Wave Motion 50 (2013) 233–245

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Wave Motion

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On the dispersion of wedge acoustic wavesElena S. Sokolova a,b, Alexander S. Kovalev b, Reinhold Timler a, Andreas P. Mayer a,∗a HS Offenburg, 77723 Gengenbach, Germanyb B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkov 61103, Ukraine

a r t i c l e i n f o

Article history:Received 17 December 2011Received in revised form 22 August 2012Accepted 25 August 2012Available online 29 October 2012

Keywords:Acoustic wavesGuided wavesWedge wavesDispersion

a b s t r a c t

Acoustic waves guided at the apex of an ideal infinite elastic wedge are non-dispersive.Weak dispersion arises due to a variety of factors. Three of them are investigated in detail:(i) Coating of one or both of the two surfaces of the infinitewedge, (ii) truncating thewedgeat its apex or replacing the tip of the wedge by a different material, (iii) slight modificationof the material constants of the wedge material in an extended spatial region of the wedgenear its tip. These three cases have been analysed within the perturbation theory, andthe third case, in addition, with the help of semi-analytic finite element calculations. Thedependence of the frequency on wavelength is derived for all three cases, and quantitativeresults are presented for the dispersion laws of example systems.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Acoustic waves guided by the apex of an ideal homogeneous elastic wedge have been discovered in the early nineteenseventies [1,2] and have recently found renewed interest in connection with potential applications in non-destructivetesting [3,4], ultrasonic motors [5,6], aquatic propulsion [7–9] and acoustic streaming [10]. They are also expected to giverise to interesting nonlinear effects [11–14]. Within the elastic continuum theory, an ideal homogeneous wedge does notdefine any length scale. Consequently, acoustic waves propagating in such systems and having wavelengths much largerthan the interatomic spacing are non-dispersive. This fact causes nonlinear effects to be cumulative over the propagationdistance.

Obviously, there are many ways to modify the ideal wedge geometry that give rise to dispersion of wedge acousticwaves. Such modifications may be undesired and due to damage or degradation, for example at the edges of turbine blades.Consequently, the dispersion of wedge acoustic waves can be made use of in non-destructive testing. On the other hand,weak dispersion can be introduced on purpose and the dispersion law tailored such that certain nonlinear effects like theformation of solitary pulses are favoured. Andersen, Datta and Gunshor discussed the possibility of modulational instabilityand the formation of envelope solitons in a regime of strong dispersionwhich impedes the growth of higher harmonics [15].In both cases, a detailed understanding of the conditions leading to the dispersion of wedge acoustic waves is needed.

A considerable amount of experimental work was devoted to the investigation of the dispersion of wedge acousticwaves [16–31], including dispersion due to coating of a surface [28–30] or due to truncation of the wedge [18,20–23,25,26,31]. Theoretical studies were carried out mainly on the basis of thin plate theory [32] and the geometrical acousticsapproximation [33–37] or by using the finite element method (FEM) [3,6,22,27,29,38,39]. Our investigations, which arebased on perturbation theory and extend the early work of Lagasse, Cabus and Verplanken [40], supplement the results ofray theory, since the latter are strictly valid only in the regime of the geometrical lengths of the system much larger thanthe wavelength, while here, we are mostly dealing with the opposite limit.

∗ Corresponding author. Tel.: +49 7803 9698 4478; fax: +49 7803 9698 454478.E-mail address: andreas.mayer@hs-offenburg.de (A.P. Mayer).

0165-2125/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.wavemoti.2012.08.015

234 E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245

In this paper, the focus is on small dispersion, and among the many possible ways of modifying the wedge to generatedispersion, three different possibilities are analysed: (i) Coating of one or both of the two surfaces of the infinite wedge,(ii) truncating the wedge at its apex or replacing the tip of the wedge by a different material, (iii) slight modification of thematerial constants of the tip material in an extended spatial region of the wedge. Our goal is to establish the dispersion law,i.e. the dependence of frequency or phase velocity on inverse wavelength, in these three cases, and to produce quantitativeresults for the wavelength-dependent velocity shift of wedge acoustic waves with respect to the wedge wave velocity inideal wedges. For this purpose, we use perturbation theory within the framework of an expansion of the displacement fieldin Laguerre functions [1,41–44]. In the third case, we also compare our results with FEM calculations. For investigationsof small dispersion, the Laguerre function method has the advantage of not introducing any artificial length scale like theelement size or the system size in FEM, that need to be controlled as they give rise to spurious dispersion of acoustic waves.

The paper is organised in the following way: in the next section, the theory of acoustic waves guided by perfecthomogeneous elastic wedges, based on the Laguerre function method, is briefly reviewed, and notation is introduced that ismade use of in the subsequent sections, where the perturbation analysis of the three dispersion cases is outlined, and resultsof analytic and numerical calculations are presented. A short discussion of the results concludes the paper.

