decaying (including Stoneley- like) waves are type 3 modes; leaky modes arise from the solutions to any of the other branches. Because of the exponential increase in the radial direction, leaky modes have infinite energy but if truncated outside of an energy flow curve, as discussed below, such modes become wave beams of finite energy, with bounded displacements that are exponentially decaying away from the flow curve and that almost satisfy the wave equation.

Other symmetries of Eq. (5) could be exploited to re- strict v to the first quadrant of the complex plane. If v is replaced by •, its complex conjugate, the matrix M becomes M since the Bessel functions have this property. Further- more, v only occurs squared except in combination with A m or B m (rn = C or R). Thus, if v is a solution to the determinant equation, so is •; - v is also a solution, provided that the mode type, s, is replaced by 3 - s. It is possible to select, then, that solution for which both the real and imaginary parts of v are non-negative. This gives rise to a (possibly) attenuating (strictly, not growing in the z direction) mode with positive phase velocity. However, as will be shown, there are certain "backward-leaking modes" whose energy flow in the rod is in the reverse direction to the phase velocity at the interface (such modes have also been discussed in the context of the optics of planar interfacesS). For such modes the phase velocity is negative even though the attenuation occurs in the positive z direction; the use of a positive phase velocity and type 3-s mode for these cases reverses the sense of time, producing an "absorbing" rather than "leaking" mode. Our convention, then, is to use a non-negative imaginary component for v, precluding exponential growth along the interface in the positive z direction and, at large distances from the interface, to only permit energy to flow parallel to, or away from, the interface.

B. Expressions for particle displacement

The determinant of the matrix M that occurs in Eq. (5) has a range in its order of magnitude which is prohibitively large. In order to solve this equation over a wide range of parameters, we make use of the asymptotic expressions (Reft 6, p. 203 ):

e • - _gg io ( •. ) • q- i e •-1/2 T/' 3rr

e g+ie- ' 2 < argO< 2 ' I1 (•') •" •172 K m (•) • e- •1/2, m • 0,]. (8)

We note that the exponential behavior of the I•'s can be accommodated by the use of cosh(•) and we divide each of the columns of M by an appropriate exponential factor while dividing the entire determinant by •/B •. The latter factor removes the square root of z dependence of the Bessel functions while the B • term removes a spurious v -- B• root.

These operations have no effect on the phase velocity determined from the vanishing of the determinant, but they multiply each of the coe•cients a• - g• (which are obtained as 3 • 3 minors of M across any row of M) so that:

1063 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons otal.: Clad rods. I 1063


at = Car'to 1/2 exp(ActC), /3= c/3 'tc 1/2 exp(Bctc), 7/= c•'•d/2 cosh (A • •c ),

(9)

6 = c6'•c 1/2 cosh(Ba•c). The complex factor c only affects an arbitrary uniform change of magnitude and starting phase, so that at any value of r, the expression for the displacement will be in the form:

Um, ( r,z,t) = Re(qm,e i(ø't- •'•) ) , Um• (r,z,t) = Re(qm•ei(ø't-•'•)),

(10)

where Re(•) is the real part of • and where

qc• = -- a + il• , tc •/2 exp( -- Acre) td/2 exp( -- Bctc)

qcz = -- fl kBcKø( Bctcr/rø ) kKø( Actcr/rø ) -- ia

•c 1/2 exp( -- Bc•c) tc 1/2 exp ( -- A ctC)

(11)

The particle orbits are then elliptical with Iqmrl and I qa• I being the magnitudes of the maximum excursions in the r and z directions, respectively. The negative phases of the q's determine the positive phase at which this maximum Occurs.

We have observed in Eq. (5) that v is a function of rof (or v/ba is a function of • = rof/ba ). From Eq. ( 11 ) one can see that if ro f is fixed (so that both to and k vary as 1/to), then roqmr and roqmz are functions of r/to.

Unlike ordinary modes whose velocities are real and

whose elliptical orbits have axes always oriented parallel and perpendicular to the z axis, the ellipse orientation for leaky modes changes as a function of r (but not z), and is inclined at angles 0m and 0m _ •r/2, where

( qmz -[- qmr ) ) (12) Om= ----1 tan-1 Im ( 2 2 2 Re (q2•z + q2m•) '

The direction of rotation about an elliptical orbit can also change with r. This change of direction is signaled by the minor orbit of the ellipse degenerating to zero, i.e., inclined rectilinear motion. Such changes are also a common feature of ordinary rod modes and Stoneley waves.

Examination of the determinant in Eq. (5) also shows that the root v = b• exists for every value of rof. This root comes from column 4 of the matrix and arises because of the

asymptotic expansions of Io (z) --. 1 and I1 (z) • z. The mode determined by this root is identically zero, so we shall re- move it by dividing the determinant by B•.

C. Limiting planar interface modes

When the interface is planar, translational symmetry in the two orthogonal directions parallel to the interface gives rise to potentials:

q• = a exp( -- kAcx + icot- ikz),

ß = • exp( - kBcx + icot- ikz) in the cladding and

(13a)

ß = 7/exp( -- kAax + icot- ikz), (13b) ß = 6 exp( - kBax + icot -- ikz)

in the rod.

The boundary condition matrix equation then has the form:

=0. (14)

All sixteen branches of the matrix equation can produce solutions to the planar interface problem. Consequently we use a double index notation for a planar interface mode, referring to s,t modes, where s and t range from 0 to 3 as in the cylindrical convention. The first index will refer to the cladding material and the second index to the rod material. The planar interface equation has the additional symmetry that if v is a solution for an s,t mode, it is also a solution for a 3 - s, 3 - t mode. Only those modes for which s or t equals 3 can be limiting cases from the cylindrical geometry.

Thus, for example, although the bare rod solution, corresponding tOpc = 0, has only one branch due to the bounded nature of the rod, the corresponding limiting planar solution has two branches--the 0 or 3 mode and the 1 or 2 mode.

Only the 3 mode corresponds to the limiting case obtained

from a rod, while both modes correspond to the limiting case obtained from the tunnel-shaped cladding with no rod inside.

For the cylindrical case, when the radius of curvature ro approaches infinity while the frequencyf remains fixed, the results might be expected to converge to those of the planar case. This is not so, however, as there are infinitely many modes in the cylindrical case for which there are no planar analog.

The key to this difficulty lies in the fact that Io and I• have different asymptotic behaviors as seen in (10). Exami- nation of the columns of M in Eq. (5) shows that Ko and Kl both appear in some columns while Io and I1 both appear in other columns. The first-order asymptotic expansions of Ko(•) exp(•)/(•) 1/2 and K• (•) exp(•)/(•)1/2 are the

1064 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons ot a/.: Clad rods. I 1064


same, but the first-order asymptotic expansions of Io(•)/(•) 1/2 cosh(•) and I1 (•)/(•)1/2 cosh(•) have the forms [ 1 + i exp( - 2•) ]/[ 1 q- exp( - 2•) ]. In order for these expressions to agree, then:

Re(•) >• 1. (15)

This condition suffices to ensure that a particular cylindrical mode is converging to a planar interface mode.

If Eq. (15) holds, a simple correspondence can be made between the a- 6 constants in Eq. (8) and those in Eq. (13). One can see that for large rof, the cosh(•) terms become essentially exp(•) and the complex displacement am- plitudes qmr and qmz have a local r dependence dominated by exp [ cf(r -- ro) ], where c is a constant depending only on v and, therefore, essentially independent of rof Thus, for planar type modes, iffis held fixed and ro tends to infinity, the orbit shape depends only on the distance from the interface, as in the planar case. However, for any mode if rofis held fixed at some value, not necessarily large, whilef and ro are varied reciprocally, then, except for an arbitrary amplitude constant, the q's depend on r/ro, i.e., reciprocally on for directly with "wavelength."

Planar interface modes exist whose phase velocity exceeds both aR and ac. The exponential form of the displacements then requires that these modes should consist of the superposition of a longitudinal and a shear plane wave in each medium. Such modes, which we may call decomposable modes, then consist of a special plane wave reflection/ refraction situation in which at least one input or output plane wave is missing.

D. Energy velocity

The elastic power (EP) flow in and out of an arbitrary volume Vis described by the Poynting vector:

•- d• ( Co•u,,•u•,• + p•oi•,• ) dx

= f•, (P6oii• - Co•lU•,l• )iti dx -l- fa•,(roit, ds• -fa •oh•ds•fo •ds•. (16) v v

Here, •r is the stress and c• V is the boundary of V. The first expression of the second equation of (16) con-

tains • (u) and, consequently, vanishes. The component P•. describes the energy per unit area and unit time flowing through a surface with normal in the x• direction. To the three components P•. of dimension l - 2t - l, we can include the energy density E as a fourth component Po. This component has dimension l - 3 and can be thought of as the time component of a space-time energy flow vector P:

E, j=0, P; = tro/t,, j = 1,2,3. (17) Equation (16) then states that in a conservative system

P is divergence-free, P;j = 0. In the case of a monochromatic elastic mode, it becomes simpler and more convenient to use energy quantities which have been time-averaged over one period. To this end, we note that the general representation

of eigenfunctions in terms ofexp(icot -- kz), where k may be complex for leaky waves, allows us to represent all field quantities (•ru,ui,P i,E,etc.) in the form f (r,z,O), where 0 = cot - kR z and k• is the real part of k. Heref is periodic in 0 with period 2zr. We then define the time-averaged quan- tity:

!Or) = f ( r,z,O)dO. (18)

Since spatial differentiation is interchangeable with time integration, Eq. (16) holds for the time-averaged Poynting vector. The vector with components (P•)/(Po) defines a velocity field E, called the energy propagation velocity (or simply energy velocity) field:

(P• (r,z)) E•(r,z) = . (19)

(Po(r,z)) For the clad rod, the cylindrical components of the

space-time energy flow vector energy velocity may be calculated:

=0,

(Pr) =pw exp[2z Im(k)] [a 2 Im(F:/;qr) q- (a 2 -- 2b •)

X Re(kFtrqz)+b2Re(kqr7qz) +b2Im(O•qz)], (Pz) =pw exp[2z Im(k) ] [a 2 Re(k)lqzl'• + (a 2 - 2b 2)

X Im(F:/;qz) + b2Re(k)lqrl 2 q- b2Im(7q•qr) ], (Po) = (Ekin) .ql_ (Epo t )

=p exp[2z Im(k)](q (Iqrl 2 + Iqzl •) a2 [2 2 b 2 2 2 d- T( [q;. d- Ikqz I ) 4- -(Iql 4- Ikqr I )

+ (a 2 - 2b 2) Im(k•;qz ) + b 2 Im(k•qr )

2b 2 Re (•q•.qr) q'- 2b 2 r r (20)

Here, the qr and qz coefficients are those for the appropriate medium as given in Eq. ( 11 ) but with the asymptotic correction terms removed. The symbol" "represents complex conjugate, q• = dqz/dr and q• = dqr/dr. If rofis held constant, then both fro q• and fro q• are functions of r?ro, so that r• P is a function of r?ro. The energy velocity is determined by (20).

The concept of energy velocity has been used in electro- magnetic theory as well as being introduced for plane waves in elasticity and fluid mechanics, where it is seen to agree

9 1 with the group velocity. - 2 However, while the energy velocity is constant and coincides with the group velocity for a plane wave, the energy velocity field becomes variable when the translational symmetry of the mode is limited. For axisymmetric leaky modes, for instance, the energy velocity field is a function of r (the exponential decays in the Pi's canceling out the z dependence). It has been shown for rod modes that the group velocity in the axial direction is the quotient of the total time averaged energy flow over the rod cross section with the total time-averaged energy density over the rod cross section. 13 Thus, even for modes in a bare



rod, the more physically fundamental energy velocity field displays local information not contained in the group velocity. Indeed, the energy velocity field can vary with r by an order of magnitude for some rod modes. This results can easily be extended to type 3 modes in a clad rod which have real k, where the energy velocity at large distances from the cylinder axis is asymptotically equal to the phase velocity of the mode. The situation for leaky waves is more complex, and while the energy velocity retains its physical validity, the physical interpretation of group velocity is unclear. Yet we are unaware of any work utilizing the variable aspects of the energy velocity for monochromatic modes with fixed wave number.

