The symmetry and coupling properties of solutions in general

anisotropic multilayer waveguides.

F. Hernando Quintanillaa, M. J. S. Lowea and R. V. Crasterb

aDepartment of Mechanical Engineering, bDepartment of Mathematics,

Imperial College, London SW7 2AZ, UK

(Dated: February 2, 2017)

Abstract

Multi-layered plate and shell structures play an important role in many engineering settings

where, for instance, coated pipes are commonplace such as the petrochemical, aerospace, and

power generation industries. There are numerous demands, and indeed requirements, on Non

Destructive Evaluation (NDE) to detect defects or to measure material properties, using guided

waves; to choose the most suitable inspection approach, it is essential to know the properties of

the guided wave solutions for any given multi-layered system and this requires dispersion curves

computed reliably, robustly and accurately.

Here we elucidate the circumstances, and possible layer combinations, under which guided

wave solutions, in multi-layered systems composed of generally anisotropic layers in flat and

cylindrical geometries, have specific properties of coupling and parity; we utilise the partial wave

decomposition of the wave field to unravel the behaviour. We introduce a classification into five

families and claim that this is the fundamental way to approach generally anisotropic waveguides.

This coupling and parity provides information to be used in the design of more efficient and robust

dispersion curve tracing algorithms. A critical benefit is that the analysis enables the separation

of solutions into categories for which dispersion curves do not cross; this allows the curves to be

calculated simply and without ambiguity.

[]Accepted for publication in the The Journal of the Acoustical Society of America 141(1), January 2017,

DOI: 10.1121/1.4973543.

PACS numbers: 43.35.Zc; 43.58.Ta; 43.20.Mv

1

I. INTRODUCTION

Guided waves, and their physical properties, constitute an essential tool in Non-

Destructive Evaluation (NDE) or Structure Health Monitoring (SHM). In recent years, due

to the increasing use of composites in aerospace structures, much attention has been given to

anisotropic waveguides, both in flat and cylindrical geometry. In particular, understanding

the properties of guided waves in systems with multiple-layers is of paramount importance

since composite structures are very often encountered as aggregates of several differently

oriented anisotropic plies.

The properties of the analytical solutions for a free isotropic plate (Rayleigh-Lamb

and SH waves) are now classical and appear in many elastic wave books, see for instance

[1] and [2]. Analytical solutions in anisotropy are somewhat harder to find, but none the

less for free plates with higher anisotropic symmetries such as cubic [3] and orthorhombic

and monoclinic [4], solutions do exist. More recently, and by means of the Stroh formalism,

dispersion relations have been obtained for generally anisotropic plates and some general

properties of the modes derived analytically as well as the asymptotic behaviour of the first

fundamental modes at high and low frequencies, see [5, 6]. Some computational aspects

of this problem are to be found in [7]. Additionally, various numerical algorithms exist to

compute explicit dispersion curves, see for instance Nayfeh [8, 9] and Lowe [10]. In [8, 9], the

Transfer Matrix Method is used to compute SH and generalized Lamb waves respectively in

multi-layered anisotropic systems, but only up to monoclinic symmetry. Other relevant ref-

erences, studying different aspects of the multiple layer problem in flat geometry are [10–15].

A parallel development has taken place for cylindrical geometries with earlier work

mainly focused on single isotropic free cylinders, see [16–21]; detailed accounts of guided

waves in cylinders are in books, i.e. [1, 2, 22–26]. The investigation of dispersion curves

in anisotropic single- and multiple-layer waveguides in cylindrical geometry by means of

various methods are in [14, 27–38].

Here we present a complete systematic study of the interplay between the symmetry

and parity properties of guided waves in multiple layered systems of arbitrary anisotropy in

2

both flat and cylindrical geometries; all materials previously considered are subsets and we

extend our approach up to and including triclinic symmetry. New cases that are essential

to the developments for multiple-layer systems, and that also have intrinsic interest in

their own right as they display some uncommon combinations of coupling and symmetry,

in single-layer flat systems are briefly but thoroughly addressed. Numerical examples are

computed using the Pseudo-Spectral Collocation Method (PSCM) which is a very robust

approach to treat waveguides with arbitrary anisotropy; a detailed account of the algorithm

itself is given in [35]. On the other hand, to help convey physical insights, our theoretical

analysis of flat waveguides, single- and multi-layer, is entirely based on the partial wave

decomposition of the fields. Without loss of generality, we restrict our attention to systems

with 2 and 3 layers; these two cases are sufficient as they allow for quick and simple

generalization to arbitrary n-layered systems. For cylindrical waveguides the analysis is

much simpler and it will be shown how the Christoffel equation, boundary and interface

conditions are sufficient to derive the desired properties of the solutions in a given multiple

layer cylindrical waveguide.

We exploit the results to obtain more efficient algorithms and determine whether the

dispersion curves for the guided wave modes cross, or not, for a given multi-layered

anisotropic system. This helps reduce, or eliminate, mode jumping occurring when a

tracing algorithm links the points belonging to one mode to those belonging to a different

nearby mode. Regardless of the computing algorithm used, the knowledge of parity and

coupling allows us to better understand the complicated dispersion curves arising in the

context of these multi-layered systems.

The paper is organized as follows. Section II presents the study, dispersion curves

and mode shapes, of new cases in free anisotropic plates; these complement [4, 5] and

provide the classification of single layer anisotropic waveguides that is necessary to move

on to the study of multiple-layer systems. In section III, the properties of the guided

waves in 2 and 3 layer systems are studied in detail using partial wave decomposition

(PWD). In particular, it is shown under which material combinations, arrangement,

axes orientation etc. the guided wave solutions possess certain symmetry and coupling

properties. Moreover, we show that 2 and 3 layer systems are sufficient to exhaust all

3

possibilities arising in n-layered aggregates. Section IV provides an analogous study for the

simpler free anisotropic cylindrical single or multi-layer waveguides; the analysis is mostly

reduced to examining the coupling of the equations for each particular case which is done

systematically. Finally, section V presents a discussion of the results and their implications

for dispersion curve computing algorithms.