2. Theory of wedge acoustic waves in anisotropic media

Calculations of the velocities of acoustic waves guided by wedges consisting of anisotropic elastic media were carriedout with the help of the FEM (see e.g. [3]), the geometrical acoustics approximation [37] and on the basis of the Laguerrefunctionmethod [43,44]. Here, we briefly describe the latter, starting with a coordinate systemwhere the x-axis is along theapex of the wedge, the y-axis is parallel to one of the two surfaces, pointing from the apex in the direction of the medium,and the z-axis is normal to that surface, pointing in the inward direction of the medium (Fig. 1(a)). The fourth-rank tensor(Cαβµν) of elastic moduli refers to this coordinate system. Here and in the following, Cartesian indices running from 1 to 3are denoted by lower-case Greek letters, and summation over repeated Cartesian indices is implied. Cartesian indices thatonly run over 1 and 2 are denoted by upper-case Greek letters.

The equation of motion for the Cartesian components of the displacement field, uα, α = 1, 2, 3, reads

ρuα =∂

∂xβ

Tαβ (2.1)

with density of the material ρ and stress tensor

Tαβ = Cαβµν

∂uµ

∂xν

. (2.2)

At the two surfaces, the boundary conditions

NβTαβ = 0 (2.3)

have to be satisfied, where Nα, α = 1, 2, 3, are the three components of a vector normal to the corresponding surface. Inaddition, the displacement field has to decay to zero at large distances from the wedge tip.

Because of translational invariance along the x-direction (i.e. the material properties and the geometry of the wedge areindependent of x), the displacement field may be set up in the form

uα(x, y, z, t) = exp(i(kx − ωt)) wα(y, z|k) (2.4)

involving the one-dimensional wavevector k and the frequency ω. The modal functions wα, α = 1, 2, 3, are approximatedby an expansion in a set of functions

fJ(y, z), J = 1, 2, . . .

with expansion coefficients aα

J . The latter depend on the elasticmoduli and the density of the wedge material as well as on the wedge angle,

wα(y, z) =

J

aαJ fJ(y, z). (2.5)

Inserting (2.4) with (2.5) in the equation of motion (2.1), projecting on function fI(y, z) (i.e. multiplying by fI(y, z) andintegrating over the wedge’s cross section) and making use of the boundary conditions (2.3), one is led to the generalizedeigenvalue problem

ω2K

NIKaαK =

J

Mαµ

IJ aµ

J , (2.6)

where (NJK ) is the mass matrix:

NJK =

Adydz f ∗

J (y, z)fK (y, z)ρ (2.7)

E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245 235

Fig. 1. Original wedge geometry (a), geometry after rotation of coordinate system (b), geometry after conformal mapping (c).

and (Mαµ

JI ) is the stiffness matrix (the complex conjugate of a quantity is denoted by a star).

Mαµ

JI =

Adydz[Dβ(k)fJ(y, z)]∗Cαβµν[Dν(k)fI(y, z)], (2.8)

and we have defined the operator

D(k) =

ik∂/∂y∂/∂z

. (2.9)

The integrals in (2.7) and (2.8) have to be extended over the infinite cross section A of the wedge.

Let↔

R = (Rαβ) be the rotation matrix corresponding to transformation of coordinates x, y, z to coordinates x, y′, z ′

(Fig. 1(b)), i.e.xy′

z ′

=

↔

R

xyz

, (2.10a)

↔

R =

1 0 00 cos(π/4 − θ/2) − sin(π/4 − θ/2)0 sin(π/4 − θ/2) cos(π/4 − θ/2)

, (2.10b)

where θ is the wedge angle.

Let↔

S = (Sαβ) be the symmetrical matrix corresponding to the conformal mapping introduced in [41] that transformsthe wedge with angle θ into a rectangular wedge in a coordinate system with axes x, η, ζ (Fig. 1(c)), i.e.x

ηζ

=

↔

S

xy′

z ′

, (2.11a)

236 E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245

↔

S =

1 0 00 1 − tan(π/4 − θ/2)0 − tan(π/4 − θ/2) 1

. (2.11b)

Defining the quantities

Cαβµν = Sββ ′Sνν′Rβ ′βRν′νCαβµν, (2.12)

the operator

D(k) =

ik∂/∂η∂/∂ζ

, (2.13)

and the functions

fJ(η, ζ ) = fJ(y, z), (2.14)

the mass matrix takes the form (det(↔

S ) is the Jacobian determinant of the coordinate transformation)

NJK =

Adηdζ f ∗

J (η, ζ )fK (η, ζ )ρ1

det(↔

S ), (2.15)

and for the stiffness matrix, we obtain

Mαµ

JI =

Adηdζ [Dβ(k)fJ(η, ζ )]∗Cαβµν[Dν(k)fI(η, ζ )]