If one draws an arrow with components E• throughout a medium, these arrows not only point in the direction of maximum energy flow, but no energy flows perpendicular to the arrows (henceforth flows will be interpreted as time-averaged over a period). Thus if one draws a small circle, or other closed surface, at time to, one can use the energy velocity field together with the outer perimeter of the circle to devel- op a tubular surface inside of which energy is conserved and whose velocity is governed by the energy velocity within the tube. An illustration of this is shown in Fig. 1 for the case of a wave mode in a rod entering a clad half-space. Since the techniques we employ here are only valid for a clad rod system of infinite axial extent, it is assumed that there is a small reaction zone near the edge of the cladding following which the rod mode is converted into a number of leaky modes, only one of which is shown. The energy flow curves for the leaky mode are obtained by integrating the energy velocity (or Poynting vector) field. Translational material homo- geneity requires that all flow curves should be congruent under shifting, although the amount of energy between any two equally spaced curves (actually surfaces of revolution about the cylinder axis) decreases exponentially with increasing z. Since the boundary conditions require that Pn, the normal component of P, be continuous across the interface, the discontinuity in direction at the interface is dictated

FIG. 1. Energy flow curves (a, b, c, c', and d ) for a leaky wave starting at the axis-perpendicular edge of the cladding. Curves a, b, c, and d are rotational- ly equivalent. Curve c', which shows the energy flow starting nearer the edge of the rod, is translationally congruent to flow curve c. This is indicated by the shifted inner circle. The self-parallel feature and the propagation of energy tubes are shown.

by the discontinuity in the tangential component of P. When Pis more parallel to the interface, the flow and energy densities are increased by the factor 1/Pn. In the case of nonleak- ing modes, all energy flow vectors are parallel to the interface, and this area correction is unnecessary. For leaky modes, however, the energy flow curves bend away from the axis and become asymptotically inclined at some "leakage angle," aL to the cylinder axis. (Uniqueness and shift invar- iance of the flow curves preclude a change in sign of slope along the flow curve.) This angle and the associated limiting energy velocity can be found from Eq. (20) by using the dominant asymptotic form of the q's at large values of r. The two asymptotic growth exponents of the q's are given by Re(kA c ) and Re(kBc ). Thus, if Re(kA c ) < Re(kBc ), the substitutions

qr •-'

q; ( _ c ), qz'"'-' --i, (21a) q; • ikA c,

may be used, while if Re(kAc ) > Re(kBc ), the substitutions

qr""--i,

q;'"-' -- ikBc,

qz'• --Bc, (2lb) q; .-.. k ( 1 -- v•/b • )

are appropriate. If Re(kA c ) = Re(kBc ), or if the dominant growth exponent is zero and v is real, the asymptotic form is ambiguous.

It is useful to simplify the expression for the asymptotic leakage angle. After some algebraic manipulation it can be shown that if the conditions in (21 a) are applicable,

tan 0L = Im(k)/Re(kAc), (22a)

while if those in (21 b) are applicable,

tan 0• = Im (k)/Re ( kBc ), (22b) where 0• is the asymptotic leakage angle. [The asymptotic expressions (22) can also be found by a different argument: The asymptotic amplitude of leaky waves exponentially increases away from the cylinder axis with an exponent linearly dependent on r, while exponentially decaying with an expon.ent z Im (k) along the cylinder axis. The limiting leakage angle a• is that at which these two exponential trends exactly cancel each other.] In the special case that I Ira(k) I '•l Re(k)[ [or [ Im(v)],•[ Re(v)l], we have

0•0, for Re(v)<a c (23a) tan OL•[Ac[, for Re(v) >ac,

when Re(v)> =0 and conditions (21a) are applicable, while

0•0, for Re(v)<bc

tan 0• -• IBc[, for Re(v) > bc, (23a)

when (22b) applies. {Strictly speaking the expression as derived for Re(v)<ac (or bc) requires that [ Im(v)/

1066 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons eta/.: Clad rods. I 1066


Re(v)],•(1 -- [ac/Re(v)]2} or {1 -- [bc/Re(v)]2}, but this latter inequality is rather forgiving, especially when Re(v) is near ac or bc in which case the leakage angle is small anyway. ) These expressions point towards the distinction between leaking cylindrical waves at high values of • [defined in Eq. (?)] having phase velocities with vanish- ingly small imaginary part and possible planar interface limits where v would be real. This distinction will be discussed

below. They also show the need in computation to avoid underflowing the imaginary part of v to zero.

If one draws an energy flow curve starting from the junction of the rod and the half-space, there can be no energy detected between the edge of the half-space and this limiting curve except for a fringe field associated to elastic nonequi- librium along the limiting curve (as could also be seen from the nontime-averaged energy flow). The lacuna thus formed is a characteristic of leaky modes and offers one experimental means for their detection. TM For values of z beyond the limiting flow curve, the wave amplitude decreases exponentially for any fixed r. (Since there may be a "mixing zone" near the junction, the peak energy may be shifted slightly in a way that cannot be predicted from this theory. The major effect of this shift will be on the apparent asymptotic leakage angle. If required, it can be removed by using peak measure- ments at two different values of r with the same transmitter. )

Because of interference effects between shear and longitudinal components, the energy flow curve is not a straight line and fluctuates in some manner near the interface. If one

were to draw a straight line inclined at crL from the intersec- tion of the rod with the half-space to any distance r, this line would be shifted some distance from the limiting energy flow curve. If rofis fixed, the dependence of (P) on r/ro shows that the magnitude of this shift will vary inversely with f.

Such a shift also occurs with leaky modes at planar interfaces. It is particularly illuminating to think of a mode where one of the type indices is 3 and the other is 0, 1, or 2, since these modes have leaky cylindrical analogs. In such cases, there is also a limiting "approach" angle in the type 3 medium with an associated asymptotic approach wave. Generally this wave is of mixed type, but if the energy velocity is close to either a• or b•, the mode will be essentially a plane wave of longitudinal or shear type. Near the interface inhomogeneous interface waves are generated on either side of the interface, and the interference of these waves with the asymptotic incoming and outgoing waves produces the bending of the energy flow lines, with a characteristic shift of the expected position of the asymptotic output wave. This shift, which is inverse frequency dependent, is analogous to the Goos-Hanchen shift in optics or the Schoch effect at liquid-solid interfaces. 8'is (When several leaky modes are simultaneously present, they will interfere and produce changes in the energy flow curves, which will affect the amount of the shift. However, for many cases of experimental interest the exponential decay with z and the relative in- sensitivity of the flow curve to small disturbances allow us to treat the flow curves of the leaky modes individually. )

In the case of decomposable modes representing a plane wave scattering reaction at a planar interface, the vector P• which represents the total energy flow•as well as the energy

flow lines---oscillate with r, indicating the interaction of the individual plane waves.

E. Locating leaky mode branches and calculating dispersion curves

Individual modes are found as solutions to the equatio•

IM(v) I =0 (24)

from Eq. (5) as modified by the asymptotic expression in (9). The modified Bessel functions occurring therein have a branch cut along the negative real axis and the cosh functions have zeros along the imaginary axis. Thus, I M (v) [ will be analytic inside the first quadrant of the complex v plane. Complex roots may then be located by applying the "argument principle":•6

1 f f.(,) ao= # zeros inside F, (25) 2rci f(v)

where F is any closed curve in the first quadrant. Real roots and pure imaginary roots of (24) can be found by ordinary root search techniques for functions of a real variable.

Newton's method was used for computing either real or complex roots to (24) once the approximate zero was located. Root accuracy depends on the bounds of Newton's method, but was always greater than eight places.

At chosen fixed values of• = rof/b•, an automated im- plementation of the argument principle was employed to de- tect roots where

0.01<•<10. (26)

Similar bounds were used for root searches on the real and

imaginary axes. Once an individual root was found, the dispersion curve

for the mode branch containing that root was obtained by varying rofover a desired range and employing a Newton's method algorithm with forward interpolation, which was double nested to allow three orders of magnitude in 1 (rof) to squeeze through difficult spots in the dispersion curve. This method was sufficiently robust that one could follow the dispersion curve of a rod mode near the cutoff to a phase velocity of 1000 times that of the longitudinal wave.

The roots associated to numerous values of rof were found, the dispersion curves traced and the redundant curves eliminated. Nonetheless, there is always the possibility that individual anomalous modes were overlooked.

Once the root v is found, the real part of v directly gives the mode's phase velocity along the interface. In dimensionless form, this velocity is v = Re(v/b• ). The attenuation along the interface is determined from the imaginary part of the wave number k and can be expressed in dimensionless form in dB by

attenuation = log 10 (e) 40•r Im (v)

-•40•r lOglo(e)b• Im(v)/[vl 2. (27)

The units of b• are in distance times frequency, and the dimensional form of the attenuation (without the b• ) ex- presses the fact that the attenuation depends linearly on frequency as well as on distance traveled down the interface. This attenuation could also be dedimensionalized to mea-

1067 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons et al.: Clad rods. I 1067


sure distance along the interface in terms of units of %, but such a form would not be extendible to the planar interface, so the units will be left in those laboratory units used to express be.

In order to assess the effects of material parameters on leaky modes, a nested Newton's method was used to vary an individual elastic constant or density or groups of such parameters. By means of these parameter homotopics, for instance, it was possible to connect leaky modes with rod modes as well as with modes in the cladding without a rod present.

This code also tests for change of mode type. The complex velocity v has been chosen so that Im(v)>•0, implying that the wave cannot exponentially grow along the interface, nor can energy flow into the interface. However, at certain values •b, the character of a mode can change. This can only happen when A • or B • are real, and when the displacements do not jump discontinuously. In particular, the asymptotic displacements cannot undergo a change of sign in the real part of the exponent, which must then be zero at a branch change. Since these exponents are proportional to the real parts of A c /v andBc/v, and since Bc < Ac, we have that 1/v 2 must be real and that 2/v 2 < 1/A • is a necessary condition for a change of mode. Dispersion curves satisfying this condition are checked for mode changes at apparent cutoffs.

II. RESULTS

A. Dispersion curves

Since a clad rod consists of the conjunction of a bare rod with an infinite cladding, it seems natural to seek the rela- tionship between leaky modes and the modes that exist in a bare rod or in an infinite material with a cylindrical hole (tunnel modes).

Qualitatively speaking, as long as the density of the rod

is not zero, most leaky modes are leaky rod modes. Figures 2 and 3 show the dispersion relations for the case of a silicon carbide rod in an aluminum cladding. The plots in these figures have been given in dimensioned form. The dimensionless parameters for this example are then p -- 0.866, ac -- 3.21, •c -- 1.68, as -- 1.92. All bare, real, or partly real SiC rod modes with cutoff value of •b- rof/b• <6 are shown as dotted lines. The associated leaky modes are plotted as solid or dashed lines. The dispersion curves in these figures have been arbitrarily truncated at their maximum values for clarity of graphical presentation; they actually continue to a value of •b--0.

The type 0 modes are the most abundant and are shown in Fig. 2. As can be seen, there is one such mode for each rod mode. In Fig. 3 are shown both type 1 and type 2 modes. The number of type 1 and type 2 modes combined in this figure is one more than the number of rod modes. These relationships will be discussed below.

The modes that exist in a bare rod have been studied by Onoe et al. '7 and reviewed by Thurston.' In the study by Onoe, the frequency (dimensionless) was appropriately calculated as a function of wave number (dimensionless), rather than the velocity versus radius-compensated frequency as is done herein. From the v vs •b point of view, rod modes show a cutoff at k -- 0 (infinite velocity) or df/dk - 0. Be- yond the k -- 0 cutoff the velocity becomes pure imaginary as does k. The df/dk - 0 cutoff represents a node in k space at which a complex branch contacts either a pure imaginary or pure real branch at a local minimum. Only the real minima have been studied herein, and in that case the complex branch has been connected with that part of the w vs k curve that continues down to where k becomes zero, since the discontinuity in slope is less and that is the path followed by the Newton's method algorithm. In Figs. 2 and 3 the real part of the lowest such complex rod branch has been included, since

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. Dispersion curves for the modal phase velocity for type 0 leaky modes and rod modes as a function of rod radius times frequency (to f) in the aluminum-silicon carbide system (aluminum: density- 2.77 g/cm 3, a c --6.323 mm//•s, bc -- 3.1 mm//•s. Silicon carbide: density -- 3.2 g/½m 3, a R -- 9.649 mm//•s, bR ---- 5.193 mm//•s). The leaky modes have been truncated at their peak value for clarity. The extra-wide spaced dotted rod mode is the complex part of the second rod mode.