II. SINGLE-LAYER ANISOTROPIC FLAT WAVEGUIDES

Before moving to multiple-layer systems we first need to develop a classification of the

single layer anisotropic flat waveguides based on their symmetry and coupling properties.

Three examples, that have not been studied before, are chosen to exemplify the ideas

and simplifications; their dispersion curves and mode shapes are discussed and computed

with the PSCM. These examples show that mode crossing is avoided altogether using the

symmetries and coupling properties, thus making numerical algorithms much more robust

and efficient. It is worth emphasizing that this is not restricted to PSCM so any algorithm

for computing dispersion curves can benefit.

PWD, used extensively [3, 10, 39, 40], provides a very intuitive physical approach;

the symmetry and coupling properties of guided waves in isotropic and various anisotropic

materials, including monoclinic symmetry, have been investigated using this method

[39, 40]. More recently, as an alternative, Shuvalov et al. [5] used the Stroh formalism to

study guided waves in free anisotropic plates and derived a formula for the symmetries of

the solutions in arbitrarily anisotropic plates in agreement with [39, 40]. To complete the

classification of materials and axes configurations according to their symmetry and coupling

properties for a single layer, we could extract formulae from [5] for each case. However, we

chose to use PWD because, firstly, it has the advantage over the Stroh formalism of being

more intuitive and accessible and, secondly, the single plate cases generalize readily to the

multiple-layer systems where larger dispersion determinants arise.

Two sets of axes are used: spatial axes fixed to the plate (or system of plates), de-

noted with small letters {x,y,z} and crystal axes denoted by capital letters {X,Y,Z} that

are rotated about the spatial axes. The fixed spatial axes {x,y,z} are defined as follows:

4

the plane {x,z} is contained in the plane of the plate (or system of plates) and the {y}

axis is normal to the plate; propagation takes place invariably along the {z} axis. For each

anisotropic crystal class, three independent axes orientations with respect to the spatial

axes are possible. These are given in Table I below. Each of the crystal configurations

Crystal Configuration Axis Normal to the Plate Propagation Axis

Y configuration {Y} {Z}

Z configuration {Z} {X}

X configuration {X} {Y}

TABLE I: The table with the three different crystal axes configuration possible for each crystal

class in cartesian coordinates. In the Y configuration crystal and fixed spatial (plate) axes {x,y,z}

are aligned.

will give rise to a solution with, generally, different parity and symmetry properties. They

also exhaust all possibilities one can encounter with regard to the symmetry and coupling

properties of the solutions. Importantly, the stiffness matrices in two given configurations

can differ in their values and even structure and yet the solutions can have the same

symmetries and coupling properties.

To illustrate the analysis, without becoming overburdened by algebra, we present it

for a monoclinic plate with {Z} as principal axis (see [26] for the stiffness matrices of each

crystal class) in the Y configuration. The symmetry plane of the crystal is perpendicular

to both the plane of the plate and the propagation direction. Usually, see [5, 39, 40], the

crystal axes are oriented with the symmetry plane of the monoclinic material contained in

the plane of the plate corresponding to the Z configuration.

The displacement vector field is expressed in the following time-harmonic form:

uj = Uj exp[i(ξy + kz − ωt)] (1)

and the Christoffel equation is easily derived by substitution of the above ansatz and its

terms analysed in order to determine the symmetries, or lack thereof, of the solutions; this

is a standard procedure and details are in [1, 39–41]. For the present case, the elements of

5

the Christoffel equation are given by:

Γ11 = ρω2 − (c55k2 + c66ξ

2)

Γ12 = −(c45k2 + c26ξ

2)

Γ13 = −(c36 + c45)kξ

Γ22 = ρω2 − (c44k2 + c22ξ

2)

Γ23 = −(c23 + c44)kξ

Γ33 = ρω2 − (c33k2 + c44ξ

2)

(2)

This Christoffel equation yields a polynomial equation of sixth degree in ξ, its roots are

labelled as ξn for n = 1, 2...6, and the solution (1) is the superposition of six independent

partial waves:

uj =

(6∑

n=1

U(n)j eiξny

)ei(kz−ωt). (3)

As usual, amplitude ratios for each n: V (n) ≡ U(n)y

U(n)x

and W (n) ≡ U(n)z

U(n)x

, are defined in terms

of the Christoffel equation’s elements; general formulas for them are in [40, 41] and these

ratios are readily computed given the six remaining unknown amplitudes U(n)x .

The first classification criterion is the parity, or lack thereof, of the solution in equa-

tion (3). This is determined by checking if the polynomial in ξn derived from the

determinant of the Christoffel equation has the symmetry: ξ ↔ −ξ. If it does, (3) will

have a definite parity and is decomposed into symmetric and antisymmetric modes after

imposing the appropriate, stress free boundary conditions. If it does not, (3) will not have

any definite parity and it will not be possible to decompose it into two independent families

of solutions. When (3) has a definite parity, the dispersion determinant is factorizable

(reducible) into as many subdeterminants as different parities exist, that is two: even or

odd. Both sets of solutions are then independent of each other, mutually orthogonal, and

the dispersion curves present no crossings amongst modes of the same parity. When (3) has

no parity, the dispersion determinant does not factorize and no crossings occur amongst

the dispersion curves of the only family of modes, see [3, 37] for details.

The second classification criterion is coupling or decoupling of the solution (3) into

in-plane ({x} axis) and sagittal-plane (yz-plane ) motions and this is determined by the

Christoffel equation: if the elements Γ12 and Γ13, responsible for the coupling between

6

the x component and the y and z components are zero, the solution will decouple into

SH and Lamb waves. The solution (3) will be coupled otherwise. When the solution

decouples, each can be studied separately and being independent the SH and Lamb

modes are free to cross regardless of their symmetry. However, within the same fam-

ily of modes, be it SH or Lamb, crossings amongst dispersion curves for modes with

the same parity do not occur. For instance, symmetric Lamb modes will not cross with

each other but can cross with any of the other sets of antisymmetric modes or any SH modes.