1

det(↔

S ). (2.16)

In (2.15) and (2.16), the integrals have to be extended over the cross section A of the rectangular wedge (Fig. 1(c)).The functions fJ(η, ζ ) are chosen as products of Laguerre functions,

fJ(η, ζ ) = Φm(kη)Φn(kζ ). (2.17)

In this case, J is a composite index representing the indicesm, n of two Laguerre functions. They are defined as

Φm(ξ) = exp(−ξ/2) [Lm(ξ)/m!] , m = 0, 1, 2, . . . (2.18)

where Lm(ξ) is them-th Laguerre polynomial with normalization

Φm(0) = 1, m = 0, 1, 2, . . . . (2.19)

This together with the orthogonality relation for the Laguerre functions implies∞

0Φm(ξ)Φn(ξ) dξ = δmn, (2.20)

(δnm is the Kronecker symbol) and the mass matrix reduces to

NJK =1k2

ρ δJK1

det(↔

S ). (2.21)

3. Perturbations generating weak dispersion

The three effects considered here for the generation of weak dispersion are treated in the framework of perturbationtheory. The square of the frequency of a wedge acoustic wave with one-dimensional wavevector k may be represented asan expansion in powers of a dimensionless small parameter. In the first case (coating a surface with a thin film) this is theproduct of k and the film thickness d (Fig. 2(a)), in the second case, it is the product of k and a length d characterizing thesize of the part of the wedge’s cross section that has been removed or modified (Fig. 2(b)). In the third case, it is the relativedifference of the densities of the two media or of their corresponding elastic moduli. Confining our analysis to first-orderperturbation theory, we obtain the dispersion law of wedge waves for the three cases in the form

ω2(k) = vW2k2(1 + f (kd)), (3.1)

where vW is the phase velocity of wedge waves propagating in the unperturbed wedge.In the first two cases, the dimensionless function f depends on its argument via a power law, while in the third case, its

behaviour is more complex.

E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245 237

Fig. 2. Coating of one of the surfaces (a), modification of part of the wedge (b).

In the following, the displacement field of a wedge wave with wavevector k for the unperturbed case will be denoted byu(0)

α and the corresponding modal functions by w(0)α . Its frequency is ω0. For the modified geometry, the displacement field

of a dispersive wedge wave having the same wavevector k is denoted by uα , the modal functions by wα , and the frequencyby ω. For the unperturbed modal functions, the representation

w(0)α (y, z|k) =

J

aαmn Φm(kη(y, z)) Φn(kζ (y, z)) = Wα(kη, kζ ) (3.2)

follows from Section 2.

3.1. Surface coating

In this subsection, we consider the effect of coating of one of the two semi-infinite surfaces of the wedge by a thin filmof thickness d, consisting of a material different from that of the bulk of the wedge (Fig. 2(a)). Its elastic moduli are denotedby C ′

αβµν and its density by ρ ′. At first order in the product of k and the film thicknesses, the effects of the thin films on topof the two surfaces are additive. Therefore, we consider here only coating of one surface, and without loss of generality, wechoose the one which is perpendicular to the z-axis. The coated surface is the semi-infinite plane defined by z = 0, y > 0.

It has been shown in [45] that the film can be correctly accounted for by an effective boundary condition at z = 0, whichreads

Tα3 = d(y)ρ ′uα −

∂

∂xΦ

C ′αΦµΘ

∂uµ

∂xΘ

, (3.3)

where

C ′αΦµΘ = C ′

αΦµΘ − C ′

αΦγ 3Γγ νC ′

ν3µΘ (3.4)

and (Γαβ) is thematrix inverse of the 3×3matrix (C ′

α3µ3). The function d(y) is equal to the film thickness d for y > ε, kε ≪ 1,and approaches zero smoothly in the interval [0, ε]. The tip of the wedge is assumed to be rounded with a radius ofcurvature that will become zero together with ε after the integral in (3.5) will have been carried out. In this way, theboundary conditions can be uniquely imposed and Green’s theorem may be applied. Multiplying the equation of motionby u(0)∗

α , integrating over the volume V of the wedge, doing an integration by parts and using the equation of motion for theunperturbed wedge, one is led to

VdVρ(ω2

0 − ω2)u(0)∗α uα =

∂V

dS u(0)∗α TαβNβ . (3.5)

The integral on the right-hand side has to be carried out over the two surfaces of the wedge. However, only the coatedsurface yields a non-zero contribution. Inserting in (3.5) the boundary condition (3.3) and keeping only terms of first orderin kd, we obtain

(ω2− ω2

0)ρ

Aw(0)∗

α (y, z) w(0)α (y, z)dydz = d ×

∞

0

ρ ′ ω2

0w(0)∗α (y, 0) w(0)

α (y, 0)

−DΦ(k)w(0)

α (y, 0)∗

C ′αΦµΘ

DΘ(k)w(0)

µ (y, 0)

dy, (3.6)

where A is the cross section of the unperturbed wedge. For the modal functions of the wedge wave in the unperturbedwedge, we use the representation (3.2), transform to variables η, ζ and obtain from (3.6) the dispersion law in the form

ω2= (vWk)2(1 − 2kd × F1). (3.7)

238 E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245

Fig. 3. Dispersion coefficient F1 (dimensionless) as function of wedge angle for a silicon wedge with one surface coated with a thin SiO2 film. Propagationdirection:

112

, coated surface normal: ⟨111⟩.