1068 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons eta/.: Clad rods I 1068


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


FIG. 3. Dispersion curves for the model phase velocity for types 1 and 2 leaky modes as well as rod modes as a function of rod radius times frequency (raf) in the aluminum-silicon carbide system (aluminum: density = 2.77 g/½m 3, a c --6.323 mm//•s, bc = 3.1 mm//•s, silicon carbide: density = 3.2 g/cm, ac = 9.649 mm//•s, bc = 5.193 mm//•s). The leaky modes have been truncated at their peak value for clarity. The extra-wide spaced dotted rod mode is the complex part of the second rod mode.

it seems to be related to the lowest type 0 leaky mode. For the silicon carbide rod, only two df/dk minima were found.

Rod modes have been classified into two kinds by the particle motion at the cutoff frequency.' In the one kind of mode the particle motion is radial, while in the second it is axial. (While this particle motion exists in the limit, it is not very useful as a practical mode signature, even in these bare rod modes, since to obtain elliptical trajectories with an aspect ratio more than, say, 10-1 in the radial or axial direction for all values of r often requires a phase velocity of more than 100bR.) For each of the two kinds of rod modes, the cutoff values of •b, which are solutions of transcendental Bessel

function equations, are approximately equally spaced; and near a Poisson ratio of 1/3, there are about twice as many axial modes as radial modes. Using a homotopic technique on densities to be discussed below, one can show that, with the exception of one type 1 leaky mode, in the ranges of raf somewhat above the cutoff value for the appropriate rod mode, the type 1 leaky modes in Fig. 3 correspond to radial modes and the type 2 leaky modes correspond to axial modes.

Figures 4 and 5 show the type 0 leaky mode dispersion curves with the low-frequency segments included. Those in Fig. 4 correspond to the axial modes and those in Fig. 5 to the

30.0.

2•.0

18.0

12.0

8.0

0.0 0

E

0 •1.0 8.0 12.0

F

K L

6.0 20.0 2ti.0 28.0 32.0 36.0 U, 0


FIG. 4. Dispersion curves for the modal phase velocity for the two, axial rod mode related subfamilies of type 0 leaky modes as a function of rod radius times frequency (r a f) in the aluminum-silicon carbide system (aluminum: density = 2.77 g/½m 3, a c = 6.323 mm//•s, bc = 3.1 mm//•s; silicon carbide: density = 3.2 g/cm 3, a c = 9.649 mm//•s, bc = 5.193 mm//•s). The low-frequency portions of each curve are shown here and each curve is identified by a letter for future reference.



•.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 curves for the modal phase velocity for the radial related subfamily of type 0 leaky modes as a function of rod radius times frequency (rof) in the aluminum-silicon carbide system (aluminum: density = 2.77 g/cm 3, a c = 6.323 mm//•s, bc -- 3.1 mm//•s; silicon carbide: density = 3.2 g/cm, ac -- 9.649 mm//•s, bc = 5.193 mm//•s). The low-frequency portions of each curve are shown here and each curve is identified by a letter for future reference.

radial modes. For future reference each of the modes in these

figures has been assigned a letter. Based on their peak behavior, the axial modes can be divided into two alternate families of modes, which we designate as the lesser and greater axial- related modes, while the other family of type 0 modes will be called the radial-related modes. This third class of modes is

the only one observed to exhibit negative phase velocities, signaling backward-leaking modes. In this case, stopping at rof= 30, there are 6 modes in each family. Figures 6 and 7 show the equivalent dispersion curves for type 1 and type 2 leaky modes, again with a designation letter assigned to each mode. There are seven type 1 modes shown in Fig. 6, and all but the lowest type 1 mode, labeled A in Fig. 6, correspond to radial-related type 0 modes. The extra mode in family 1 can

be seen in Fig. 17 (a)--although the elastic constants used there differ slightly--to correspond not to a rod mode, but to a mode arising in a hollow aluminum cladding with rigid boundary. This correspondence holds for values of rofbe- tween about 6.5 and 22.5 mm//•s. Outside this range different correspondences hold. Two families of type 2 modes, shown in Fig. 7, correspond to the equivalent axially related type 0 although the difference in peak behavior is much less pronounced than for type 0 modes. There are six modes shown in each family. There are no type three modes for this material combination. [Though not strictly obeyed at every value of ½,, we state a general "rule of thumb" that the number of 0 modes plus the number of 3 modes equals the number of 1 modes plus the number of 2 modes. This arises heur-

30.0

2LI.0

18.0

12.0

6.0

0.0

F G E

D

C

0 ti. 0 8.0 12.0 16.0 20.0 2U,. 0 28.0 32.0 36.0 El0

Radius X Frequency (rnrn/t.t, rn )

FIG. 6. Dispersion curves for the modal phase for the subfamily of type 1 leaky modes as a function of rod radius times frequency (rof) in the aluminum- silicon carbide system (aluminum: density -- 2.77 g/½m 3, a c = 6.323 mm//•s, bc = 3.1 mm//•s; silicon carbide: density = 3.2 g/cm, ac = 9.649 mm//•s, bc -- 5.193 mm//•s). The low-frequency portions of each curve are shown here and each curveis identified by a letter for future reference.



2u• O.

E •8.o i v

6.0

E

D

C

0 •.0 8.0 •2.0 0.0

0

G

•---• --- / .............. 20.0 2U,.0 28.0

Radius X Frequency (mm/l•m)

L

32.0 36.0 L10.0

FIG. 7. Dispersion curves for the modal phase velocity for the subfamily of type 2 leaky modes as a function of rod radius times frequency (to f) in the aluminum-silicon carbide system (aluminum: density = 2.77 g/cm 3, a c = 6.323 mm//ts, bc = 3.1 mm//ts; silicon carbide: density = 3.2 g/cm 3, as = 9.649 mm//ts; bR = 5.193 mm//ts). The low-frequency portions of each curve are shown here and each curve is identified by a letter for future reference.

istically from the two type wave terms (longitudinal and shear) canceling for pair of branches and combining for the other pair. ]

It is apparent from these figures that for this clad rod system k (or v) is a well-defined function of the frequency-- there are no points for which k = 0 or d.f/dk = 0•and there is no interconnection between modes as occurs for the bare

rod. '7 It appears that for vanishing Pc the configuration occurring in the bare rod is topologically unique and that the leaky modes separate at points where d.f/dk = 0 in the bare rod diagram. Indeed, one might expect that modes leaking into an acoustic or elastic cladding (where there are four distinct types of leaky modes versus only one type for the rod) would better model rod mode experiments (where the environment of the rod is not a perfect vacuum).

Another distinction occurs in the mid-frequency range of the dispersion curves, where rod mode curves exhibit terracing. Each rod mode is known to reach a plateau just above the longitudinal velocity of the rod before continuing down to the asymptotic shear velocity.' For the systems we have studied, only type 0 leaky modes exhibit the terracing feature; but, there, an important additional feature occurs, which we call mode knitting (see Fig. 8 ). If one increases the density of the cladding, for instance, starting from some value "near" zero, one can see, starting at "very high" values of •b, the end of the terrace of a certain mode approaching the corner of the next higher adjacent mode. At a "critical" density these modes actually touch, the complex velocities be- coming equal at a particular value of •b. This is shown in Fig. 8 (a) for the 11 and 12 modes from Fig. 2 (o) (or modes H and/in Fig. 4) at a value of rof•27.520 218 076 87 mm//•s (•b = 5.299 483 55) over a range of densities close to the critical density. Above that density the modal lines inter- change; the terrace of the higher mode, together with those of previously knit modes, are transferred to the lower mode. Figure 8 (b), for instance, shows a blowup of the phase velocity curves for the same two modes at three different densities

in the range 27<rof<28. The intermediate density chosen is just below the crossover density. As a consequence, there seems to exist a subclass of modes in infinitely clad rods whose dispersion curves have, as their asymptotic limit, the longitudinal velocity of the rod, even though no longitudinal velocity mode exists in the limiting planar interface case. This feature is quite distinct from the case of bare rods where only the shear or Rayleigh velocity is an asymptotic limit. For low cladding density, p•0, the number of modes that exist at a particular value of•b increased with increasing density. The same situation obtains for low rod density, p• with the number of modes increasing with increasing rod density. There, as will be discussed, the limiting case consists of Dirichlet boundary conditions for the rod with no displacements at the outer surface. Thus, there is some intermediate value ofp where mode knitting is maximized. Note that, however, since we have not identified which particular modes stay at the higher longitudinal velocity, we cannot rigorously conclude that these high velocity "tails" remain on any one mode.

At high values of •b some aspects of the clad rod solutions approach those of the planar interface. Of course, there always exist infinitely many clad rod solutions while there are only a finite number of planar interface solutions, and plane interface modes are often approximated as the limiting case of a given clad rod mode; but this does not always occur. Whenever a leaky planar interface wave or a Stoneley wave of type m,3 (m = 0 -- 3 ) exists, it is the limit of a similar type m cylindrical mode; but this is not the case for decomposable planar interface modes. For instance, in the case of an aluminum matrix encasing a steel rod, there exist two limiting decomposable planar interface modes with velocities signifi- cantly higher than the longitudinal velocity of either material (7.41 mm//•s for a type 1,3 mode and 6.78 mm//•s for a type 2,3 mode as opposed to VL = 6.32 mm//•s in A1 and VL = 5.92 mm//•s in steel). (See Fig. 32 caption for the materi-

al constants of the aluminum-steel system. ) (There is also a

1071 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons et aL: Clad rods. I 1071


10.970

10.968

E 10. 966

._•

> 10.96q

10.962,

10 96o

{a!

• REAL PART OF MODE H IMAGINARY PART OF MODE H

.... REAL PART OF MODE I •-- IMAGINARY PART OF MODE I/

/ /

i I i I !

\,

•,,•

2.85660 2.85662 2.8566q 2.85666 2.85668 2

Density (gm/cma)

0.620

0.619

0.618

0.617

0.616

0.615 85670

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 imaginary parts of modes Hand IofFig. 4 plotted as a function of density near the critical crossover density --- 2.856 645 108 7 g/cm 3 at a value of rof= 27.520 218 076 87 mm/ps. This is just past the critical value of ro f, and here the real part ofv has already crossed over while the imaginary part has not. (b) Blowup of the dispersion curves showing mode knitting for modes H and I of Fig. 4 plotted as a function of rof at three different densities: one below the critical density (2.77 g/cm 3 which is that of aluminum), one just below the critical density (2.856 g/cm 3) and one above the critical density (2.97 g/cm3).

2,1 type decomposable mode that does not have a cylindrical analog. )

The first of these decomposable modes is made up of three input planar waves (one shear wave in A1 at -- 65 ø, one (weak) shear wave in steel at 61 ø, and one longitudinal wave in steel at 29 ø) and only one output wave (one longitudinal wave in A1 at 31ø). (The reverse wave with a single longitudinal input and three outputs is of 2,0 type and does not have a cylindrical analog. Generally speaking, 0 type waves have both planar modes going away from the interface, 1 type waves have the longitudinal wave going away and the shear plane wave going toward the interface, 2 type waves are just the reverse, and 3 type waves have both plane waves going toward the interface.) However, there is no type 1 mode

whose asymptotic limit is 7.41 mm/fts. Rather, for very high modes numbers, as the velocity of an individual mode passes from its peak value near the associated rod mode cutoff down to the asymptotic value of 5.92 mm/fts (in this case), it takes on the character of a decomposable mode for values of ½ where v• 7.41 mm/ps.