For the case we illustrate, the monoclinic material, the Christoffel equation (2) does

not decouple and has the symmetry ξ ↔ −ξ. Therefore, the solution will split into

two independent families of modes: symmetric and antisymmetric. The solutions have

analogous properties to those in the Z configuration studied in [5, 39, 40].

To build the dispersion determinant of the problem, one needs to substitute solution

(3) expressed in terms of the ratios into the six stress-free boundary conditions:

Tyx|y=±h =∑6

n=1 U(n)x Dxne

±iξnh

Tyy|y=±h =∑6

n=1 U(n)x Dyne

±iξnh

Tyz|y=±h =∑6

n=1 U(n)x Dzne

±iξnh

(4)

where, in this case, the coefficients Din are given by:

Dxn ≡ i{c66ξn + c26ξnV

(n) + c36kW(n)}

Dyn ≡ i{c26ξn + c22ξnV

(n) + c23kW(n)}

Dzn ≡ i{c45k + c44kV

(n) + c44ξnW(n)} (5)

giving a system of six equations for six unknown amplitudes U(n)x and to obtain non-trivial

solutions, the determinant of the coefficients must vanish which gives the dispersion relation.

From the symmetry ξ ↔ −ξ of the roots, the ratios V (n) and W (n) as well as the

coefficients Din will possess symmetries, see [40, 41] for details. Taking these symmetries

into account, and performing the following row operations, to obtain sines and cosines:

(n) = 12((n) + (n+ 1))

(n+ 1) = 12i

((n)− (n+ 1))(6)

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where (n), (n) = 1, 3, 5 denote the old and new rows of dispersion determinant D respectively

and the following shorthand for cosines and sines is used sn ≡ sin(ξnh) and cn ≡ cos(ξnh),

the determinant reads:

D(ω, k, ξn(ω, k)) =

Dx1c1 −Dx1c1 Dx3c3 −Dx3c3 Dx5c5 −Dx5c5

Dx1s1 Dx1s1 Dx3s3 Dx3s3 Dx5s5 Dx5s5

Dy1c1 −Dy1c1 Dy3c3 −Dy3c3 Dy5c5 −Dy5c5

Dy1s1 Dy1s1 Dy3s3 Dy3s3 Dy5s5 Dy5s5

Dz1c1 Dz1c1 Dz3c3 Dz3c3 Dz5c5 Dz5c5

Dz1s1 −Dz1s1 Dz3s3 −Dz3s3 Dz5s5 −Dz5s5

(7)

The following column operations must be performed in order to obtain zeroes:

(N) = 12((N) + (N + 1))

(N + 1) = 12((N)− (N + 1))

(8)

where (N), (N) = 1, 3, 5 denote the old and new columns of D respectively. The determinant

now is greatly simplified

0 Dx1c1 0 Dx3c3 0 Dx5c5

Dx1s1 0 Dx3s3 0 Dx5s5 0

0 Dy1c1 0 Dy3c3 0 Dy5c5

Dy1s1 0 Dy3s3 0 Dy5s5 0

Dz1c1 0 Dz3c3 0 Dz5c5 0

0 Dz1s1 0 Dz3s3 0 Dz5s5

(9)

and after rearranging one is left with a factorized determinant which is lengthy and omitted

for brevity.

Each of the subdeterminants obtained by factorization, yield each of the generalized

Lamb dispersion relations (cf. [40, 41] and [39]) for symmetric and antisymmetric modes.

As explained above and in [37], since each subdeterminant is not further reducible, crossings

amongst modes with the same parity do not occur. Although illustrated for a specific

case, this can all be done for completely general anisotropy. It can also be extended to

multiple-layer systems with the added complication that one needs to take into account

interface conditions and the interplay between the properties of the individual solutions of

8

different layers to yield the properties of the global solution.

To see how this is used in practice we now consider three cases which, to the best

of our knowledge, have not been presented elsewhere. Rather than repeat the theory above

for each example, with lengthy algebraic changes, we note that the interested reader can

find all the details in [42].

Our first example is again a monoclinic plate, but now in the X configuration; the

decoupling of the modes in Lamb and SH for this configuration is well-known, see [6, 43].

The analysis of the polynomials in ξ, derived from the decoupled Christoffel equations,

further shows that the symmetry ξ ↔ −ξ is absent and thus neither the Lamb nor the SH

modes split into symmetric and antisymmetric modes. The dispersion determinants for

Lamb and SH modes are therefore non-reducible and only two sets of independent modes

exist with no definite parity. Crossings can occur between Lamb and SH modes, but not

amongst modes of the same family. The dispersion curves computed with the PSCM for

this plate, with typical parameters given in the appendix, are shown in figure 1 where Lamb

modes are plotted in blue solid lines and SH modes are plotted in dot-dashed red lines; we

see that crossings occur between SH and Lamb, but crossings do not occur within each

mode family.

The displacement fields’ mode shapes for the fifth Lamb mode, label A in figure 1

at ∼ 3 MHz-mm, and SH mode, label B in figure 1 at ∼ 6 MHz-mm, are presented

separately in figures 2.A and 2.B respectively. In order to analyse the mode shapes’ features

it must be noted that only the real part of solution (1) is physically meaningful:

Re uj(y, z; t) = ‖Uj(y)‖ e−βz cos(αz − ωt+ φj(y)) (10)

where k = α + iβ. The phase of the complex displacement vector field can in general

depend on the coordinate {y}. In the two-dimensional plots in figures 2.A (i and 2.B (i,

the norm, normalized to one, of the displacement vector components is plotted in black

circles whereas the phase is plotted in a solid red line. In this and the subsequent mode

shape plots, the horizontal axis gives the radians of the phase in terms of π, no axis or scale

has been chosen for the norm which has been normalized so that the point with maximum

9

amplitude has value one, only the origin of the horizontal axes is the same for both norm

and phase. The vertical axis corresponds to the thickness of the plate.

In figures 2.A (ii and 2.B (ii, the norm of the vector and the phase are plotted in a three-

dimensional space where the vertical axis corresponds to the spatial {y} axis running across

the thickness of the plate and the horizontal plane is just the complex plane in polar coor-

dinates. Each point in the graph has coordinates {‖Uj(y)‖cos(φj(y)), ‖Uj(y)‖sin(φj(y)), y}.