The dimensionless constant F1 depends on the ratio of the two densities, ρ ′/ρ, and on ratios of elastic moduli of the film andwedge material, i.e. on the acoustic mismatch between these two materials. In Fig. 3, the coefficient F1 is shown as functionof the wedge angle. The wedge is made of silicon with one of its two surfaces coated with an isotropic silicon oxide film. Thepropagation direction of the wedge wave is the (−1−1 2) direction, and the coated surface is a (111) surface.

The data in Fig. 3 as well as all other numerical results displayed in this paper refer to the wedge mode with lowestvelocity. All numerical calculations based on the Laguerre function method were done using the twenty lowest Laguerrefunctions, only. This number was found to be by far sufficient to obtain convergent results for the phase velocity and modalfunctions in the range of wedge angles considered here.

Interesting features in Fig. 3 are the change of sign of the dispersion coefficient, which implies that at wedge anglesaround 50°, the presence of the film has only a small influence on the dispersion of wedge waves in this system, and theappearance of a maximum around 70°. Consequently, it depends on the wedge angle whether a film leads to an increase ordecrease of the phase velocity of wedge waves with increasing frequency.

Clearly, for constant film thickness, the effect of the film on the wedge wave velocity rapidly increases as the wedgebecomes very sharp.

Eq. (3.7) implies a linear dependence of the phase velocity of wedge acoustic waves on wavenumber k and likewise onfrequencyω. The experimental work and finite-element simulations by Tang and Yang [29] on the frequency dependence ofthe phase velocity of wedge waves in coated wedges refer to the regime 0 < kd ≤ 0.5, approximately. Their results indicatethat even in this regime of small values of kd, the plots of the phase velocity as function of frequency show a considerablecurvature in some cases, and it is difficult to extract numerical values for the slope at zero frequency.

3.2. Modification of the wedge tip

We now consider the modification of a small part of the cross section of the wedge close to its tip, assuming that theextension of this part is much smaller than the wavelength of wedge waves. In Fig. 2(b), the modified part has a triangularcross section. This need not be the case. In fact, the interface between the modified and unmodified part of the wedge mayhave any smooth single-valued shape. The modification may consist in removal of material (truncation or rounding of thetip) or in a replacement by material with density or elastic properties different from that of the bulk of the wedge. Bothcases, removal and replacement of material, may be treated in parallel since the removal of material may be considered as areplacement with material having vanishing elastic moduli and density. The unmodified part of the volume occupied by thewedge will be called V0, while the modified part will be denoted by V1 (Fig. 2(b)). The elastic moduli and the density in V0are denoted by Cαβµν and its density by ρ, while the modified part has elastic moduli C ′

αβµν = Cαβµν + ∆Cαβµν and densityρ ′

= ρ + ∆ρ.Similar to our treatment of the coating of the wedge, we shall project the equation of motion for the displacement field

in the unmodified part of the wedge on the displacement field u(0)α of the perfect wedge and integrate over V = V0 ∪ V1.

Integrating twice by parts and making use of the fact that the surface of the wedge is traction-free, we obtainVdVρ(ω2

0 − ω2)u(0)∗α uα =

V1

dV∆ρω2u(0)∗

α uα −∂u(0)∗

α

∂xβ

∂uµ

∂xν

∆Cαβµν

. (3.8)

The following derivation refers to a triangular shape of the cross section of the modified part of the wedge. It is based on theassumption that the displacement field and its gradients vary little in the volume V1. This assumption is corroborated by a

E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245 239

Fig. 4. Modal functions associated with acoustic modes of a rod having triangular cross section with one surface being fixed to a rigid wall. Coordinatesystem as given in the inserts. Shown are the real quantities w3 (upper), iw1 (middle) and w2 (lower) as defined in (2.4). Integer values at the abscissacorrespond to FEM nodes. Material of the rod: quartz (dashed) and quartz with upper part being replaced by an artificial material with large acousticmismatch (solid). The thin vertical line (horizontal line in the inserts) indicates the boundary between the twomaterials. Displacements of material pointson a surface (a) (dashed arrow in the insert) and on the symmetry plane (b) (dashed arrow in the insert).