To complete the discussion of dispersion, we wish to describe the rod modes satisfying the Dirichlet--zero displacement--boundary conditions as well as the tunnel modes that exist in an infinite cladding without a rod present. The real rod mode structures for both the Dirichlet and

Neumann boundary conditions are shown in Fig. 9. The Dir- ichlet rod mode structure, while somewhat distorted from that of the Newmann (traction-free) rod mode structure, can still be connected to that structure for modes in the inter-

mediate to high values of • except for the traction-free surface modes, which cannot exist with Dirichlet boundary conditions. (Note that neither of these limiting rod mode structures depends on density.) However, for those modes that are in their low • range--near or below their peak--the correspondence between Neumann and Dirichlet boundary condition modes cannot be simply made, but must be in- ferred from the part of the dispersion curve for higher •. The displacement for all Dirichlet modes appears to be axial near the cutoff frequency.

In the traction-free Neumann case, there exist only four pure tunnel modes--usually one of each type for most values ofac/l•c = at/be; this is illustrated in Fig. 10 for the case of the aluminum cladding used in the previous examples, where ac/l•c • 2.04. In general, type 0 and 1 modes appear to exist at all values of ac/l•c and •c = rof/bc. (Values of •c less than 0.01 and values of ac/lgc greater than 10 were not investigated. ) The type 2 and 3 modes exist for all values of •c > •maC/•C, rn = 2,3. Just below these "switching" frequencies, the type 2 mode changes to a type 1 mode and the type 3 mode changes to a type 2 mode. In the type 2 to type 1 mode transition, the exponential term controlling asymptotic growth switches from the shear to longitudinal type. Just above the value of •2, the phase velocity is real and exceeds 2bc, the asymptotic growth exponent is zero and the asymptotic energy velocity is b c, while just below the switching frequency the asymptotic group velocity changes to a c. The "low-frequency" type 1 mode also has a lower cutoff value Of•c that occurs for a value of o slightly less than ac, and ceases to exist for values Ofac/lgc somewhere above 2.35, where the type 2 mode merely exhibits a lower cutoff when v=a• (e.g., in the case when ac/]?c = 10, •: •0.0636). The switching frequency for the type 3 to type 2 transition occurs at that frequency where the exponent for exponential decay of the type 3 mode becomes zero and then turns positive producing a type 2 mode. At the transition point both the phase and asymptotic energy velocities are bc. The variation of the cutoff frequency with ac/]•c is quite small [e.g., •b3(1.1 ) = 0.224 and ½3(10.0) = 0.378]. Below the transition point, the type 2 mode is a leaky mode with nonzero imaginary part.

The zero displacement Dirichlet boundary conditions appear to give rise to only one tunnel mode (of type 1 ) in the infinite cladding. This mode, also shown in Fig. 10, exists

1072 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons et al.' Clad rods. I 1072


30.0

.................................................................................................................................. • ...... ' i Ill , I ' r '1 ......... iii

.

.

..' .

.

.

2L•.0

E E ]8.o i

._

• ]2.0.

8.0

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. Dispersion curves for the modal phase velocity for both ordinary rod modes (zero tractions at the surface -- Neumann modes) and Dirichlet rod modes (zero displacements at the surface) as a function of radius times frequency (rof) in a silicon carbide rod (silicon carbide: density -- 3.2 g/cm 3, a c -- 9.649 mm/lts, bc -- 5.193 mm/lts). The extra-wide spaced dotted rod mode is the complex part of the second Neumann rod mode.

over the whole range of •bc values investigated when 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 forbidden range begins is about 0.41. The existence of this single mode, as well as anomalies occurring at various values of •b near mode cutoffs, are examples where the number of type 0 plus 3 modes is not the same as the number of type 1 plus 2 modes.

We are now in a position to attempt a catalog of all

8.0

_

•.6

0.0 0.0 •.0 8.0 12.0 16.0 20


FIG. 10. Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod radius times

frequency (rof) in an aluminum tunnel (an infinite space-filling solid of aluminum with a cylindrical hole bored out) (aluminum: density -- 2.77 g/cm 3, a c -- 6.323 mm/lts, bc = 3.1 mm/lts). The crossover from type 2 to type I mode occurs at rof-•0.833665 mm/lts, and the type 3 cutoff occurs at about 1.067 202 4 mm/s, where the phase velocity becomes bc.

axisymmetric leaky modes. As mentioned earlier, when the density of either the rod or the cladding goes to zero (p goes to infinity or 0), the determinant of the matrix in Eq. (5) essentially factors into the product of two subdeterminants, one associated to the tractions and the other to the displacements at the interface. The condition for a free rod or clad-

ding is the vanishing of the traction subdeterminant, while the vanishing of the displacement subdeterminant describes a rod or cladding bonded to an infinitely rigid material.

There are always infinitely many modes for any clad rod as long as the density of the cladding is nonzero. Thus, in any clad rod with p near 0 or infinity, the set of clad rod modes can be identified with all rod modes of one boundary condition type together with all tunnel modes of the other boundary condition type. As we have already seen, near small values of p the network of complex and imaginary rod modes becomes greatly simplified when cladding material on nonzero density is present, leading to a single family of mod. es which can be identified with the real rod modes both above

and below their cutoff frequency. This same situation appears to occur with Dirichlet rod modes for large values ofp. In such circumstances even though the rod has very small density, it has sufficient stiffness to balance the motion of the cladding. The rod displacements, then, are very large vis-h-vis those of the cladding and the tractions at the interface are non-negligible.

The question then occurs: Can one map all leaky modes for p near zero, say, onto all leaky rod modes with large values ofp and thus completely encompass all leaky modes, or are there modes that occur at intermediate values ofp that cannot be linked to any extremal modes? While we cannot answer this question in general, which requires the variation of four other parameters, we shall address specific aspects of it.

Calculations were carried out for a clad rod consisting of a pair of materials with the same elastic constants as the aluminum and silicon carbide used herein, but with p rang- ing from 0.001 to 1000 while •b -- rof/bR was held at a fixed



value of 5.78 (to f= 30 mm//•s). At the value ofp = 1.16, corresponding to the A1/SiC example used, there were 37 modes, as seen in Figs. 2 and 3 showing significant real phase velocity somewhere below • = 5.78. Between p = 1.16 and 0.001, as discussed earlier, two clad rod modes could be correlated with each of the 17 real SiC rod modes with a cutoff

below • = 5.78 as well as with the lowest rod mode. This gave rise to 36 modes. The 37th mode, of type 2, was found to cross over to a type 1 mode at a value ofp•0.176, and this mode continued down to the Dirichlet tunnel mode. This

situation differs from that obtained at lower values of •, where the lowest type 1 mode was found to transform, with changing values of p, to the Dirichlet tunnel mode. At this value of •, that lowest mode crosses over at a small value ofp to a type 2 mode and then continues down to a rod mode. At still higher values of • the lowest type 1 mode smoothly transforms into a rod mode and the Dirichlet tunnel mode is

found to correspond to a type 2 mode after a crossover. These results show that homotopic correspondences are by no means unique, or equivalently, that homotopies in • (dispersion curves) and p do not commute.

Corresponding to high values ofp we note that there are 16 Dirichlet rod modes with cutoffs below • = 5.78 and one Neumann tunnel mode of type 2 to which there should be correspondents. [ Examination of Fig. 15 (a) shows that the other three Neumann modes will have no correspondent mode at values of p as low as 1.16. In investigating these three tunnel modes for large values ofp, both the type 0 and 1 modes were found to cut off near zero phase velocity at values ofp• 2.8 at which the shear modulus of the rod is approximately equal to that of the cladding, and the type 3 tunnel mode cut off at p-• 7.5, since such surface modes do not exist over most of the range ofp when/3c < 1. ] In going from p = 1.16 to p = 1000, every real valued Dirichlet rod mode was again the limit of two leaky modes, one of type 0 and the other of type 1 or type 2; and the type 2 Neumann mode was the limit of the lowest type 2 leaky mode. Three of the type 1 or type 2 modes crossed over to be type 2 or type 1 modes at values ofp exceeding 3.35; and the lowest type zero mode as well as the highest modes of types 0-2 all tended to complex Dirichlet rod modes.

The appropriate clad rod dispersion curves at p = 1000 were compared with those for rod modes obtained using Dir- ichlet boundary conditions. They were seen to almost coincide down to near the appropriate rod mode cutoff where the clad rod mode velocity acquires a large imaginary component. Of course, unlike the rod modes, there are an infinite number of leaky modes at any fixed value of •; almost all of these modes, however, are associated to the low end tails of higher rod modes. These other modes have small phase velocity and tend to complex Dirichlet rod modes.

Thus, in this case, while no unusual modes were found to exist only for intermediate values of p, the complete homotopic correspondence among modes from p = zero to infinity has to be qualified at any value of • by the possible oversight of clad rod modes near their peak value as well as a type zero mode corresponding to the lowest order traction free rod mode. Further, the correspondence between clad rod modes and rod or tunnel modes may change with different values of •.

A second specific aspect of the homotopic evolution of clad rod modes with density was investigated. Using ordering relations among the elastic velocities in a manner similar to that suggested by Thurston, • a set of 12 examples was chosen and the evolution of the tunnel modes with changing p was calculated. Table I gives the characteristics of the 12 ordering types. The properties of the cladding were kept constant through all examples, with the ratio ac/bc fixed at 2, a value closely approximating that of aluminum, and the value of rofwas fixed at 15.5 ( = 5be ). The 12 order types were broken into size classes involving relationships among the longitudinal and shear velocities with two subclasses in each type determined by the relative magnitude of an appropriate surface wave and bulk velocity. Figures 11-16 show the change with p, forp greater than 0.2, in the five tunnel related modes (four of Neumann and one of Dirichlet type) for each of these cases. The aluminum/silicon carbide example used throughout much of this study is similar to case 5a. In the first three classes, where bR • be, the mode behavior appears to break into a tunnel dominated and rod dominated region with a boundary occurring approximately at the value ofpR where the shear impedance (lob) of the two media

TABLE I. Characteristics of the 12 ordering types.

Class sn bn an Sc bc a c Example system

la:a n <sc<b c 1.4 1.5 2.7 2.89 3.1 6.2 AI-Th (p = 4.2) lb: s c < a n < bc 1.56 1.67 3.0 2.89 3.1 6.2 Al-graphite (p..•0.54) 2a:bn <sc<bc<an <ac 2.61 2.8 5.04 2.89 3.1 6.2 AI-W (p--•6.9) 2b: s c < bn < bc < an < ac 2.80 3.0 5.4 2.89 3.1 6.2 3a: b n < sc < bc < ac < an 2.56 2.75 6.6 2.89 3.1 6.2 3b: s c < bn < bc < ac < an 2.8 3.0 7.2 2.89 3.1 6.2 4a: a c < sn < bn 7.46 8.0 14.4 2.89 3.1 6.2 AI-B (p--•0.85) 4b: sn < ac < bn 5.97 6.4 11.52 2.89 3.1 6.2 5a: bc < sn < bn < ac < an 4.85 5.2 9.36 2.89 3.1 6.2 AI-SiC (p--• 1.2) 5b: s n < bc < bn < ac < an 2.98 3.2 6.4 2.89 3.1 6.2 6a: bc<sR <b n <a n <ac 3.17 3.4 6.12 2.89 3.1 6.2 6b: s t < bc < bn < an < ac 3.03 3.25 5.85 2.89 3.1 6.2 Al-steel (p--• 2.8)



10.0

8.0

8.0

2.0

0.0

LEGEND CASE 1A:TYPE 0 NEUMANN CASE 1A:TYPE 1 NEUMANN CASE 1A:TYPE 2 NEUMANN CASE 1A:TYPE 3 NEUMANN CASE 1A:TYPE 2 DIRICHLET

p-1 (Rod Density / Cladding Density)

10.0

8.0

8.0

2.0

0.0 0

(8)

/

/

LEGEND CASE 2A:TYPE 0 NEUMANN CASE 2A:TYPE 1 NEUMANN CASE 2A:TYPE 2 NEUMANN CASE 2A:TYPE 3 NEUMANN

........ CASE 2A:TYPE 1 DIRICHLET

I

p-1 (Rod Density/Cladding Density)

10.0

8.0

8.0

0.0

{b)

LEGEND CASE lB:TYPE 0 NEUMANN CASE lB:TYPE 1 NEUMANN

•SE I•:•YPE • NEUMANN SE TPE NEUMANN ....... CRSE lB TYPE 1 DIBICHLET

............................................................... :• ..... .•.Z ...........................................................................