Since the phase depends on {y} the curve does not lie within a two-dimensional plane but

twists around in space. Note however, that this plot does not correspond to the actual

physical process and movement of the structure but it is just presentational so that the

properties of the solution become more apparent. The norm is symmetric and the phase

antisymmetric with respect to the middle plane of the plate, as a result, the solution given

by equation (10) has no definite parity which is in agreement with the previous analysis. In

cases where the solution is split into symmetric and antisymmetric parts, the phase shows

a {y} dependence such that the appropriate sign changes occur across the thickness of the

plate to ensure that the modes possess the correct symmetry or antisymmetry.

We can use the analysis and interpretation above more generally, for instance for

guided waves for the most general class of Trigonal crystals with 7 independent constants

[26] which has analogous properties to those of the Triclinic class in any of the above config-

urations: modes do not decouple and they do not have any definite parity. The Christoffel

formula for the Triclinic case can be found in [40], the corresponding formula for the Trigonal

class is obtained from it by setting the appropriate entries equal to zero. Using an analysis

similar to that above their properties can be derived easily: since the solutions have no def-

inite parity the mode shapes feature similar properties to those of the monoclinic case above.

For the Trigonal classes 32, 3m and 3m the equations are simpler since c15 = 0 due

to an extra symmetry, see [26]. For the X and Z configurations, the solutions have the

same properties as those of the previously analysed Monoclinic crystal in Z configuration:

coupled modes which split into symmetric and antisymmetric. Our second example, shown

in figure 3, displays the dispersion curves for a Trigonal 5 mm thick plate with c15 = 0

in Y configuration. Here the problem decouples and neither the SH (dot-dashed red

10

lines) nor the Lamb (blue solid lines) modes have any definite parity so the mode shapes

are qualitatively the same as those just presented for the Monoclinic plate in X configuration.

Finally, Tetragonal crystals, with up to 7 independent elastic constants [26], are in-

vestigated. The Y and Z configurations have solutions similar to those of the monoclinic

crystal in Z configuration studied above. The most interesting and unusual situation is

encountered in the X configuration where the solution decouples into SH and Lamb modes.

In turn, due to the Christoffel equation symmetries, the in-plane SH modes split into

symmetric and antisymmetric sets whereas the Lamb modes do not have any definite parity.

This is the only combination of material and axes orientation where this hybrid situation

takes place. Tetragonal crystals of the classes 4mm, 422, 42m and 4/mmm have c16 = 0

due to an extra symmetry, [26]. As a consequence of this, their solutions in any of the three

configurations possess the same properties as those of Isotropic or Orthorhombic materials

which are well-known and will not be pursued here.

The dispersion curves of a Tetragonal 5 mm thick plate in X configuration are plot-

ted in figure 4 where SH modes are shown in dot-dashed red lines and Lamb modes in

blue solid lines. The mode shapes of the fourth Lamb mode, label A in figure 4 at ∼ 6.5

MHz-mm, are shown in figure 5.A and those for the fourth SH mode, label B in figure 4 at

∼ 6 MHz-mm, in figure 5.B. The norm is plotted in black circles and the phase in solid red

line in both figures 5.A (i and 5.B (i. The three-dimensional plots in figures 5.A (ii and 5.B

(ii have the same meaning as the ones presented previously. Note the difference between

the {y} dependence in the Lamb mode with no definite parity and that of the SH mode

which is antisymmetric and hence only the sign changes across the thickness as predicted.

These examples complete, and exhaust, all possibilities that can be encountered, those not

presented can be found in the literature and references cited above. By restricting attention

to the solutions’ symmetry and coupling properties of each material and configuration we

group them into fewer families. This facilitates the task of studying the solutions’ properties

in the different possible combinations that may arise in the case of multiple layer systems.

This is achieved by focusing on what the solutions for materials of different symmetry

class have in common. For instance, Triclinic and Trigonal materials in any configuration

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have in common that their solutions are coupled and have no definite parity, thus they

will be grouped into a single family since, in terms of parity and coupling, they are the same.

The following 5 families emerge and the materials and configurations they comprise

are given along with the solutions’ parity and coupling properties.

• TC : Triclinic and Trigonal (c15 6= 0) in any axes configuration. Solutions are coupled

with no definite parity.

• MD: Trigonal (c15 = 0) in Y configuration configuration. Monoclinic in X configura-

tion. Solutions are decoupled in Lamb and SH modes with no definite parity.

• MS/AC : Trigonal (c15 = 0) in X,Z configurations. Monoclinic and Tetragonal (c16 6= 0)

in Z,Y configurations. Solutions are coupled with definite parity: symmetric and

antisymmetric modes.

• Tet∗D: Tetragonal (c16 6= 0) in X configuration. Solutions decouple into Lamb and SH

modes. Lamb modes have no definite parity. SH modes are split into symmetric and

antisymmetric modes.

• OS/AD : Orthorhombic, Tetragonal (c16 = 0), Hexagonal, Cubic and Isotropic crystals

in any configuration. Solutions decouple into Lamb and SH modes each of which has

a definite parity thus splitting into symmetric and antisymmetric modes. This gives

a total of four independent sets of solutions.

The letters for each family correspond to the first letter of the crystal class with the highest

number of independent elastic constants contained in that family; these are used to study

multi-layer systems.

Changing the configurations of the crystal axes by, say, an arbitrary rotation, normally

causes a given crystal class to move into a completely different family. For instance, consider

an orthorhombic crystal in Y configuration which belongs to OS/AD . Now, rotate the crystal

about the {Y} axis by an angle different from 90 degrees or any integral multiple of it, say

33 degrees. This gives a stiffness matrix and a plate whose solutions have the properties

of a monoclinic crystal in Y configuration with its plane of mirror symmetry contained in

the plate, see [40] and [39]. Thus, that orthorhombic plate with those given crystal axes

orientations will belong to the family MS/AC .