finite-element calculation for a finite rectangularwedge fixed to a rigid substrate (Fig. 4). The height h of the replaced triangleof the wedge’s cross section was chosen such that kh = 0.125. The wedge material is fused quartz, and the tip of the wedgehas been replaced by an artificial material that has the same density as aluminium, but elastic moduli smaller than those ofaluminium by a factor of 10. This reduction factor was introduced to increase the acoustic mismatch between the materialsat the tip and in the lower part of the wedge. The (real) displacement components iw1(z, z), w2(z, z), w3(z, z) (Fig. 4(a))and iw1(0, z), w2(0, z), w3(0, z) (Fig. 4(b)) associated with an acoustic mode in this system are shown as functions of z andcomparedwith the displacement components of the correspondingmode in awedge consisting of quartz only. (w1(0, z) andw3(0, z) vanish for symmetry reasons). The relative difference in frequency of the twomodes is (ω−ω0)/ω0 = −0.015. Thecoordinate system is chosen as indicated in the figure. Obviously, the displacement field components change slope abruptlyat the interface, but their variation in the tip region is not much larger than in the other part of the wedge. The differencebetween the displacement components with and without tip modification are large in the tip region V1, but very small in V0up to the interface.

Therefore, the right-hand side of (3.8) may be evaluated approximately in the following way: on the assumption that thedisplacement field and its gradients do not vary rapidly in the tip region, (3.8) is approximately replaced by

ρ(ω20 − ω2)

Aw(0)∗

α (y, z) wα(y, z) dydz ≈ A1

∆ρ ω2

0 w(0)∗α (0, 0) wα(0, 0)

−Dβ(k)w(0)

α (y, z)∗y=z=0 ∆Cαβµν

Dν(k)wµ(y, z)

y=z=0

. (3.9)

In the same way as in the transition from (3.5) to (3.6), we have made use of the form (2.4) of the displacement field toreduce the volume integrals to integrals over the cross-section of the wedge.

Here, A is the cross section in the y–z plane of the total wedge filling the volume V = V0 ∪ V1, and A1 is the cross sectionof the part of the wedge filling the volume V1. With the coordinate system of Fig. 2(b), we can make use of the fact thatthe quantities wα(y, z) and D2(k)wα(y, z) are continuous across the interface at z = −h. This allows us to introduce thefollowing approximations:

wα(0, 0) ≈ w(0)α (0, 0), (3.10a)

[D1(k)wα(y, z)]y=z=0 ≈D1(k)w(0)

α (y, z)y=z=0 , (3.10b)

[D2(k)wα(y, z)]y=z=0 ≈D2(k)w(0)

α (y, z)y=z=0 . (3.10c)

With the help of the additional continuity condition for the stress components Tα3 across the interface, we may expressthe quantity D3(k)wα(y, −h + ε) (i.e. in the tip region above the interface) by quantities at z = −h − ε, where ε is aninfinitesimally small positive length:

C ′

α3µ3

D3(k)wµ(y, z)

z=−h+ε

= Cα3µ3D3(k)wµ(y, z)

z=−h−ε

− ∆Cα3µ2D2(k)wµ(y, z)

z=−h−ε

− ∆Cα3µ1D1(k)wµ(y, z)

z=−h−ε

. (3.11)

The quantities on the right-hand side of (3.11) all refer to the medium below the tip region and may therefore beapproximated by the corresponding quantities for a perfect wedge with unmodified tip. In addition, the arguments y and

240 E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245

z = −h ± ε may be replaced by 0 if the cross section of the tip region is sufficiently small. With these approximations andwith the matrix (Γαβ) introduced in Section 3.1, (3.11) becomes

[D3(k)wα(y, z)]y=z=0 ≈ ΓαβCβ3µ3D3(k)w(0)

µ (y, z)y=z=0

− Γαβ∆Cβ3µ2D2(k)w(0)

µ (y, z)y=z=0

− Γαβ∆Cβ3µ1D1(k)w(0)

µ (y, z)y=z=0

. (3.12)

The approximations (3.10) and (3.12) are now inserted in (3.9), and after a rearrangement of terms, the following result isobtained:

ρ(ω20 − ω2)

Aw(0)∗

α (y, z) w(0)α (y, z) dydz ≈ A1∆ρ ω2

0 w(0)∗α (0, 0) w(0)

α (0, 0)

− A1Dβ(k)w(0)

α (y, z)∗y=z=0

∆Cαβµν − ∆Cαβγ 3 Γγ δ ∆Cδ3µν

Dν(k)w(0)

µ (y, z)y=z=0

. (3.13)

Eq. (3.13) allows for the calculation of the correction to the dispersion relation ofwedge acousticwaves due to amodificationof the material properties in a spatial region at the wedge tip with an extension in the y–z plane that is much smaller thanthe wavelength of the wedge waves.