•'"-'-- ........ •' i // \/ ,!

/'\

0.0 1.0 2.0 3.0 k•.O 5

JCI-1 (Flod Density / Cladding Density)

FIG. 11. Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod density/cladding density (p - 1 ) for the clad rod of class la from Table I. The Dirichlet mode has crossed over to type 2 in this region ofp rather far removed from the Dirichlet region (p--•0). (b) Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod density/cladding density (p - ' ) for the clad rod of Class lb from Table I.

10.0

8.0

8.0

2.0

0.0

LEGEND CASE 2B:TYPE 0 NEUMANN CASE 2B:TYPE 1 NEUMANN CASE 2B:TYPE 2 BOTH KINDS CASE 2B:TYPE 3 NEUMANN

0.0 1.0 2.0 3.0 Lt.0 5.0

p-1 (Flod Density / Cladding Density)

FIG. 12. Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod density/cladding density (p - • ) for the clad rod of class 2a from Table I. (b) Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod density/cladding density (p- ') for the clad rod of Class 2b from Table I. The Dirichlet and type 2 Neumann modes coincide for this range ofp.

are equal. In the last three classes, where b• > bc, the type 0 and 1 modes exhibit a cutoff at a value ofp• where the shear moduli of the cladding and rod are equal and the type 3 modes cutoff when v = bc. Only the type 2 Neumann related and the Dirichlet related modes exist for all classes and all

values ofp > 0.2. In four of the cases these modes coincide, giving rise to a totally tunnel related mode; in the other eight cases these modes tend to rod modes at the other end of thep scale. Both the type 1 and type 2 related modes can exhibit

crossovers. More detailed studies are required to determine whether the ordering classes used here give an exhaustive breakdown of wave mode types.

B. Attenuation

Figures 17-20 display the attenuation in dB/mm MHz for the A1-SiC system whose dispersion relations were given previously in Figs. 4-7. These figures use the dimensional



10.0

8.0

8.0

2.0

0.0

LEGEND

CRSE 3R:TTPE 0 NEUMANN C•SE 3R:TTPE ! NEUMANN C•SE 3R:TTPE 2 BOTH KINDS CRSE 3•:TTPE 3 NEUMANN

I) -1 (Rod Density / Cladding Density)

10.0

8.0

8.0

LEGEND CRSE 3B:TTPE 0 NEUMRNN CRSE 3B:TTPE ! BOTH KINDS CRSE 3B:TTPE 2 NEUMRNN CRSE 3B:TTPE 3 NEUMRNN

2.0

o o o.o i.o •.o •.o ,'0 ....... .•

D-1 (Rod Density / Cladding Density)

FIG. 13. (a) Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod density/ cladding density (p - ' ) for the clad rod of class 3a from Table I. The Dir- ichlet and type 2 Neumann modes coincide for this range ofp. The type 1 mode for this case required a somewhat larger error bound on the value of the determinant arising from (5) due to a section of apparently real roots lying between --•0.0859 and --•0.0862. (b) Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod density/cladding density (p - ' ) for the clad rod of Class 3b from Table I. The Dirichlet and type 2 Neumann modes coincide for this case.

10.0

0.0 l.O 2.0 3.0 •.0 5 (8) I3-1 (Rod Density / Cladding Density)

10.0

8.0

8.0

2.0

0.0 0

LEGEND CRSE NB:TTPE 0 NEUMRNN CRSE NB:TTPE ! NEUMRNN CRSE NB:TTPE 2 NEUMRNN CRSE NB:TTPE 3 NEUMRNN CRSE N:TTPE 2 DIRICHLET CRSE NB:TTPE ! DIRICHLET

13 -1 (Rod Density / Cladding Density)

FIG. 14. (a) Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod density/ cladding density (p - ' ) for the clad rod of class 4a from Table I. The cutoffs for the type 0 and 1 modes occur at --• 0.1501 and the cutoff for the type 3 mode occurs at --•0.0499. The Dirichlet and type 2 Neumann modes coincide for this range ofp. (b) Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod density/cladding density (p - ] ) for the clad rod of Class 4b from Table I. The cutoffs for the type 0 and 1 modes occur at --• 0.2346 and the cutoff for the type 3 mode occurs at --• 0.0806. The crossover of the Dirichlet mode from type 1 to type 2 (with decreasing rod density) occurs at --• 3.3936.

form of the attenuation that does not have the factor bR shown in Eq. (27). Corresponding curves from the appropriate pairs of figure are labeled with the same letter. To find the relative attenuation in dB between two points, (r, zl ) and (r, z2), located at equal radial distances from the center of the rod, one must multiply the value given by the frequency of the mode and the axial distance z2-zl.

C. Energy flow and displacement plots

Even considering the relative simplicity of radial axial modes in an infinitely clad rod with perfectly bonded interface, a large variety of wave phenomena can occur as the five parameters describing a given mode are changed. We show here only some of these phenomena. The plots as given show

1076 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons ot a/.' Clad rods. I 1076


20.0

18.0

E 12.0

._

O '

> 8.0,

g o,

o.o o

LEGEND œRSE 5R:TYPE 0 NEUMRNN œ•SE 5R:TYPE 1 NEUMRNN œ•SE 5•:TYPE 2 NEUMRNN œ•SE 5R:TYPE 3 NEUMANN

.... œ•SE 5R:TYPE 1DIBIœHLET

0 ].0 2.0 3.0 u,.0 5

p-1 (Rod Density / Cladding Density)

10.0

8.0

6.0

2.0

0.0 0

LEGEND œRSE GR:TTPE 0 NEUMRNN œRSE 6R:TTPE I NEUMRNN œRSE 6R:TTPE 2 NEUMRNN œRSE 6R:TTPE 3 NEUMRNN œRSE 6R:TTPE 2 DIBIœHLET œRSE 6R:TTPE I DIRIœHLET

0 .... 1.0 2.0 3.0 I-1.0 5 lb-1 (Rod Density / Cladding Density)

]0.0

8.0

6.0

2.0

0.0 0

LEGEND œRSE 5B:TTPE 0 NEUMRNN œRSE 5B:TTPE 1 NEUMRNN œRSE 5B:TYPE 2 NEUMRNN CRSE 5B:TYPE 3 NEUMRNN

....... CRSE 5B:TTPE 1 DIRIœHLET

0 1.0 2.0 3.0 q.O 5

•-l(RodDensity/CladdingDensity)

FIG. 15. (a) Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod den-

sity/cladding density (p - •) for the clad rod of class 5a from Table I. The cutoffs for the type 0 and 1 modes occur at .•0.3554 and the cutoff for the type 3 mode occurs at -0.1213. The Dirichlet and type 1 Neumann modes coincide for this case. (b) Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod density/cladding density (p - • ) for the clad rod of Class 5b from Table I. The cutoffs for the type 0 and 1 modes occur at --•0.9385 and the cutoff for the type 3 mode occurs at --•0.5308. The Dirichlet and type 1 Neumann modes coincide for this case.

10.0

8.0

6.0

ti.0

2.0

0.0 0

LEGEND

CRSE 6B:TTPE 0 NEUMRNN CRSE 6B:TTPE 1NEUMRNN CRSE 6B=TTPE 2 NEUMRNN CRSE 6B•TTPE 3 NEUMRNN

....... CRSE 6B:TTPE ! DIRIœHLET

D-1 (Rod Density / Cladding Density)

Lt.0 5

FIG. 16. (a) Dispersion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod den-

sity/cladding density (p - •) for the clad rod of class 6a from Table I. The cutoffs for the type 0 and 1 modes occur at --• 0.8313 and the cutoff for the type 3 mode occurs at --•0.4070. The crossover of the Dirichlet mode from type 1 to type 2 (with decreasing rod density) occurs at •0.0535. (b) Dis- persion curves for the modal phase velocity along the fiber axis for both Neumann and Dirichlet modes as a function of rod density/cladding density (p - • ) for the clad rod of class 6b from Table I. The cutoffs for the type 0 and 1 modes occur at • 0.9098 and the cutoff for the type 3 mode occurs at ..•0.4905.

the elliptical particle trajectories at a variety of points throughout the clad rod. Since only the relative size of the ellipses is important in a given plot, the scale varies from plot to plot for ease of graphical presentation. A large arrowhead is shown on each trajectory to indicate particle location throughout the clad rod at one moment of time, thus indicating positions of equal phase. The direction of the arrowhead

indicates the sense of motion of the particle. Two smaller arrowheads visible on some of the larger ellipses show the positions at two other equally spaced periods of time as the particle proceeds in its orbit. If the size of the ellipse is too large or too small to permit convenient graphing, it is not plotted. From the center of each ellipse extends an arrow which gives the energy velocity at that point.



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


FIG. 17. Attenuation in dB/(mm MHz) along the fiber axis for the two, axial rod mode related subfamilies of type 0 leaky modes as a function of rod radius times frequency (rof) in the aluminum-silicon carbide system. These attenuation curves correspond to the phase velocity curves of Fig. 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

Radius X Frequency (mm/l•s)

FIG. 18. Attenuation in dB/(mm MHz) along the fiber axis for the radial related subfamily of type 0 leaky modes as a function of rod radius times frequency (rof) in the aluminum-silicon carbide system. These attenuation curves correspond to the phase velocity curves of Fig. 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 in dB/(ram MHz) along the fiber axis for the family of type 1 leaky modes as a function of rod radius times frequency (to f ) in the aluminum-silicon carbide system. These attenuation curves correspond to the phase velocity curves of Fig. 6.



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 in dB/(mm MHz) along the fiber axis for the family of type 2 leaky modes as a function of rod radius times frequency (rof) in the aluminum-silicon carbide system. These attenuation curves correspond to the phase velocity curves of Fig. 7.

Starting from a convenient location within the rod, an integrated curve shows the flow of energy proceeding from that point. Since all energy flow curves are parallel with respect to a shift in the axial direction, all other energy travel- ing through a tube of a given cross-sectional area will de- crease exponentially with increasing z. To better indicate graphically the nature of the lacuna that lies to the left of the "leading" energy flow curve, the plotting of particle trajectories to the left of the energy flow curve has been suppressed. Usually this curve is not the leading curve, but represents that curve shifted to the right to give an indication of energy flow within the rod. Rigorously, the actual position of the leading energy flow curve is unknown. If one considers a rod in a cladding of infinite radius, but only half-infinite axially (such as >•0), then the modes described in this study represent only an approximation to the actual modes of such a configuration, although we can expect that the approximation becomes better the further one moves away from the edge of the cladding. Even if one were to accept that the infinite rod modes were an accurate approximation to those of a rod with a truncated infinite cladding, especially far from the edge of the cladding, the presence of multiple modes near the edge would produce mode interference with a resulting complicated energy flow curve.

Only in regions where one mode predominates could one expect the curves shown to give an accurate representation of the energy flow. In particular, one might expect that even with one mode predominating in regions away from the face of the cladding, the presence of many highly attenuated modes with high leakage angle could cause a shift of the leading edge of the energy curve of the predominant mode away from the junction point of the rod and the cladding. In experiments thus far conducted in steel rods clad with aluminum and with SiC rods clad with aluminum, seven leaky modes have been detected (of types 0 and 2), but the above mentioned possible shift was not observed. •4'•8

Figures 21-27 show a series of displacement plots for

modes of types 0-2 for the aluminum-clad silicon carbide rod. The modes in Figs. 21 and 22 are ones which have been generated from waves sent down a bare rod into a cladding. 19 They are calculated at rof= 9 mm/ps, which is just beyond the peak of these two modes. The displacement and energy plots of the rod mode associated with the leaky mode of Fig. 21 is shown in Fig. 28, also for rof= 9 mm/ps. At this value

s.ø 1 q O•

3.0

1.0

0.0 0 0

/'/ I / I

(.// / / / ............... ..................... ................ ................ ................. .............. .................... 1.0 2.0 3.0 •.0 •.0

Z / •adius

FIG. 21. Displacement orbits and energy velocity diagrams for mode D of Fig. 4 at a value of rof-- 9 mm//ts. For this mode, v = 18.41 + il 1.09 mm//ts, the asymptotic leakage angle -- 75.58 ø and the asymptotic energy velocity = 6.25 mm//ts. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position (equal phase). The dimensioned inset arrow scales the energy velocities which run from the center of each orbit (unless suppressed for graphical reasons). Those energy velocity arrows with hollow heads are associated to flows inside the fiber and those with solid heads to flows in the cladding. Because of the tangential discontinuity in energy flow at the interface, two energy velocity arrows appear for orbits centered along the interface.



