International Journal of Mechanical Sciences 55 (2012) 42–49

Contents lists available at SciVerse ScienceDirect

International Journal of Mechanical Sciences

0020-74

doi:10.1

n Tel.:

E-m

journal homepage: www.elsevier.com/locate/ijmecsci

Group velocity of elastic waves as eigensensitivity of a parameter dependenteigenvalue problem

Atul Bhaskar n

School of Engineering Sciences, Aeronautics & Astronautics, University of Southampton, Highfield, Southampton SO17 1BJ, UK

a r t i c l e i n f o

Article history:

Received 3 December 2009

Received in revised form

14 January 2011

Accepted 7 December 2011Available online 14 December 2011

Keywords:

Group velocity

Eigensensitivity

Rayleigh’s quotient

Elastic waves

03/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ijmecsci.2011.12.001

þ44 2380 593825.

ail address: A.Bhaskar@soton.ac.uk

a b s t r a c t

We consider the problem of calculating group velocity of elastic waves along different branches of the

dispersion curves without carrying out direct numerical differentiation along these curves. We obtain

an expression for the rate of change of eigenvalues with respect to a parameter for a parameter-

dependent ‘non-linear eigenvalue problem’. Eigenvalues have the meaning of frequency (or its square)

and the role of the parameter is played by the wave number. We then define a quotient which shows

Rayleigh-quotient-like behaviour in that it differs from unity by a quantity that scales as � E2, where Eis the magnitude of a parameter used to perturb an eigenvector. By carrying out first order perturbation

analysis of a generalised eigenvalue problem involving two Hermitian matrices, at least one of which is

positive definite, we obtain an exact expression for group velocity as a rate of change of an eigenvalue.

This form is particularly suited for elastic waves. The connection of the eigensensitivity relationship

with the stationarity of the Rayleigh’s quotient is explored. Two specific examples of mechanical

waveguides are presented: (i) the Timoshenko waveguide and (ii) the bending-torsion coupled elastic

waveguide.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In his classic treatise on acoustics and structural dynamics [1],Rayleigh considered various aspects of the generalised eigenvalueproblem including the changes in eigenvalue when the matricesthat appear in the eigenvalue problem depend on a parameter.Typically this parameter could be a geometrical parametersuch as a length, or a thickness, or a material parameter such asdensity or the modulus of elasticity. Despite complete in hismathematical details, Rayleigh did not use the language ofmatrices while obtaining his results on the perturbation ofeigenvalue. These results have been frequently used in structuraldesign-sensitivity [2], optimisation [3], and design search con-texts [4] when one needs to calculate the change in frequency dueto structural modification [5]. We explore the use of this knownmathematics to the calculation of group velocity for wavepropagation problems.

The wave phenomenon is of great interest in many branches ofphysics such as optics, acoustics, electromagnetism, elastodynamics,and geophysics. The fundamental relation that describes the propa-gation behaviour is the so-called dispersion relation which relateswave numbers to the compatible values of frequency for propaga-tion. The dispersion relation can often be thought of the dependence

ll rights reserved.

of frequency o upon the wavenumber k as symbolically given by

o¼oðkÞ: ð1Þ

This relationship needs to be respected for a disturbance thatpropagates harmonically in space and time. The above equationcannot often be written explicitly. The ratio cp ¼oðkÞ=k is the phasevelocity which is the rate of change of phase and it has very littlephysical significance. The velocity of interest is the group velocitywhich is the gradient of the dispersion curves on the o�k plane andis given by cg ¼ @oðkÞ=@k for each ‘branch’ of the dispersion curves.The group velocity is also known as the velocity of propagation—thespeed with which an approximately monochromatic wave packettravels. It can be shown that important physical quantities such asenergy, signal and information travel at the speed of the relevantgroup velocity [6].