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III. MULTIPLE-LAYER ANISOTROPIC FLAT WAVEGUIDES

We utilise the PWD approach, as for the single layer, but now with the added complica-

tion of interface conditions between the different layers that must be taken into account in

addition to the usual stress-free boundary conditions. The procedure to reduce the disper-

sion determinant for any given system is analogous, all be it more unwieldy, to that of the

single layer and will not be given here. Without any loss of generality, we consider systems

with 2 or 3 layers as they fall into one of the 5 families according to symmetry and coupling

properties; it is key to realize that the whole 2 or 3 layer aggregate is effectively a single

plate belonging to one family. By adding one or two extra layers one obtains systems with

4 or 5 layers equivalent to a 2 or 3 layer system whose solutions’ properties are known; the

process can then be continued inductively to as many layers as one wishes.

A. Systems with even number of layers

In general, for a system of n even layers, there are two possibilities:

n = 2m m ∈ N

Symmetric =⇒ Always reduces to a 2m− 1 system

Non− symmetric =⇒ Cannot be reduced(11)

For symmetric systems of even layers the two middle layers are the same and they can be

regarded as a single layer of double thickness. For a system to be considered as ”symmetric”,

the thicknesses of the different layers must be the same on both sides of the middle plane

of the system. This means that a system such as At/Bt/Bb/Ab, where t stands for top

and b stands for bottom, where layers At and Ab have different thicknesses, would not be

symmetric even though its crystal sequence A/B/A is.

The simplest case is a system composed of two layers. Symmetric bilayer systems

are trivial, they are equivalent to a single layer system of double thickness treated in section

II. Adding layers to both sides preserving the symmetry of the system leads invariably to

systems with odd numbers of layers studied in section III B. If only one layer is attached,

one obtains a non-symmetric system and these, whether with two or more layers, are easy

to deal with. First, assuming that the problem does not decouple, once the boundary

and interface continuity conditions for stresses and displacements have been imposed and

13

the corresponding coefficients’ determinant set to zero, it is not possible to simplify and

factorize the determinant into a product of smaller non-reducible determinants due to

the lack of symmetry, be it because of the choice of materials and/or thicknesses of the

different plies. This agrees with the intuitive notion that a system with no symmetry will

not have symmetric solutions. If the problem is decoupled into SH and Lamb modes, the

above results hold separately for each decoupled solution and can be studied independently.

It must be remembered that, in these cases where decoupling occurs, crossings will be

allowed amongst SH and Lamb modes regardless of their symmetry properties. A four-layer

non symmetric system with the two middle layers of the same material is reduced to an

equivalent three-layer system, which might or might not be symmetric depending on the

combination of thicknesses of the top and bottom layers. This and similar configurations

will be regarded as odd-n multilayer systems from the very beginning.

Finally, in order to extend this analysis to a higher numbers of layers, the classifica-

tion for single layer systems presented in section II can also be used for multilayer systems

if attention is restricted to the parity, or lack thereof, and coupling properties of the system

solutions. To illustrate this, consider a bilayer system whose layers are of different materials.

This automatically rules out any family whose solutions might present any parity (even if

both individual layers do have), thus one is only left with the two possibilities: TC and

MD. The system will only decouple if both layers present decoupling. If this is the case, the

bilayer system will belong to MD (one will actually have two simpler problems for SH and

Lamb modes each in a bilayer system). If any of the layers fails to present decoupling the

global solution will be coupled and belong to TC .

After this, one can regard the two layers as one single layer belonging to one of the

two families above and add a third layer; depending on the type of layer, position (top or

bottom) and thickness, the new system can be regarded as either another non-symmetric

two-layer system or as a three-layer system to be described next. This analysis can be

repeated inductively for any number of layers and it is very useful when one wants to know

what to expect from a particular multilayer problem.

14

B. Systems with odd number of layers

For a system of an odd number of n layers, one has the following two possibilities:

n = 2m+ 1 m ∈ N

Nonsymmetric (N)

Symmetric (S)(12)

We move straight to considering n layers, as for even-n layers these will have solu-

tions with no symmetry with respect to the middle plane of the plate. Non-symmetric

arrays are achieved not only by choosing a nonsymmetric arrangement of materials but also

by choosing different thicknesses for layers of the same material symmetrically disposed.

As a result, assuming first a coupled system, the boundary and interface continuity

conditions will yield only one non-reducible dispersion determinant for the amplitudes of

the displacement vector field. Since the modes are coupled and the non-symmetric array

yields solutions with no definite parity, the system belongs to the TC family. Should a

non-symmetric system present decoupling, one would obtain a dispersion determinant for

each family of SH and Lamb modes which would also have no definite parity. These systems

belong to the MD family.

For symmetric systems, conclusions cannot be drawn so straightforwardly. For ex-

ample, consider a simple system such as TC/OS/AD /TC which have a typical Orthorhombic,

or even Isotropic, layer sandwiched between two equally thick Triclinic or Trigonal layers.

We anticipate that the global solution will not be decoupled due to the coupling effect

of the Triclinic layers. Regarding parity, on the one hand, one could argue that due to

the presence of the coupling TC layers, their lack of symmetry spreads throughout the

whole system and its solution has no definite parity. On the other hand, one could expect

the extrinsic symmetry of the array (B/A/B) to be somehow reflected in the system’s

solution. In other words, do extrinsic symmetries and properties of a system prevail over

the individual properties of its components?

To provide an answer, due to the infinite number of possibilities, where possible,

counterexamples will be given to rule out certain combinations and narrow the path

15

towards a final solution; this rules out all possible combinations except for one. For

this, it is possible to show the reduction of the dispersion determinant analytically

but for the n-layer case this is extremely unwieldy and a repetition, albeit on a much

larger scale, of the reduction procedure carried out for the single layer; an example for

SH modes in a 5-layer system can be found in [42]. More general derivations for Lamb

and coupled modes follow exactly along the same lines, but the size of the matrix is increased.