5.0

q.0

3.0

'" I / /

/ /

/ /

•.o .:.. ...... •• ............ • .............. • .................. • ...................

o.o •.o •'0 ..... •'0 ..... •'0 ..... •.o Z / Radius

FIG. 22. Displacement orbits and energy velocity diagrams for mode N of Fig. 5 at a value of to f= 9 mm//•s. For this mode, v = 12.05-4-i3.21 mm//•s, the asymptotic leakage angle = 61.01 ø and the asymptotic energy velocity - 6.25 mm//•s. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position (equal phase). The dimensioned inset arrow scales the energy velocities which run from the center of each orbit (unless suppressed for graphical reasons). Those energy velocity arrows with hollow heads are associated to flows inside the fiber and those with solid heads to flows in the cladding. Because of the tangential discontinuity in energy flow at the interface, two energy velocity arrows appear for orbits centered along the interface.

/ / / . \ \ \

......................... ..................... ......................... ........................ ................ 0.0 ....... • • ................ , ...........

-1.0 0.0 1.0 2.0 3.0 •.0 Z / •adius

FIG. 24. Displacement orbits and energy velocity diagrams for mode D of Fig. 4 at a value of to f-- 4.5 mm//•s. For this mode, u- -- 1.18 -F i 6.4 mm//•s, the asymptotic leakage angle - 94.46 ø and the asymptotic energy velocity -- 2.81 mm//•s. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position (equal phase). The dimensioned inset arrow scales the energy velocities which run from the center of each orbit (unless suppressed for graphical reasons). Those energy velocity arrows with hollow heads are associated to flows inside the fiber and those with solid heads to flows in the cladding. Because of the tangential discontinuity in energy flow at the interface, two energy velocity arrows appear for orbits centered along the interface.

of rof the phase velocity for this mode -•29.36 mm/Fs-• 5.65 bR. Since there is no dissipation for this mode, the orbits are unchanged along the z axis, and only one set was plotted. As can be seen, there are regions in the rod where the energy flow is reversed indicating that no energy will be conducted through that region of the rod for that mode. Through the remainder of the rod, the energy velocity ranges from zero to about 1.27 bR. The peak energy velocity

occurs at about two-thirds of the rod radius; and the overall group velocity-•orbital rotation changes sign are clearly seen from the figure.

Figure 23 shows the second lowest type. 0 mode, also at rof-- 9 mm/Fs. This is a relatively large value of •b for this mode, and although the energy velocity is beginning to approach the b•, the orbits far from the interface remain ellip- tically shaped and oriented in the direction of energy glow.

q.0

3.2

1.6

0.8

o. 8 ........ ::-'1::'-.'::. . ........ .......... ..... _ ............ ............. : ............ :... 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 and energy velocity diagrams for mode A of Fig. 4 at a value of to f= 9 mm/Fs. For this mode, v -- 5.66 -4-/0.19 mm//•s, the asymptotic leakage angle -- 75.58 ø and the asymptotic energy velocity -- 5.65 mm//•s. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position ( equal phase). The dimensioned inset arrow scales the energy velocities which run from the center of each orbit ( unless suppressed for graphical reasons). Those energy velocity arrows with hollow heads are associated to flows inside the fiber and those with solid heads to flows in the cladding. Because of the tangential discontinuity in energy flow at the interface, two energy velocity arrows appear for orbits centered along the interface.



8.0

0.0' O.

:'ddd

Radius

5.0

FIG. 25. Displacement orbits and energy velocity diagrams for mode C of Fig. 4 at a value of to f= 40 mm//zs. For this mode, v -- 9.689 + t0.0066 mm//zs, the asymptotic leakage angle -- 49.26 ø and the asymptotic energy velocity -- 6.323 mm//zs. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position (equal phase). The dimensioned inset arrow scales the energy velocities which run from the center of each orbit (unless suppressed for graphical reasons). Those energy velocity arrows with hollow heads are associated to flows inside the fiber and those

with solid heads to flows in the cladding. Because of the tangential discontinuity in energy flow at the interface, two energy velocity arrows appear for orbits centered along the interface. (In this case the larger size of the orbits inside the rod prohibit their inclusion on the figure.)

FIG. 27. Displacement orbits and energy velocity diagrams for mode C of Fig. 7 at a value of rof= 9 mm//zs. For this mode, v- 7.44 + •0.059 mm//zs, the asymptotic leakage angle -- 65.55 ø and the asymptotic energy velocity -- 3.098 mm//zs. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position (equal phase). The dimensioned inset arrow scales the energy velocities which run from the center of each orbit (unless suppressed for graphical reasons). Those energy velocity arrows with hollow heads are associated to flows inside the fiber and those

with solid heads to flows in the cladding. Because of the tangential discontinuity in energy flow at the interface, two energy velocity arrows appear for orbits centered along the interface.

5.0

3.0

2.0

1.0

0.0 0 0 1.0 2.0 3.0 LI.0 5

Z / Radius

FIG. 26. Displacement orbits and energy velocity diagrams for mode B of Fig. 6 at a value of rof= 9 mm//zs. For this mode, v- 10.29-t-•O.35 mm//zs, the asymptotic leakage angle -- 52.16 ø and the asymptotic energy velocity -- 6.32 mm//zs. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position (equal phase). The dimensioned inset arrow scales the energy velocities which run from the center of each orbit (unless suppressed for graphical reasons). Those energy velocity arrows with hollow heads are associated to flows inside the fiber and those

with solid heads to flows in the cladding. Because of the tangential discontinuity in energy flow at the interface, two energy velocity arrows appear for orbits centered along the interface.

0 82 -%L%•... [

0.62

0.•12

0.22 ß

O. 02! ............. :--..•. -- -0.5 -0.3 -O.l 0.1 0.3 0

Z / Radius

FIG. 28. Displacement orbits and energy velocity vectors for the rod mode at a value of rof= 9 mm//zs associated to mode D of Fig. 4. This is the fifth rod mode counting from the top left in Fig. 2. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position (equal phase). The dimensioned inset arrow scales the energy velocities that run from the center of each orbit (unless suppressed for graphical reasons). The phase velocity is 29.357 for this case and, since there is no attenuation, orbits and energy vectors are unchanged under translation.



This feature is retained even up to very high values of •, where the imaginary component of the phase velocity and leakage angles are very small, the energy velocity is almost exactly bR, and the asymptotic orbits have a major axis/ minor axis ratio • 1.75. Figure 24 shows the mode of Fig. 22 at to f-- 4.5 mm/lts, where it is backward leaking; and Fig. 25 shows the third lowest type 0 mode at to f= 40 mm/lts that lies on one of the almost longitudinal velocity tails arising after mode knitting.

The energy flow curves for these modes exhibit some of the more common characteristics encountered for leaky modes. The waveforms in each of the rod and cladding are made up of the sum of two components, one arising from a shear potential and one from a longitudinal potential. In the cladding, in particular, if only one of these components is present, the waveform described appears to have its angle of leakage and energy velocity almost independent of r, depending only on v. When both components are present at approximately equal magnitudes, the waveform produced has an energy flow curve (actually a conical surface in three dimensions) in the overall direction determined by the predominant component but with overlying oscillations produced by interaction with the second component. When dealing with waves of very low exponential growth, this os- ciliatory character in the energy flow curve can continue for a substantial distance from the interface, as in Figs. 25 and 32, before the exponentially predominant term takes over to produce an essentially straight line. In other cases the relative decay of one of the components may be of the order of a wavelength producing a flow curve that just appears to bend as in Fig. 23, or one of the components may predominate in regions very close to the interface to produce an effectively straight energy flow curve. An extreme case of the curvature of the energy flow curve is discussed below for an example drawn from the aluminum-steel system.

Figures 26 and 27 show examples of type 1 and type 2 modes (here mode B in Fig. 6 and mode C in Fig. 7) also taken at values of ro f-- 9 mm/lts. The predominantly longitudinal character of the type 1 mode and the predominantly shear type character of the type 2 modes are seen from com- paring the energy flow direction with the orientation of the particle orbits.

Type 3 modes, which are not leaky modes, do not exist in the A1-SiC system. We show instead the displacements and energy velocity for two type 3 modes for the system listed as 2a in Table I using to f= 15.5 mm//•s and a density of 19.32 g/½m 3. This is similar to the aluminum-tungsten clad rod withp - 1 -• 6.9. For type 3 modes all elliptical trajectories have their major and minor axes either perpendicular to or parallel to the cylinder axis. In Fig. 29 is shown a plot of the lowest type 3 mode, which tends to the Stoneley mode as rof tends to infinity. Figure 30 shows the displacements for that type 3 mode shown in Fig. 12 (a), except at a value of p- 1 = 6.9, which lies slightly outside that figure. These figures use the customary plots of the relative maximum displacements in the radial and axial directions. Those points where the r and z displacements are of the same sign indicate a clockwise particle motion and those where they are of op- posite sign indicate a counterclockwise motion. Changes in

! • RADIAL DISPLACEMENT .... AXIAL DISPLACEMENT

.... ENERGY V•LOCITY ......... INTERFRC

3.0

2.0

1.0

0.0 •\'•.L Tungsten i Aluminum

0.0 0.• 1.6

3.0

0.8 1.2

R / Radius

2.8

2.• E

o --

2.L; •

2.2

2.0 2 0

FIG. 29. Displacement and energy velocity plots for the Stoneley-like mode for the system listed as 2a in Table I. The density was taken as 19.32 g/cm 3 and rof-- 15.5 mm//ts. That makes this system somewhat similar to an aluminum clad tungsten rod. The phase velocity and overall group velocities for this mode are 2.71 mm//ts. When the radial and axial displacements are of the same sign, the orbital motion is clockwise; otherwise it is counterclockwise.

the direction of rotation are usually separated by a point of rectilinear motion. Examination of several of these modes

shows that, as the phase velocity increases, the number of changes in rotational direction increase, and the relative minima of the energy velocity curves tend to lie near points of rectilinear motion. Positions of equal phase are not indi-

3.0 .

-• •.o

._>

• 1.o

,• -o.o ._•

E i ' "\ /.•..i• RRDIRL DISPLACEMENT :m • ' \ J' • ' •'i-"• -x L o•õ•L•C•E•T E 1 0'• \•J .... ENERGY VELOC I TT .I - ß /

-2.0 0.0 O.L$ 0.8 1.2 1.6 2

R / Radius

'3.0

2•

2 2

2 •

FIG. 30. Displacement and energy velocity plots for the type 3 Neumann tunnel related mode for the system listed as 2a in Table I. The density was taken as 19.32 g/cm 3 and rof-- 15.5 mm//ts. This mode is shown in Fig. 12(a) for values ofp- • up to 5, while here the value ofp- • -- 6.9 to better match the aluminum-tungsten system. But the curve is almost flat over the intervening values ofp- •. The phase velocity and overall group velocities for this mode are 2.88 mm//ts. When the radial and axial displacements are of the same sign, the orbital motion is clockwise; otherwise it is counterclockwise.



cated in this figure; doing so would require an additional curve.