For elastic waveguide problems, relationship (1) can be calcu-lated analytically, when possible, but for many practical pro-blems, one needs to resort to numerical calculations. Therelationship cg(k) has to be often calculated by numericallydifferentiating (1), given o�k data at a discrete set of points.This is not always very satisfactory—particularly when severalbranches of the dispersion curves cross and/or veer against eachother. In this paper, we present cg(k) as an outcome of anappropriate eigensensitivity analysis. Having cast the problemas a parameter-dependent eigenvalue problem, the eigenvaluesand the eigenvectors (eigenfunctions) become dependent on the

A. Bhaskar / International Journal of Mechanical Sciences 55 (2012) 42–49 43

parameter. In the present context, the wavenumber plays the roleof this parameter.

The wave propagation problem presented here is limited toelastodynamics [7]—except the initial general treatment inSection 2. The approach can be adapted to problems in otherbranches of physics by modifying the analysis appropriately to adifferent set of field equations. Apart from the simplicity, a reasonfor choosing example from the domain of elastodynamics is thehighly dispersive nature of the bending elastic waves.

Expressions for the rate of change of eigenvalues with respectto a change in structural parameter have been known in theliterature. The motivation has been to calculate the frequencychange due to structural modification [8–10]. The approachof [8] was generalised for nonsymmetric systems in [11].The derivatives of the eigenvector have been studied too (see,e.g., [12]). Sensitivity results have been used in experimentalmodel updating [13]. More general matrix eigensensitivity hasbeen studied by many authors [14–18] which includes non-linearproblems and higher order sensitivity. The case of repeated rootshas attracted much attention [19–21] because of the specialstructure of the eigenspace associated with it which is a subspacespanned by the same number of independent eigenvectors asthe number of the repeated roots. Non-selfadjoint problems havebeen studied by some authors [22–24] which differ essentially intheir use of a different set of orthogonality relation. The literaturein the area of frequency changes due to structural modification islarge and a comprehensive review is beyond the scope of thepresent work.

This paper is organized as follows. We present the two-parameter ‘non-linear eigenvalue problem’ in the next sectionand derive the associated sensitivity when the matrices involvedare Hermitian. This is followed by an illustrative example. InSection 3, we present generic field equations for elastic wave-guides and cast the wave propagation problem in the form of aparameter dependent eigenvalue problem. We then derive aneigensensitivity expression from the dispersion relation for thegroup velocity of elastic waves. Finally, we relate the eigensensi-tivity result to the stationarity of the Rayleigh’s quotient. InSection 4, we present two examples of elastic waveguides toillustrate the theoretical development: (i) the Timoshenko wave-guide which involves general complex Hermitian matrices in thejoint wavenumber-frequency domain and (ii) the bending torsioncoupled waveguide which involves two real symmetric matrices.Finally, we present concluding remarks in Section 5. A part of thispaper (mostly Section 3) was presented in the 7th EUROMECHSolid Mechanics Conference [25].

1 We will refer to an eigenvalue problem as non-linear when the set of

algebraic equation is a non-linear function of the unknown l. As opposed to this,

problems that have the coefficient matrix A dependent on the eigenvector /,

i.e. A¼Að/Þ are sometimes referred to as non-linear eigenvalue problems.2 Note that for a specified value of k, Eq. (2) is a linear function of the vector /.

2. The parameter dependent non-linear eigenvalue problemand its eigensensitivity

A standard eigenvalue problem of the form A/¼ l/ becomes aparameter dependent eigenvalue problem when the coefficientmatrix depends on a parameter, say k, such that AðkÞ/¼ l/. Theeigensolutions then depend on this parameter, i.e. /i ¼/iðkÞ andli ¼ liðkÞ. Two common generalisations of the standard eigenva-lue problem are (i) when the eigenvalue problem is expressed interms of two matrices—the resulting problem is known as thegeneralised eigenvalue problem or the matrix pencil eigenvalueproblem, and (ii) when the eigenvalue appears as a polynomialwhere the coefficients are matrices—this problem is known asthe l-matrix problem. For many wave propagation problems, theeigenvalue problem appears in a form not identifiable in oneof the above two categories. This can happen, for example,when we seek non-trivial solutions of a set of homogeneousalgebraic equation such that the expressions involved are general

non-linear functions of an unknown parameter l in them.1

Consider the following mathematical structure:

AðkÞ/iðkÞ ¼ BðoiÞ/iðkÞ ð2Þ

in which A is a Hermitian positive definite matrix for all the values ofthe parameters k, and o is assumed to admit only real solutions.2

Real values of the first parameter k are prescribed and the values ofthe second parameter oiðkÞ for which Eq. (2) admits non-trivialsolutions are sought. The eigenvectors can then be assumed to benormalisable with respect to a matrix which depends on k. Considerincrements in k and the corresponding increment in o compatiblewith Eq. (2). This results in increments in the two matrices and alsothe vector /. Substituting these increments into Eq. (2), and ignoringthe second order terms, we have

DA/iþAD/i ¼DB/iþBD/i: ð3Þ

Expanding the finite increments using a Taylor’s series with the firstderivative shown explicitly as

DA¼@A

@kDkþOðDk2

Þ, DB¼@B

@oiDoiþOðDo2

i Þ ð4Þ

and ignoring second order terms, pre-multiplying Eq. (3) by /Hi ,

where the superscript H denotes the conjugate transpose of theconcerned quantity and O is the order symbol, we have

/Hi

@AðkÞ

@k/iDk¼/H

i

@BðoiÞ

@oi/iDoi�/H

i ðA�BÞD/i: ð5Þ

Because we have assumed that both AðkÞ and BðoiÞ are Hermitian,the last term in the above equation equals its conjugate transposeD/H

i ðA�BÞ/i which must be equal to zero due Eq. (2). Dividing Eq.(5) throughout by Dk and taking the limit of vanishing Dk and Doi,we have

@oi

@k¼

/Hi

@AðkÞ

@k

� �/i

/Hi

@BðoÞ@o

� �o ¼ oi

/i

: ð6Þ

The eigensensitivity result, in the above form, does not seem tobe available in the literature. With the interpretation of o and k asthe frequency and wavenumber of a propagating wave, the aboveexpression is that of group velocity along the i-th branch ofdispersion relation. However, as far as eigenderivative results areconcerned, it is still a special case of more general results such asthose of Jankovic [14] who calculated an expression for the nthderivative of the eigensolutions of a non-linear eigenproblem.One needs to specialise the result of Theorem I there to first ordersensitivity, pre-multiply by the inverse of the first term, and againpre-multiply by the Hermitian of the eigenvector to obtain Eq. (6).Despite being narrower in scope, we prefer Eq. (6) since formEq. (2) is frequently found in many physical problems and becauseof a property of the quotient discussed below. The expression

Qið/Þ ¼/HAðkÞ/

/HBðoiÞ/¼

nð/Þ

dð/Þð7Þ

is not a Rayleigh’s quotient associated with the eigenproblem (2).This is because the ‘eigenvalue’ oi does not appear as a linearterm in Eq. (2). Indeed the value of this quotient must alwaysbe equal to 1 for /¼/i. One of the well known properties of theRayleigh’s quotient is its stationarity around the eigenvectors—the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

k

∂ ω

i/∂ k

Fig. 2. Gradients @oi=@k of the three eigenvalues in Fig. 1 as functions of k.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

ε

Qi

Fig. 3. Quotients Qi ,i¼ 1;2 as functions of the perturbation parameter E. Note the

parabolic nature of the quotient in the neighbourhood of unity. Randomly

generated 40 values of the perturbation vectors were used in this simulation.

A. Bhaskar / International Journal of Mechanical Sciences 55 (2012) 42–4944

stationary values being the eigenvalues. What analogous property, ifany, does the quotient above possess?