Attention is restricted to three layers: B/A/B, where A and B denote the layer

type, for instance TC . All the layers are assumed to have the same thickness since

otherwise the system would be non-symmetric (N). The results are summarized in table

II. The analysis of a given configuration proceeds in two steps. Firstly, one analyses

whether the multiple-layer system decouples or not. This involves checking that the

equations of motion as well as the boundary and interface continuity conditions yield two

decoupled sets of PDEs. If any of these sets of equations fails to be decoupled, be it the

equations of motion or any of the boundary or interface conditions, the problem will be

coupled. In practice this is equivalent to: The solution of a given multilayer system will

be decoupled if and only if the solution in each layer is also decoupled when considered

individually. Note that, this is valid for any multilayer system in flat or cylindrical geometry.

Once it is known whether decoupling occurs or not one can proceed to study the

symmetry of the solutions. If the problem decouples one simply solves two separate and

independent problems; if it does not, one has only one problem, albeit bigger, to investigate.

The goal is to find under which circumstances a system classified as symmetric, that

is, with a symmetric array of layers and thicknesses, has solutions with a definite parity:

symmetric and antisymmetric modes. In order to show this, one can naively assume the

following:

Assumption 1. The global solution of a 3-layer symmetric system, with n odd, has a definite

parity, odd or even, giving rise to antisymmetric or symmetric modes respectively.

The strategy is to provide counterexamples to the above in order to rule out certain

symmetric combinations B/A/B of the 5 families above thus narrowing the path towards

16

the suitable combination for the existence of solutions with definite parity. This can be

generalized to arbitrary odd n by a similar inductive argument to that used for the even n

case.

Counterexample 1. Assumption 1 does not hold for 3-layer symmetric systems whose

middle layer belongs to any of the following families: TC, MD or Tet∗D.

For this counterexample a system such as OS/AD /TC/O

S/AD suffices. O

S/AD is taken to be an

orthorhombic crystal and TC a triclinic crystal, both in Y configuration: {Y} orthogonal to

the plane of the plate and propagation along the {Z} axis. Further details of their physical

and elastic properties are given in the appendix. Dispersion curves for this system are shown

in figure 6 and the displacement field’s profiles for the fifth mode at ∼ 1 MHz-mm in figures

7.a and 7.b in two- and three-dimensional spaces respectively as in the previous section.

From the shape of the norm, phase and equation (10) the solution has no definite parity

with respect to the middle plane of the plate system as expected.

Counterexample 2. Assumption 1 does not hold for 3-layer symmetric systems whose

outer layers belong to any of the following families: TC, MD or Tet∗D.

To illustrate this counterexample the dispersion curves and mode shapes of a system

TC/MS/AC /TC are presented. The physical properties and further details are given in the

appendix. Dispersion curves for this system are shown in figure 8 and the displacement

field’s profiles are similar to the previous counterexample where solutions have no definite

parity.

Note that the family Tet∗D is quite unusual, it only contains one crystal class in

X configuration and displays symmetric and antisymmetric SH modes whereas the

Lamb modes have no definite parity. Thus, in systems such as OS/AD /Tet∗D/O

S/AD or

Tet∗D/OS/AD /Tet∗D, SH modes will split into symmetric and antisymmetric sets but Lamb

modes will not. Thus, only part of the complete solution, the SH modes, has definite parity.

These cases and similar ones are subsumed in the two counterexamples above.

The two counterexamples leave only one possibility left to examine:

Conclusion 1. Assumption 1 holds only for systems whose layers, no matter in which

position, belong to the following families: MS/AC or O

S/AD .

17

In other words, all layers must have solutions with definite parity to ensure that the

system’s dispersion determinant can be reduced and all the modes split into symmetric and

antisymmetric.

The above results for the three-layer system B/A/B are summarized in table II be-

low:

@@@@@

B

A (I) (II) (III) (IV)

TXC MS/AC MX

D ,Tet∗D OS/AD

TXC

Coupled Coupled Coupled Coupled

X X X X

MS/AC

Coupled Coupled Coupled Coupled

X S/A X S/A

MXD ,Tet∗D

Coupled Coupled Decoupled Decoupled

X X X X or *

OS/AD

Coupled Coupled Decoupled Decoupled

X S/A X or * S/A

TABLE II: A summary of the properties of the global solution for a flat three layer system B/A/B

depending on the properties of each individual component. The symmetry with respect to the

diagonal cells reflects the change in the crystal types of the array B/A/B ↔ A/B/A. S/A stands

for solution with definite parity, X stands for solutions with no definite parity and the asterisk for

solutions with properties analogous to Tet∗D.

Columns labelled with (I) and (III) correspond to systems in counterexample 1. For

columns labelled with (II) and (IV) one has: cells which are not highlighted corresponding

to systems in counterexample 2, and cells which are highlighted corresponding to those

systems for which the assumption 1 holds. Cells in which the ”X or *” appears correspond

to the cases described in the paragraph right before conclusion 1 about systems containing

Tet∗D materials.

The statement of conclusion 1 can be proved for a system of n-layers. However,

18

even for the simplest case of SH modes the notation is extremely unwieldy and the

procedure is analogous to that presented for single layer systems. The interested reader

can find a detailed proof for SH modes in a 5-layer system in [42]. The proof for SH modes

in the n-layer case is exactly the same but with a higher but finite number of cycles un-

til the system’s dispersion determinant is completely factorized, see also [42] for a discussion.

Conclusion 1 is illustrated with an example of a system with the following sequence

of materials: OS/AD /M

S/AC /O

S/AD , the physical and geometrical properties are given in

the appendix. The dispersion curves are shown in figure 9. Due to the presence of

the monoclinic layer the solution is coupled but will be split into symmetric (solid

blue lines) and antisymmetric modes (dot-dashed red lines). The mode shapes for the

fourth symmetric mode, label A in figure 9 at ∼ 1.5 MHz-mm are shown in figures 10

(i and 10 (ii. The mode shapes of the antisymmetric mode are not presented here for

brevity though they display the expected parity. It is clear that the {y} dependence of the

phase is such that both families of modes have definite parity: symmetric and antisymmetric.