We have indicated that it is possible for type 3 modes to make a transition to type 2 modes when the decay exponent becomes zero. If the phase velocity of the resultant type 2 mode becomes greater than b c, then, in the cylindrical geometry it always seems to have an imaginary component, and, by formulae (23) must be a leaky wave. However, in the planar interface geometry it may be possible to have a purely real phase velocity. Such a mode would be a combination of a plane shear wave and an exponentially decaying wave, and, thus, be partially decomposable on the appropriate side of the interface. On that side of the planar interface, the asymptotic energy flow is governed by the energy flow of the plane wave and is not governed by the asymptotic leakage angle formulae (22), which are ambiguous in this case. If, on the other hand, the phase velocity of the resultant type 2 mode becomes less than b c and its imaginary part is very small, then, in either the cylindrical or planar interface geometries, it has exponential growth away from the interface, but the leakage angle is essentially zero. Strictly speaking, such waves do not leak energy, and we refer to them instead as divergent waves. In addition to divergent waves of type 2, divergent waves of types 0 or 1 exist whenever the phase velocity is less than a c. Since the energy flow for such modes is parallel to the cylinder axis, the only way to propagate a beam Of finite radius of divergent type is to bring the energy in from the edge of the cladding. We are not aware of the experimental detection of such modes. When type 3 modes of almost the same phase velocity as a divergent type 1 or type 2 mode exist, then all 3 types of divergent waves, although possible are not likely to be physically present. Di- vergent waves have been discussed elsewhere for the planar interface. :ø

There are ten nonredundant planar interface modes for the aluminum-silicon carbide system. Four of these modes are decomposable. The energy flow patterns for the two type 2,3 modes are shown in Fig. 31 (a) and (b). The shift at the interface is apparent in the nondecomposable mode of Fig. 31 (a) and has a magnitude of about 1.1 mm for a 2-MHz wave. This is about 70% of the shortest of the four wave-

lengths involved in this system. In this mode the waveforms are essentially of shear type on both sides of the interface, but in the types 0,3 and 1,3 they change from shear to longitudinal type producing a 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

wavelength in aluminum and equal to the shear wavelength in silicon carbide. Although there are four plane waves mak- ing up the decomposable mode in Fig. 32 (b), the longitudinal component in the aluminum is very small giving rise to an almost pure shear mode in the aluminum.

Five of the 10 modes (those with the second type index equal to 3) can be approximated by modes in the aluminum- clad silicon carbide rod. Two of the five are decomposable modes and, therefore, not the limit of a single leaky mode branch. The other three leaky modes are high • limits of an associated branch of each of the three leaky mode types (0- 2). For the type 1 and 2 modes, this mode has the lowest limiting phase velocity, but the type 0 mode has a phase velocity between bR and ac. Table II lists these ten modes.

We conclude this section with an example of possible technical interest in its own fight drawn from the aluminum-steel system (steel: density = 7.9 g/cm 3, aR = 5.92 mm//zs, bR = 3.25 mm//zs). We have noted before in dis- cussing changes of density that type 2 modes can shift over to type 1 modes. In some systems, such as aluminum-steel, this transition can occur with changing • while at a fixed density.

TABLE II. The ten planar interface modes for the aluminum-silicon carbide system.

Nondecomposable modes

Type v= Re(v) + iIm(v)

Asymptotic Asymptotic Asymptotic Asymptotic leakage energy leakage energy

angle (A1) velocity (A1) angle (SIC) velocity (SIC) in degrees in mm//zs in degrees in mm//zs

0,3 5.56 + 1.1 li 1,3 8.64 + 2.46i 2,3 4.73 + 0.23i 1,1 4.62 + 3.42i 3,1 10.48 + 2.45i 3,0 5.56 + 1.1 li

21.03 5.40 31.82 4.92

49.01 6.13 56.89 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

Decomposable modes

v = Re(v) Type in mm//zs

Longitudinal Shear Longitudinal Shear plane wave plane wave plane wave plane wave

(A1) (A1) (SIC) (SIC) in degrees in degrees in degrees in degrees

0,3 10.03 50.92 72.00 15.84 51.84 2,3 10.08 - 51.14 72.08 16.75 58.98 3,2 11.53 -- 56.75 -- 74.40 33.20 --63.23 2,1 16.38 -- 67.29 79.09 -- 53.90 71.51



7.0

5.0

3.0

1.0

-1.0

-3.0 0.0 2.0

./ ,/

/'

,/

• • • •---•- • • • • ,

q.0 6.0 8.0 10.0 12.0 lq.0

Distance in Tangential Direction (mm)

5.0

3.0

• 1.o / ._

z -l.O

i:5 -3 0

-5.0 ": .............. 0 0 2.0 q.O 6.0 8.0 10.0

{b) Distance in Tangential Direction (mm)

FIG. 31. (a) Displacement orbits and energy velocity diagrams for the type 0,3 nondecomposable planar interface mode listed in Table II. The frequency used for the calculation was 2 MHz, and the apparent shift along the interface for this mode is about 1.1 mm. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position (equal phase). The dimensioned inset arrow scales the energy velocities which run from the center of each orbit (unless suppressed for graphical reasons). Those energy velocity arrows with hollow heads are associated to flows inside the fiber and those with solid heads to flows in the cladding. Because of the tangential discontinuity in energy flow at the interface, two energy velocity arrows appear for orbits centered along the interface. (b) Displacement orbits and energy velocity diagrams for the type 0,3 decomposable planar interface mode listed in Table II. The frequency used for the calculation was 6 MHz. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position (equal phase). The dimensioned inset arrow scales the energy velocities which run from the center of each orbit (unless suppressed for graphical reasons). Those energy velocity arrows with hollow heads are associated to flows inside the fiber and those with solid heads to flows in the

cladding. Because of the tangential discontinuity in energy flow at the interface, two energy velocity arrows appear for orbits centered along the interface.

In this system, for instance, one such transition occurs at rof--• 10.177 mm//•s, where type 2 mode transforms to a type 1 mode at a velocity --•6.32358 mm//•s. The type 1 mode continues for only a short interval down to rof--• 10.1348 where it has a cutoff at the longitudinal velocity of the aluminum cladding. The important thing to ob- serve in this case is that the leakage angle of the type 2 mode, which obeys (23b), is about 60 ø right down to the crossover point while that of the type 1 mode, which obeys (23a), is less than 1ø. Just above the crossover point the phase velocity and the two leakage exponents tend to zero. As one moves above the crossover point, the discrepancy between the leakage exponents increases, but because it's still small, and because the mode near the rod should look like an almost non-

attenuating longitudinal wave, one can expect that in a range

of rofabove 10.177 mm//•s, there exists a mode which is essentially confined to the rod for some distance, which is a function of rof, and then turns rather sharply--due to the takeover of the type 2 exponent--to leak away at about 60 ø to the interface. This distance will be a very sensitive function of rof and, in principle, can produce a wave shift of any magnitude. Figure 32 shows the displacement and energy

velocity profile for such a wave at a value of rof-- 10.5 mm//•s, where the magnitude of the apparent shift is about 14.6 times the radius of the rod.

D. Asymptotic leakage angle

The distribution of asymptotic leakage angles for the A1-SiC example again corresponding to Figs. 4-7 is present-



20.0

16.0 ,

12.0.

8.0,

LJ O'

0.0 0

/ ß

//

/

0 ..... ,'0 ..... 8'0 .... i2 'ø .... i6'0 .... 20'o Z / Radius

FIG. 32. Displacement orbits and energy velocity diagrams for type 2 mode in the aluminum-steel system at ro f= 10.5, which lies near a crossover to a type 1 mode (aluminum: density = 2.77 g/cm 3, a c = 6.323 mm//•s, bc = 3.1 mm//•s; steel: density = 7.9 g/cm 3, an = 5.92 mm//•s, bR = 3.25 mm//•s). For this mode, v = 8.33 + i0.02 mm//•s, the asymptotic leakage angle = 60.66* and the asymptotic energy velocity = 3.09999 mm//•s. The tip of the largest arrowhead on each elliptical trajectory marks a synchronous position (equal phase). The dimensioned inset arrow scales the energy velocities which run from the center of each orbit (unless suppressed for graphical reasons). Those energy velocity arrows with hollow heads are associated to flows inside the fiber and those with solid heads to flows in the cladding. Because of the tangential discontinuity in energy flow at the interface, two energy velocity arrows appear for orbits centered along the interface. The apparent shift along the interface for this mode is almost 15 times the rod radius.

ed in Figs. 33-36. As with the dispersion curves, a uniform trend is apparent: at values of•b will below the phase velocity peak, the leakage angle is large, near 90 ø , and, in the case of the type 0 radial modes, the leakage angle actually exceeds 90 ø for the values of •b where the mode is backward leaking. As •b approaches the position of the maximum in phase velocity, the leakage angle starts to drop, and continues so until reaching an asymptotic limit. Depending on the relative values ofac, aR, bc, and bR, there may be a mode with leakage angle tending to that of an associated planar interface mode, but all other rod-related modes, whose phase velocity pos- sesses a small--but usually nonzero--imaginary component, have a leakage angle given by (23a) or (23b). Note that, since the limiting phase velocities are independent of p, so are these limiting leakage angles (although the range ofp where the limiting values can be used may depend on •b).

It is possible to analyze the leakage angle for this type of mode, with relatively small imaginary component in v, rather completely. If we set c = ac or bc and C = A c or -Bc, respectively, then, when Re(v) > c, the leaky exponent takes the form

exponent • d- Im (k) / tan ( I C I), (28a)

the plus sign occurring for types 0 or 1 when c = ac and for types 0 or 2 when c = bc. Similarly, when Re(v) < c,

exponent• ---F_ Re(k) [ 1 -- Re(l•)2/c2] 1/2 (28b) and when Re (v) • c

exponent• d- (2•rf/c)[ Im(v)/c]. (28c)

From Eqs. (28) it can be seen that for type 0 or 1 modes, for instance, and for almost all instances where a c/bc is not near one, Eq. (23a) determines the leakage angle. Similarly

(23b) determines the leakage angle for type 2 modes, thus giving rise to a possible discontinuity in leakage angle at crossovers from type 1 to type 2 modes as noted above.

At any fixed value of •b, then, there are infinitely many modes with leakage angle near 90 ø , since angles near 90 ø , together with very low phase velocities is characteristic of high-order modes well below their peak value of •b. At higher values of •b there are also many modes with leakage angle near their limiting value. In addition, one can expect a few modes with an intermediate leakage angle. Except for low values of •b, where there are no other modes with a given limiting leakage angle, it is these intermediate modes that best offer the possibility for experimental detection in the cladding. Usually, those modes with a higher leakage angle have a greater attenuation, so that their amplitude will be relatively weak at points (r,z) in the cladding where r/z is the tangent of the mode with intermediate leakage angle. (Here, z has been measured from the leading edge of the leakage and we have assumed that the initial amplitude--the "injection coefficient"--of the intermediate mode is sufficiently large compared to modes with higher leakage angles.) The intermediate mode will then appear in almost pure form over a range ofz (at fixed r) up to the leading edge of the next leaky mode. Even when pairs of intermediate modes coexist, one may have a sufficiently larger injection coefficient to make it the predominant mode.

E. Limiting energy velocity

The limiting energy velocity represents the magnitude of the energy flow at large distances from the rod/cladding



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


FIG. 33. Asymptotic leakage angle in degrees for the two, axial rod mode related subfamilies of type 0 leaky modes as a function of rod radius times frequency (rof) in the aluminum-silicon carbide system. These curves correspond to the phase velocity curves 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


FIG. 34. Asymptotic leakage angle in degrees for the radial related subfamily of type 0 leaky modes as a function of rod radius times frequency (rof) in the aluminum-silicon carbide system. These curves correspond to the phase velocity curves of 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

Radius X Frequency (mm/ixs)

FIG. 35. Asymptotic leakage angle in degrees for the family of type 1 leaky modes as a function of rod radius times frequency (rof) in the aluminum-silicon carbide system. These curves correspond to the phase velocity curves of Fig. 6.



FIG. 36. Asymptotic leakage angle in degrees for the family of type 2 leaky modes as a function of rod radius times A frequency (ro f) in the aluminum-silicon carbide system. These curves correspond to the phase velocity curves of Fig. 7.

interface. At these distances the direction of energy flow is given by the asymptotic leakage angle. The limiting energy velocity is then derived using whichever of the regimes (21 a) or (2lb) is valid. For the examples shown herein, this is usually (21 b) for type 2 modes and (21 a) for types 0 and 1 modes except for backward leaking modes, where it becomes (2lb).