The case of wedge truncation is also included as the limiting case ρ ′→ 0, C ′

αβµν → 0. In the second term on the right-hand side of (3.13), this limit has to be taken with some care. First, we note that

Cδ3µν

Dν(k)w(0)

µ (y, z)y=z=0

= 0. (3.14)

This follows from the traction-free boundary conditions for the two surfaces of the infinite wedge,

N (j)β Tαβ(x, y, z) = 0, (3.15)

where j = 1, 2 labels these two surfaces and N (j)β is a Cartesian component of a vector normal to surface j. The left-hand side

of (3.14) is a linear combination of N (1)β Tαβ(x, 0, 0) and N (2)

β Tαβ(x, 0, 0). Eq. (3.14) implies that the term ∆Cαβγ 3 Γγ δ ∆Cδ3µν

on the right-hand side of (3.13) may be replaced by C ′

αβγ 3Γγ δ C ′

δ3µν . Introducing the scaling parameter λ, setting ρ ′= λρ,

C ′

αβµν = λCαβµν and performing the limit λ → 0 (which corresponds to the absence of material at the tip of the wedge) in(3.13), the following expression is obtained for the difference between the square of the frequencies of wedge waves in atruncated wedge and a perfect wedge:

ρ(ω20 − ω2)

Aw(0)∗

α (y, z) w(0)α (y, z) dydz ≈ −A1ρ ω2

0 w(0)∗α (0, 0) w(0)

α (0, 0)

+ A1Dβ(k)w(0)

α (y, z)∗y=z=0 Cαβµν

Dν(k)w(0)

µ (y, z)y=z=0

. (3.16)

An alternative, but equivalent expression for the right-hand side of (3.16) is obtained from (3.8) in the following way:Replacing in (3.9)∆ρ by−ρ and∆Cαβµν by−Cαβµν , doing an integration by parts on the right-hand side of (3.8), performingthe limits ρ ′

→ 0, C ′

αβµν → 0 and using the equation of motion for the perfect wedge and the traction-free boundaryconditions at its surfaces, we arrive at

VdVρ(ω2

0 − ω2)u(0)∗α uα = −

∂V0

dSβ

∂u(0)∗

µ

∂xν

uαCµναβ

, (3.17)

where ∂V0 is the surface of the truncated wedge and dSβ is a Cartesian component of the surface element. In (3.17), thedisplacement field uα may be approximated by the displacement field of the perfect wedge, u(0)

α :VdVρ(ω2

0 − ω2)u(0)∗α u(0)

α = −

∂V0

dSβ

∂u(0)∗

µ

∂xν

u(0)α Cµναβ

, (3.18)

or, equivalently,VdVρ(ω2

0 − ω2)u(0)∗α u(0)

α =

∂V1

dSβ

∂u(0)∗

µ

∂xν

u(0)α Cµναβ

. (3.19)

(∂V1 is the surface of the volume removed from the wedge by truncation.) With the help of Green’s theorem (3.19) may betransformed into

ρ(ω20 − ω2)

VdV u(0)∗

α u(0)α =

V1

dV

−ρ ω2

0 u(0)∗α u(0)

α +∂u(0)∗

α

∂xβ

Cαβµν

∂u(0)µ

∂xν

. (3.20)

E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245 241

Fig. 5. Dispersion coefficient F2 (dimensionless) as function of wedge angle for a truncated silicon wedge. Propagation direction:112

, direction of one

surface normal: ⟨111⟩.

Fig. 6. Dispersion coefficient F2 (dimensionless) as function of wedge angle for a silicon wedge with the material at the tip replaced by germanium.Propagation direction:

112

, direction of one surface normal: ⟨111⟩.

Using (2.4) for the displacement field of the wedgewaves in the perfect wedge and assuming that the cross-section A1 of thetruncated volume is sufficiently small such that the variations of the quantities wα(y, z) and Dβ(k)wα(y, z) over the crosssection are small, we arrive again at the result (3.16). We note, that this derivation, and consequently the result (3.16), isindependent of the specific shape of the cross-section of the truncated volumeof thewedge. The shape neednot be triangularhere. It is only the size of the cross-sectional area that enters (3.16).

The dispersion law following from (3.9) and (3.16) is of the general form

ω2= (vWk)2(1 − 2k2A1 × F2). (3.21)

The coefficient F2 depends on the acoustic mismatch between the two materials filling the volumes V0 and V1.Fig. 5 shows the quantity F2 as function of the wedge angle for truncated wedges of silicon. In Fig. 6, the same quantity

is displayed for the case of the tip region of the silicon wedge being replaced by germanium. The orientation of the wedgerelative to the crystallographic coordinate system of Si is the same as in Fig. 3. In the case of truncation, a change of signof the dispersion coefficient is observed, similar to the case of surface coating, but unlike the situation in Fig. 6. Here, F2 ispositive, i.e. there is normal dispersion for all wedge angles. In all cases, the effect of the modification of the wedge tip withconstant area A1 strongly increases for wedges with small wedge angles.