Consider the quotient in Eq. (7) when the vector /i is

perturbed in the manner /¼/iþEni, where ni is a random vector

and the perturbation parameter E is a measure of the magnitudeof perturbation. The numerator nð/i,EÞ of the quotient Q ð/Þ then

equals /Hi A/iþ2EnH

i A/iþE2nHi Ani. Similarly, the denominator

dð/i,EÞ equals /Hi BðoiÞ/iþ2EnH

i BðoiÞ/iþE2nHi BðoiÞni. Therefore,

nð/i,EÞ ¼ aþE2nHi Ani and dð/i,EÞ ¼ aþE2nH

i BðoiÞni where a stands

for the first two terms of the numerator and the corresponding

terms of the denominator (i.e. the E0 and E1 order terms) that are

equal due to Eq. (2). After dividing both nð/i,EÞ and dð/i,EÞ by a,we have from (7):

QiðEÞ ¼ Q ð/iþEniÞ ¼ 1þOðE2Þ: ð8Þ

Therefore, if the trial vector in the quotient (7) differs from anactual eigenvector by a vector of the order E, then the quotientdiffers from unity by a quantity of the order E2. This statementis reminiscent of the property that the Rayleigh’s quotientdiffers from the corresponding eigenvalue by a magnitude of theorder E2, if the trial vector differs from the actual eigenvector by aquantity of the order E. Indeed, when B matrix can be explicitlywritten as a parameter times a constant matrix, e.g. as inBðliÞ ¼ liB, the result concerning Rayleigh quotient follows fromthe above statement as a special case.

Example 1 (An eigenproblem in terms of matrices that are

non-linear functions of k and o). Consider the following parameterdependent non-linear eigenvalue problem:

ðkþk2þk6Þ ðk�k2

�k6Þ

ðk�k2�k6Þ ðkþk2

þk6Þ

" #fð1Þ

fð2Þ

( )¼

o 0

0 2o

� � fð1Þ

fð2Þ

( ): ð9Þ

While the relationship for group velocity is correct for a generalclass of wave phenomena, we restrict ourselves to more familiarcases when the phase velocity and the group velocity are positive.

The coefficient matrix on the left side is identified as AðkÞ and

that on the right side as BðoÞ. The two eigenvalues are calculated

analytically for each prescribed value of k by solving a quadratic

equation in terms of the unknown o. This may not be possible in

more complicated non-linear algebraic equations. The results are

plotted in Fig. 1. The gradients of these two matrices, with respect

to the two parameters that they are functions of, can be

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

k

ω, ω

Fig. 1. The three eigenvalues oiðkÞ,i¼ 1;2 in Example 1 as functions of the

parameter k.

calculated as

@AðkÞ

@k¼ð1þ2kþ6k5

Þ ð1�2k�6k5Þ

ð1�2k�6k5Þ ð1þ2kþ6k5

Þ

" #,

@BðoÞ@o ¼

1 0

0 2

� �: ð10Þ

The gradients of the two eigenvalues oi, i¼1,2 are calculated using

Eq. (6) and are presented in Fig. 2. The larger eigenvalue in Fig. 1

shows a near constant slope for long waves which indicates group

velocity approximately equalling 1.5 (all quantities in this example

are non-dimensional). For greater wavenumber (shorter waves), the

slope is steeper and the group velocity increases. The second branch

shows a zero slope in the asymptotic limit of long waves indicating

zero group velocity in that limit which progressively increases for

shorter waves. These are reflected by the trends of the two branches

in Fig. 2. These values indeed correspond to those calculated by

numerically differentiating along the curves in Fig. 1.

Finally, to verify that the difference 9Q ðEÞ�19� E2 (see Eq. (8)),

we randomly generated vectors ni and calculated the quotient for

increasing value of E. A plot of 40 such runs is shown in Fig. 3.

A. Bhaskar / International Journal of Mechanical Sciences 55 (2012) 42–49 45

As expected, the dependence on the magnitude of perturbation Ehas a dominant parabolic trend. The actual values, of course,

depend on the nearness of n1, n2 and n3 to /1, /2 and /3

respectively which determines the steepness of these curves.

Note that the curves are parabolic with respect to the perturba-

tion parameter. Each curve is generated by a random perturbation

vector which is held the same along a curve as E is increased from

0 to 0.2. Therefore, the behaviour of the quotient in Fig. 3 is in

accordance with the expectation of Eq. (8) as illustrated here.