Finally, a comment concerning counterexamples 1 and 2 when systems with more

than three layers are considered. Systems in counterexample 1 can be extended by adding

layers to the three-layered system and if the inner three-layer core does not support

symmetric and antisymmetric modes, neither will any of the extended systems no matter

what combination of plates is chosen. To see this it is enough to realize that the inner

core of three belongs to TC or MD since the system’s solution has no definite parity.

Adding one layer to each side, thus making a five-layered aggregate, takes the system to a

configuration equivalent to the three-layered systems considered in counterexample 1 which

have a middle layer belonging to either TC or MD. Therefore, the solution for the five-layer

system will have no symmetry. One can continue this process to increase the number of

layers arbitrarily obtaining analogous conclusions as for the three and five layers systems

just explained. For systems in counterexample 2 the reasoning follows the same lines and

the conclusion is that when layers of triclinic-like material are inserted in a system, in a

symmetric fashion but not in the middle position, the final system’s solution will have no

definite parity.

19

IV. SINGLE-LAYER AND MULTIPLE-LAYER ANISOTROPIC CYLINDRICAL

WAVEGUIDES

We now consider single and multiple layer systems in cylindrical geometry with axial

propagation; circumferential propagation follows the same lines, and is simpler since only

one family of modes need be considered; this is omitted for brevity. As for flat guides, the

goal is to find how many independent dispersion relations exist for a given configuration

knowing that each of them will give rise to a set of modes whose dispersion curves do not

intersect and thus making it possible to avoid or understand mode crossings.

In cylindrical geometry, the existence of more than one independent dispersion rela-

tion is solely determined by the coupling or decoupling of the given problem for a fixed

harmonic order n. If the problem decouples in torsional and longitudinal modes, two

independent dispersion relations exist and therefore crossings amongst these two families

can occur. Moreover, families corresponding to different harmonic orders n1 6= n2 can

cross amongst themselves since a different dispersion relation is obtained for each harmonic

order ni which acts as a parameter. The single layer results for all crystal classes and

axis configurations are summarized in table III and the axes configurations in cylindrical

coordinates are defined in table IV:

For multilayer cylindrical systems very little changes. The interface continuity condi-

tions for stresses and displacement vector fields of the layers can lead to the coupling

of otherwise decoupled modes. Assume the system is composed of two layers and fixed

harmonic order n. For a family of modes with harmonic order n, the modes are decoupled

in torsional and longitudinal for one layer but they are not for the other. As a result of

the interface continuity conditions, the layer with coupling will also extend this coupling to

the other layer. Hence, the system’s solution of this bilayer example is coupled and only

one independent family of modes exists. This argument is valid for an arbitrary number of

layers with a fixed harmonic order n: if at least one layer of the system presents coupling

then the system’s solution will be coupled. In other words, for a fixed harmonic order n, a

system of n cylindrical layers will support decoupled Torsional and Longitudinal modes if

and only if each of its layers does, examples of various cylindrical multiple-layer systems

20

XXXXXXXXXXXXXXXXXXCrystal Class

ConfigurationR Θ Z

Isotropic n=0 n=0 n=0

Cubic n=0 n=0 n=0

Hexagonal n=0 n=0 n=0

Tetragonal c16 = 0 n=0 n=0 n=0

Tetragonal c16 6= 0 X n=0 X

Trigonal c15 = 0 X X n=0

Trigonal c15 6= 0 X X X

Othorhombic n=0 n=0 n=0

Monoclinic X n=0 X

Triclinic X X X

TABLE III: The harmonic order n for which each crystal class and axes configuration decouples

for cylindrical geometry. ”X” means that the problem is coupled for all values of n. The different

configurations are shown in table IV after this one. The stiffness matrices in cylindrical coordinates

in R, Θ and Z configurations have the same formal structure as the corresponding stiffness matrices

in Cartesian coordinates in Y, Z and X configurations respectively.

Crystal Configuration Label Axis Normal to the Cylinder Propagation Axis

R configuration {R} {Z}

Θ configuration {Θ} {R}

Z configuration {Z} {Θ}

TABLE IV: The three different crystal axes configurations in cylindrical geometry. In the default

R configuration crystal and fixed spatial (plate) axes {r, θ, z} are aligned.

can be found in the literature [27, 29, 35, 35, 35] and [42].

V. CONCLUDING REMARKS

We have obtained a new classification of single layer anisotropic waveguides according

to their solution’s properties rather than on the crystal’s symmetries and demonstrated

21

the efficiency of doing so. All crystal classes and independent axes configurations can be

grouped into only 5 different sets, each of which contains only those crystal classes axes

configurations whose solutions share some common features. This makes the analysis of

multiple generally anisotropic layer systems tractable. This insight is valuable and enables

us to rapidly delineate a problem into one of these families; this has future practical

advantages contributing to the more efficient design of numerical algorithms for dispersion

curve computation and tracing. In particular, by appropriately exploiting the symmetry and

coupling properties of the solutions when building a code, the mode jumping phenomenon

which sometimes occurs using root tracing routines can be drastically reduced or totally

eliminated. In addition, this will optimize the codes and save time particularly if one is only

interested in one particular family of modes which can be computed separately without

the need of computing any of the others. This is especially advantageous in multiple-layer

systems where the computational time can be large if many anisotropic layers are present.

Acknowledgement

F.H.Q. would like to thank Professor Dale E. Chimenti for a stimulating discussion.

APPENDIX A: NUMERICAL DATA

The physical and geometrical information used for the figures are given here. In terms

of the PSCM the number of grid points N varies from one example to another, but it is

always at least double the number of modes plotted in the figure in order to ensure high

accuracy. On a practical level N is chosen to achieve the shortest computation time, that

is, if one is interested in the first 10 modes, running a code with N = 100 is unnecessary; a

value of N between 25 and 30 has consistently been shown to be sufficient.

The parameters for the Monoclinic plate of figures 1 and 2 are as follows:

ρ = 8938.4 kg/m3 ; h = 5 mm (A1)

22

h stands for the thickness of the plate. The elastic stiffness matrix is given in GPa:

c =

201.1 89.3 122.6 −20.3

208.3 115.4 14.1

174.9 6.2

63.1 16.7

82.4

37.8

(A2)

Note that the above matrix has been rotated appropriately in order to study the X

configuration presented in figures 1 and 2.