In their calculations of the phase velocity of wedge acoustic waves in truncated wedges, Lagasse et al. [40] found aquadratic dependence of the phase velocity onwavenumber k. This relationship in the long-wavelength limit followsdirectlyfrom (3.21).

242 E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245

Fig. 7. Phase velocity V of acoustic waves guided by the apex of a rectangular edge consisting of GaAs in its lower part (y + z > d) and of Al0.1Ga0.9As inits upper part y + z < d. Perturbation theory (dashed), semi-analytic FEM (solid).

3.3. Small variations of material constants

The material constants in the volume part V1 of the wedge are now assumed to differ only slightly from those of thematerial in V0, but the size of the cross section A1 need no longer be small compared to thewavelength of thewedge acousticwave.

First-order perturbation theory with respect to small quantities |∆ρ| /ρ,∆Cαβµν

/C0 leads to formula (3.8) with uα

replaced by u(0)α . (C0 may be the bulk modulus of the material in V0, for example.) Using (2.4), we obtain the result

ρ(ω20 − ω2)

Aw(0)∗

α w(0)α dydz =

A1

∆ρω2

0w(0)∗α w(0)

α −Dβ(k)w(0)

α

∗∆Cαβµν

Dν(k)w(0)

µ

dydz (3.22)

for the frequency shift of the wedge waves. The dispersion law takes the form

ω2= (vWk)2(1 − F3(kd)). (3.23)

The function F3(kd) is a linear combination of ∆ρ, ∆Cαβµν with coefficients that depend on the material properties of themedium filling V0, on the shape of the area A1 and on kd, where d is a length characterizing the size of A1 (Fig. 2(b)).

In the limiting case of kd ≪ 1 we obtain

F3(kd) = 2k2A1F2 (3.24)

and in the opposite limit kd ≫ 1 this function tends to the constant value

F3(kd) =v2W − v2

W

v2W

, (3.25)

where vW is the velocity of wedge waves in a homogeneous perfect wedge consisting entirely of the tip material of V1. Thisis because in this limit, the displacement field of the wedge wave is entirely localized in the volume V1.

After inserting (3.2) in (3.22) and transforming to integration variables η, ζ , the integrals in (3.22) have been evaluatedsemi-analytically in the following way. Considering a rectangular wedge with A1 being the triangle with height h (Fig. 2(b)).In the coordinate system of Fig. 1(a) with wedge angle θ = 90° this triangle is characterized by the inequalities y + z <

d, y, z > 0, where d =√2h. Making use of the explicit form of the Laguerre functions,

Φn(ξ) = exp(−ξ/2)n

g=0

angξn−g , (3.26)

where

ang =(−1)n−gn!

g![(n − g)!]2, (3.27)

the right-hand side of (3.22) can be converted into a two-fold sum over incomplete Gamma functions that have beenevaluated numerically. Explicit expressions are given in the Appendix.

Fig. 7. shows the phase velocity of wedge acoustic waves propagating in such a system, where the lower part (V0) of thewedge is filled by GaAs, while the upper part (V1) consists of Al0.1Ga0.9As. The surfaces are (001)-planes. The density andelastic moduli of Al0.1Ga0.9As have been obtained from the corresponding values of AlAs and GaAs by linear interpolation.

E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245 243

To validate our perturbation-theoretical results, we have performed a calculation of the phase velocity of wedge wavesusing a semi-analytical finite element method (FEM) with second-order triangular elements [2,38]. Very good agreement isfound between the perturbation-theoretical result and the FEM data for values of kd down to 2. For smaller kd, a discrepancyis found, which is related to the way the integrals are carried out in the perturbation-theoretical method. In the sums ofincomplete Gamma functions, large factorials occur which lead to numerical inaccuracies.

We note that the FEM-value for the wedge wave velocity in the perfect GaAs wedge is identical with the one obtainedby the Laguerre function method. In the limit kd ≪ 1, (3.22) and (3.13) yield the same frequency shift.

4. Conclusions

Among various effects that give rise to weak dispersion of wedge acoustic waves, we have analysed three, which areimportant since they can be used to tailor dispersion in a relatively easy way, and which can be used to extract informationabout the system from propagation properties of wedge acoustic waves in non-destructive testing applications. In the firstcase, one surface or both surfaces of the wedge is/are coated by a thin film. In this system, the phase velocity varies linearlywith the inverse wavelength. This is the same dispersion law as for Rayleigh-type surface acoustic waves propagating inelastic half-spaces that are covered by a thin film [45]. In the second case, the wedge is truncated near the tip or thematerial parameters aremodified in the close vicinity of thewedge tip. Here, the phase velocity is found to vary quadraticallywith the inverse wavelength as in the case of bulk acoustic waves that become dispersive when the wavelength is of theorder of magnitude of inter-atomic distances, or as in the case of acoustic waves of shear horizontal polarisation that aretrapped at the surface of a half-space by a thin film deposited on it [45]. Dispersion relations of wedge acoustic waves,determined experimentally or by FEM calculations, are mostly presented by plotting the phase velocity as a function offrequency. Our results imply that in these plots, the dispersion curves approach the zero-frequency limit with non-zeroslope in the first case (surface coating), while the slope becomes zero in this limit in the second case (tip truncation ordifferent material parameters near the tip). The coefficients F1 (F2) determine the slope (the curvature) of the dispersioncurve at zero frequency.