Figures 37-40 show the limiting energy velocity for the A1-SiC example. Except for the tunnel related type 1 mode, the types 1 and 2 curves limit at high values of • near the longitudinal and shear velocities of the cladding, respectively, while the type 0 behavior is more complex. The asymptotic energy velocity curves for the type 0 modes in Fig. 37 peak at a value just below a c at values of • slightly beyond the phase velocity peaks in Fig. 4. [The type 0 mode labeled "B," which is related to type 0 Neumann tunnel mode is an exception to this. This mode, which tends at high • to a

planar mode, has a large complex component to its velocity, and does not follow (23a). ] In this region the asymptotic particle motion is described by eccentric ellipses oriented in the direction of leakage, and the values of Im (v) / Re( v ) are also rather small, so that Eq. (23a) is approximately correct. This equation can, there, be interpreted as giving the phase velocity along the z axis in terms of a projection of the natural wave motion in the direction of asymptotic energy motion, where the phase and energy velocities are equal, onto the z axis. As the phase velocity approaches that of the cladding, the asymptotic energy velocity also drops, and both the energy and phase velocities eventually approach the shear velocity of the rod. The asymptotic particle motion here remains elliptical and parallel to the interface. For those modes whose phase velocity remains near as, such as that whose displacements near the interface are.indicated in Fig. 25, the asymptotic energy velocity remains near ac. At large

]0.

2.0

0.0 ................................................................ O. 0 •l. 0 8.0 12.0 16.0 20.0 2•. 0 28.0 32.0

Radius X Frequency (mm/i•s) 36.0 •to o

FIG. 37. Asymptotic energy velocity in mm//zs for the two, axial rod mode related subfamilies of type 0 leaky modes as a function of rod radius times frequency (rof) in the aluminum-silicon carbide system. These curves correspond to the phase velocity curves of Fig. 4.

1087 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons et al.' Clad rods. I 1087


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 energy velocity in mm//•s for the radial related subfamily of type 0 leaky modes as a function of rod radius times frequency (rof) in the aluminum-silicon carbide system. These curves correspond to the phase velocity curves of 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

Radius X Frequency (mm/l•s) NO 0

FIG. 39. Asymptotic energy velocity in mm//•s for the family of type 1 leaky modes as a function of rod radius times frequency (rof) in the aluminum-silicon carbide system. These curves correspond to the phase velocity curves of Fig. 6.

10.0

.-:I. 8.0 E E

o 6 0 __o ß

c g.O

• 2.0

0.0 0

A

Radius X Frequency (mm/ixs)

FIG. 40. Asymptotic energy velocity in mm//•s for the family of type 2 leaky modes as a function of rod radius times frequency (ro f) in the aluminum-silicon carbide system. These curves correspond to the phase velocity curves of Fig. 7.

1088 J. Acoust. Soc. Am., Vol. 92, No. 2, Pt. 1, August 1992 Simmons et al.: Clad rods. I 1088


distances from the interface, where the asymptotic formula holds, the particle orbits are very eccentric and essentially that of a longitudinal wave in the cladding moving in the direction of asymptotic leakage.

The curves in Fig. 38 show similar behavior except for the discontinuities occurring when the modes become backward leaking. The discontinuity in asymptotic energy velocity that occurs upon transition from forward to backward leaking mode (or vice versa) is due to the change in the dominant asymptotic terms from (21 a) to (21 b). Just below the value of •b marking this transition, both asymptotic growth exponents are almost zero, and the distance to reach the asymptotic limit is very large. The asymptotic shape of the particle trajectories tends to be elongated in the direction of leakage perpendicular to the interface. Just above the transition value of •b, the growth exponents, which have changed sign, remain close to zero, but now the decay of the shear related exponent is less than that of the longitudinal related exponent. The consequent asymptotic shape of the particle trajectories tends to be elongated perpendicular to the direction of leakage, or parallel to the interface.

III. CONCLUSION

The class of nonattenuating guided modes in a clad rod system is restricted to those materials where the shear velocity of the rod is less than that of the cladding. In other systems all modes will leak energy from the rod into the cladding, and even in this restricted class of materials, modes leaking energy from the rod into the cladding exist.

Over a large range of elastic parameters, the wave modes in a clad rod system, which break down into four families-- three of which are leaky or divergent--can be correlated with the modes of the bare elastic rod or the cladding without a rod (tunnel), sometimes with traction-free boundary and sometimes with zero-displacement boundary conditions. There are infinitely many rod-related modes and five tunnel-related modes.

There are five dimensionless parameters needed to describe an axisymmetric mode in a clad rod with infinite cladding. This presents a variety of wave behavior that is too great to present in detail. Instead, the variation of mode behavior with change in rod radius or frequency is given in detail for one example of a silicon carbide rod embedded in an aluminum matrix.

We may summarize these results in the A1-SiC system by observing that, in addition to the correlation with rod and tunnel modes, over much of the range of•b where the attenuation is not excessive, most leaky modes in the cladding may be thought of approximately as longitudinal or shear waves in the cladding which are modified locally in the neighbor- hood of the interface by interaction with a wave of the other type. The principal wave type is longitudinal for types 0 and 1 and shear for type 2. The asymptotic orbit shapes are eccentric ellipses oriented in directions, respectively, parallel (perpendicular) to the direction of energy travel for modes of type 0 or 1 (2). Far from the interface these waves have a natural travel direction along a cone that is traced out by the energy velocity field, and they exponentially decay in a di-

rection perpendicular to this direction. The phase velocity along the z axis is essentially derived for these waves by projection from the natural direction of motion onto the z axis.

For a variety of systems other than A1-SiC, the behavior of the tunnel-related modes under change of density was studied. These systems were chosen with different ordering regimes in the wave velocities. In those systems where bR < bc, the mode behavior breaks into a tunnel dominated and rod dominated region with a boundary occurring approximately at the value ofpR where the shear impedance (pb) of the two media are approximately equal. In those systems where b• > bc, the type 0 and 1 modes exhibited a cutoff at a value ofp• where the shear moduli of the cladding and rod are equal.

The pattern of energy flow was analyzed using the detailed energy velocity field in a geometrically dispersive situation. This type of analysis does not seem to have been carried out before. It was shown that a lacuna can be expected after the start up of a leaky wave, and that the generator of this conelike lacuna, after tracing a curved path near the interface due to the interference of the two components of the leaky wave, becomes a straight line inclined at some angle to the rod axis. Even at high frequencies, when the imaginary part of the higher modes in a family is quite small, the leakage angle for a mode can be finite and large if the wave speeds in the rod are larger than those in the cladding.

Because of the interference pattern near the interface between longitudinal and shear type components, the energy flow curves and, therefore, the edge of the lacuna of a leaky mode may shift from the position estimated from the asymptotic leakage angle of the mode. This shift, analogous to similar shifts found in optics and in acoustics, can be obtained directly by following the energy flow pattern. In some leaky waves, the energy pattern actually reverses itself and flows backwards; if this occurs asymptotically in the cladding, the leaky waves are backward leaking. In other cases the shift can be arbitrarily large and adjusted as a sensitive function of frequency, perhaps allowing unusual waveguides to be con- structed.

In addition to the complex patterns found in leaky waves, the energy velocity field displays the interference between plane and/or inhomogeneous waves near a plane interface. This pattern differs from the averaged energy flow, given by the group velocity, even in bare rods, and for clad rod systems possessing guided modes, the energy velocity at large distances from the interface is equal to the phase velocity of the cladding, not, as might be expected, the group velocity of the mode.

Leaky modes were connected with similar modes occurring at planar interfaces. For the cylindrical geometry there are an infinite number of such modes, but only a finite number in the planar interface geometry. It was seen that one clad rod system mode, not always that with lowest phase velocity, tends to that of the planar interface system, if such a mode exists and has nonzero asymptotic leakage angle. Oth- erwise, such a mode is a cluster point for rod-system modes, but is not the limit of any one mode. Decomposable and partially decomposable planar interface modes were described as was high-frequency phenomenon of mode knit-



ting in clad rod modes. Indeed, it was shown that the topo- logical characteristics of the modal structure in clad rods seems to differ substantially from that of a bare rod; that is, we have found that there is a rich structure of leaky, or evanescent, modes in a clad rod system and that, depending on the particular mode and frequency, the energy leaked by such modes conveys information about both the rod and the cladding.

If the structure near the interface zone is more complex, i.e., if the elastic constants vary with r and with f, most of the methods developed herein can be extended to describe the leaky modes in that structure. 19 Among the modes of such a clad rod system with interface zone, one can expect to find some that are sensitive to the elastic structure of the interface

zone. Torsional sensitivity can be included by extending the above analysis to 0 dependent modes where n • 0. In such cases the particle orbits will no longer be planar ellipses but will follow slightly more complicated three-dimensional paths. Anisotropy in the direction of the rod axis can also be included for the clad rod system, but more complex anisotro- pies can only be treated in the context of the planar interface using the type of formulation described in this study.

ACKNOWLEDGMENTS

We are particularly grateful for the early support from the Office of Naval Research and the continuing financial support from the National Institute of Standards and Tech- nology's Office of Nondestructive Evaluation. In addition we would like to acknowledge valuable discussions with A. H. Kahn and W. L. Johnson.

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39. C. Gazis, "Three-Dimensional Investigations of the Propagation of

Waves in Hollow Circular Cylinders. I. Analytic Translation," J. Acoust. Soc. Am. 31, 568-573 (1959).

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5D. A. Lee and D. M. Corbly, "Use of Interface Waves for Nondestructive Inspection," IEEE Trans. Sonics Ultrason. SU-24, 206-212 (1977).

6G. N. Watson, ,4 Treatise on the Theory of Bessel Functions H (Cambridge U. P., Cambridge, 1966), 2nd ed.

?F. W. J. Olver, in Bessel Functions of Integer Order, edited by M. Abramowitz and I. Stegun (Natl. Bur. Stds., Washington, DC, 1964).

ST. Tamir and H. L. Bertoni, "Internal Displacement of Optical Beams at Multilayered Periodic Structures," J. Opt. Soc. Am. 61, 1397-1413 (1971).

9B. A. Auld, Acoustic Fields and Waves in Solids (Wiley, New York, !973 ), Vol. 1, pp. 221-223.

!oj. L. Synge, "Flux of Energy for Elastic Waves in Anisotropic Media," Proc. R. I. A. $8, Sec. A (1956).

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(1965) (cf. especially Fig. 6 where the 0.31 Poisson ratio is rather close to the 0.296 value for SIC).

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14E. Drescher-Krasicka, J. A. Simmons, and H. N. G. Wadley, "Propaga- tion and Dispersion of Radial Axial Leaky Modes on Cylindrical Inter- faces," submitted to J. Acoust. Soc. Am.

15A. Schoch, "Seitliche Versetzung eines total reflekzierten Strahls bei U1- traschallwellen," Acustica 18-19, 2 (1952).

16L. V. Ahlfors, in Complex Analysis, edited by G. Springer and E. H. Spen- ier (McGraw-Hill, New York, 1979), 3rd ed.

•?M. Onoe, H. D. McNiven, and R. D. Mindlin, "Dispersion of Axially Symmetric Waves in Elastic Rods," J. Appl. Mech. 29, 729-734 (1962).

•8E. Drescher-Krasicka, J. A. Simmons, and H. N. G. Wadley, "Guided Interface Waves," in Review of Progress on QNDE, edited by D. Thomp- son and D. E. Chimenti (Plenum, New York, 1968), Vol. 6B, pp. 1129- 1131.

19H. N. G. Wadley, J. A. Simmons, R. B. Clough, F. Biancaniello, E. Drescher-Krasicka, M. Rosen, T. Hsieh, and K. Hirschman, "Composite Materials Interface Characterization," NBS Internal Report 87-3630, Gaithersburg (1987).

2oj. A. Simmons, E. Drescher-Krasicka, H. N. G. Wadley, M. Rosen, and T. M. Hsieh, "Ultrasonic Methods for Characterizing the Interface in Composites," in Review of Progress in QNDE, edited by D. O. Thompson and D. E. Chimenti (Plenum, New York, 1988), Vol. 7.



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