The parameters for the Trigonal (tellurium) plate in Y configuration of figure 3 are

as follows:

ρ = 6250 kg/m3 ; h = 5 mm (A3)

The elastic stiffness matrix is given in GPa:

c =

32.7 8.6 24.9 12.4

32.7 24.9 −12.4

72.2

31.2

31.2 12.4c(11)−c12

2

(A4)

The parameters for the Tetragonal plate in X configuration of figures 4 and 5 are:

ρ = 3420 kg/m3 ; h = 5 mm (A5)

23

h stands for the thickness of the plate. The elastic stiffness matrix is given in GPa:

c =

169.10 122.2 122.2

208.08 83.22 22.50

208.08 −22.504

36.44

75.42

75.42

(A6)

The parameters for the Orthorhombic plates used in the multi-layer systems of fig-

ures 6, 7, 9 and 10 :

ρ = 3820 kg/m3 ; h = 5 mm (A7)

h stands for the thickness of the plate. The elastic stiffness matrix is given in GPa:

c =

132 6.9 5.9

12.3 5.5

12.1

3.32

6.21

6.15

(A8)

The parameters for the Triclinic plate of figures 6, 7 and 8 are:

ρ = 8938.4 kg/m3 ; h = 5 mm (A9)

h stands for the thickness of the plate. The elastic stiffness matrix is given in GPa:

c =

235.54 88.64 88.78 19.05 −2.97 −6.54

215.85 108.43 −14.14 2.35 22.16

215.77 −4.90 0.61 −15.57

58.24 −17.68 −7.02

39.69 11.66

44.02

(A10)

24

The parameters for the Monoclinic plate of figure 8 are those given for the Orthorhombic

plate in A8 rotated by π/3.3 radians about the {Y} axes of the crystal.

The parameters for the Monoclinic plate in figures 9 and 10 are:

ρ = 8938.4 kg/m3 ; h = 5 mm (A11)

h stands for the thickness of the plate. The elastic stiffness matrix is given in GPa:

c =

201.1 89.3 122.6 −20.3

208.3 115.4 14.1

174.9 6.2

63.1 16.7

82.4

37.8

(A12)

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28

FIG. 1: Lamb (solid blue line) and SH (red dash-dotted line) modes in a Monoclinic 5mm. thick

free plate in X configuration with propagation along the {Y} crystal axis. An enlargement in the

inset shows that the 8th and 9th Lamb modes do not actually cross even though on the scale of the

main figure they appear to do so.

29

FIG. 2: Two- and three-dimensional, i) and ii) respectively, mode shapes of the fifth Lamb at ∼ 3

MHz-mm for the Monoclinic 5mm. thick free plate labelled (A) in figure 1 and of the fifth SH

mode at ∼ 6 MHz-mm for the Monoclinic 5mm. thick free plate labelled (B) in figure 1. In column

i), the norm of the complex vector field normalized to one is shown in black circles and the phase

is shown in solid red lines. In column ii), the coordinate of the displacement complex vector in

complex polar coordinates is shown at each point of the thickness of the plate.

30

FIG. 3: Lamb (solid blue line) and SH (red dash-dotted line) modes in a Trigonal 5mm. thick free

plate in Y configuration with propagation along the {Z} crystal axis.

31

FIG. 4: Lamb (solid blue line) and SH (red dash-dotted line) modes in a Tetragonal 5mm. thick

free plate in X configuration with propagation along the {Y} crystal axis.

32

FIG. 5: Two- and three-dimensional, i) and ii) respectively, mode shape of the fourth Lamb mode

at ∼ 6.5 MHz-mm for the Tetragonal 5mm. thick free plate labelled (A) in figure 4 and of the

fourth SH mode at ∼ 6 MHz-mm for the Tetragonal 5mm. thick free plate labelled (B) in figure 4.

In column i), the norm of the complex vector field normalized to one is shown in black circles and

the phase is shown in solid red lines. In column ii), the coordinate of the displacement complex

vector in complex polar coordinates is shown at each point of the thickness of the plate. Note that

in this particular case, the SH modes are symmetric and antisymmetric even though the Lamb

modes are not.33

FIG. 6: Dispersion curves of coupled modes for a symmetric three-layered system composed of flat

Orthorhombic-Triclinic-Orthorhombic layers. No crossings are seen to occur amongst the modes

when zooming in. All plates in Y configuration with propagation along the {Z} crystal axis.

34

FIG. 7: Two- and three-dimensional, i) and ii) respectively, mode shapes of the fifth mode at

∼ 1 MHz-mm, labelled (A) in figure 6, for the symmetric three-layered system composed of

Orthorhombic-Triclinic-Orthorhombic layers. In column i), the norm of the complex vector field

normalized to one is shown in black circles whereas the phase is shown in solid red lines. In column

ii), the coordinate of the displacement complex vector in complex polar coordinates is shown at

each point of the thickness of the plate.

35

FIG. 8: Dispersion curves of coupled modes for a symmetric three-layered system composed of

flat Triclinic-Monoclinic-Triclinic layers. No crossings are seen to occur amongst the modes when

zooming in. All plates in Y configuration with propagation along the {Z} crystal axis.

36

FIG. 9: Dispersion curves of symmetric (blue solid lines) and antisymmetric (dash-dotted red

lines) modes for the symmetric three-layered system composed of flat Orthorhombic-Monoclinic-

Orthorhombic layers. All plates in Y configuration with propagation along the {Z} crystal axis.

37

FIG. 10: Two- and three-dimensional mode shapes of the fourth symmetric mode at ∼ 1.5 MHz-

mm, labelled (A) in figure 9, for the symmetric three-layered system composed of Orthorhombic-

Monoclinic-Orthorhombic layers. In column i), the norm of the complex vector field normalized

to one is shown in black circles whereas the phase is shown in solid red lines. In column ii), the

coordinate of the displacement complex vector in complex polar coordinates is shown throughout

the thickness of the plate. The phase of the modes is constant up to a sign throughout the thickness

as expected for this family of modes.

38