In the third case, an extended spatial region near the wedge tip is replaced by a medium that has material parametersclose to those of the material corresponding to the bulk of the wedge. Here, the dependence of the phase velocity on theinversewavelength is not a simple power law andmay bematerial-dependent since it is influenced by several elasticmoduliin anisotropic media. It also depends on the shape of the cross section of the spatial region where the material propertieshave been modified. In Section 3.3, we have not considered explicitly graded material properties in the upper part of thewedge. However, we would like to stress that Eq. (3.9) also applies when the density and elastic moduli of the upper part ofthewedge vary in the directions normal to the apex of thewedge. Such spatial variationswould strongly affect the dispersionlaw.

In addition to the three reasons for dispersion discussed here, there are many other deviations from the ideal wedge thatgive rise to dispersion of wedge acoustic waves, but which are beyond the scope of this work. These may be modificationsof the wedge’s cross section, for example the transition from a sharp-angle wedge to a plane-parallel plate [34] or a bilinearcross-section [27] or a wedge with triangular cross section on a pedestal or substrate [16,38]. Other reasons for dispersionare a non-zero curvature of the edge [24,33] or a periodic array of metal strips on a surface of a piezoelectric wedge [15].Dispersion effects that are difficult to avoid are random roughness of thewedge apex [19] or dislocation concentrations [46].In addition to influencing the velocity of wedge waves, they also give rise to attenuation. However, if the length scalescharacterizing the roughness of the tip are sufficiently small, its effect on the velocity of wedge waves may approximatelybe modelled by a modification of the elastic properties and the density of the wedge material near the tip, and our resultsof Section 3.2, especially the dispersion law, should be applicable.

Acknowledgments

We would like to thank Peter Hess and Alexey M. Lomonosov for stimulating discussions. Financial support by theDeutsche Forschungsgemeinschaft (Grant No. MA 1074/9) is gratefully acknowledged.

Appendix

For the function F3, defined in (3.24), for a rectangular wedge with volume V1 having slightly modified density/elasticmoduli, the following expression is obtained:

F3(kd) × ρ

∞n,m=0

aα∗

nmaαnm = ∆ρ

∞n,m,l,p=0

aα∗

nmaαlp

ng=0

mj=0

angang′a

mj a

mj′ I(2n − g − g ′, 2m − j − j′, kd)

−k2

ω20

∞n,m=0

∞l,p=0

aα∗

nmaµ

lp

ng=0

mj=0

lg ′=0

pj′=0

angamj a

lg ′apj′{∆Cα1µ1I(f , f ′, kd)

+ ∆Cα2µ3[0.25I(f , f ′, kd) − 0.5(p − j′)I(f , f ′− 1, kd)

244 E.S. Sokolova et al. / Wave Motion 50 (2013) 233–245

− 0.5(n − g)(f − 1, f ′, kd) + (n − g)(p − j′)I(f − 1, f ′− 1, kd)]

+ ∆Cα3µ2[0.25I(f , f ′, kd) − 0.5(l − g ′)I(f − 1, f ′, kd)− 0.5(m − j)(f , f ′

− 1, kd) + (l − g ′)(m − j)I(f − 1, f ′− 1, kd)]

+ ∆Cα2µ2[0.25I(f , f ′, kd) − 0.5fI(f − 1, f ′, kd) + (l − g ′)(n − g)I(f − 2, f ′, kd)]

+ ∆Cα3µ3[0.25I(f , f ′, kd) − 0.5f ′I(f − 1, f ′, kd) + (p − j′)(m − j)I(f − 2, f ′, kd)]}, (A.1)

where

ang =(−1)n−gn!

g![(n − g)!]2, f = n − g + l − g ′, f ′

= m − j + p − j′, (A.2)

I(a, b, x) = a!b!Γ (a + b + 2, x)Γ (a + b + 2)

, (A.3)

and Γ (a + b + 2, x) is the upper incomplete gamma-function,

Γ (a, x) =

∞

xexp(−y)ya−1, (A.4)

Γ (a + b + 2) is the regular gamma function:

Γ (a) =

∞

0exp(−y)ya−1. (A.5)

V1 has a shape as shown in Fig. 2(b).

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