3. Group velocity of elastic waves from the sensitivityof an eigenproblem

The form of the non-linear eigenproblem discussed in theprevious section is suitable for many physical problems involvingwave propagations (e.g. those in spin waves in micro-magnetism).In this section, we derive a special result for the group velocity asan eigensensitivity of a generalised eigenvalue problem which issuitable for elastic waves. The material presented in this section ismostly on the lines of the work presented in the conference(ESMC, 2009 [25]) except that the conference contribution waslimited to real matrices). This difference changes the orthogon-ality relations for the eigenvectors.

The dynamics for conservative mechanical systems is gov-erned by the classical Lagrangian L¼ T�U where the kineticenergy T and the potential energy U are given by the quadraticforms [26] T ¼ 1

2/_w,M _wS and U ¼ 1

2/w,LwS where the expres-sions inside angular brackets separated by a comma represent anappropriate inner product. Here wðr,tÞ is the vector of fieldvariables as a function of the spatial co-ordinates r and time t;M represents the inertia operator (usually a scalar for one fieldvariable and a matrix for a vector of field variables); L representsthe stiffness operator (usually a linear differential operator invol-ving differentiation with respect to the spatial co-ordinates forproblems with one field variable and a matrix of differentialoperators for problems with a vector of field variables); a dotrepresents differentiation with respect to time.

We are not considering terms such in the kinetic energy suchas 1

2/_w,GwS, that are linear in velocities, or terms such as

12/w,HwS in the kinetic energy that are independent of velocities.Such terms are often encountered while writing the equations ofmotion in a rotating frame of reference or for elastic medium thatis convecting. The sensitivity of eigenproblems resulting fromsuch gyroscopic problems has been discussed elsewhere [24] inthe context of a design problem.

The field equations governing the propagation of elastic wavescan be obtained by applying Hamilton’s principle d

R 21 L dt¼ 0

where d represents the first variation of the action integral. Theresulting partial differential equation can be expressed as

M €wþLw¼ 0: ð11Þ

As a concrete example of propagation in the x-direction of ataught string M¼m, the mass per unit length of the string;L¼ T@2=@x2 where T is the tension in the string. For a beam,L¼ EI@4=@x4 where EI is the bending rigidity of the beam.Reciprocity theorem ensures that the operators L and M areself-adjoint. In addition, we assume that M is at least positivesemi-definite which follows from the non-negativity of thekinetic energy associated with the operator.

For simplicity, we look into the cases with only one direction ofpropagation, so the problem can be viewed as one of a waveguide.Eq. (11) implicitly assumes that the inertia and the stiffness effectsare separable, i.e. there are no operators that involve differentiationwith respect to time and space simultaneously. When we look for a

propagating wave solution of Eq. (11) in the form:

wðx,tÞ ¼ w exp½iðkx�otÞ� ð12Þ

the condition for having non-trivial solutions leads to the dispersionrelation:

Dðk,oÞ ¼ 0 ð13Þ

which relates the wave number k to the corresponding admissiblevalue(s) of the frequency o. This relationship is fundamental to theunderstanding of the propagation behaviour as it carries informationabout the dispersion characteristics (i.e. how the wavespeed isrelated to the wavenumber, for example); it is also essential forresponse calculations. When the relationship (13) is analyticallytractable, one obtains the group velocity as a function of thewavenumber by direct differentiation. For most complex problems,this is not the case and one can compute the dispersion relation oðkÞnumerically. The group velocity can then be obtained by numericaldifferentiation. In the next section, we develop an expression toobviate this numerical differentiation.

3.1. Perturbation analysis and eigenderivative

with respect to wavenumber

We now restrict the forms of the two matrices A and B to thosethat are frequently encountered in wave problems in elasticity.The wave number-dependent matrix A is Hermitian; it nowdesignated by KðkÞ and has the following general form

AðkÞ ¼KðkÞ ¼K0þ ik ~K1þk2K2þ ik3 ~K3þk4K4þ � � � , ð14Þ

where the matrices with even subscripts are symmetric and thosewith odd subscripts are skew-symmetric–thus rendering KðkÞHermitian. This matrix originates from the potential energyexpression. For standing wave (i.e. normal mode) problems,this is the well-known stiffness matrix which is symmetric.The frequency dependent matrix BðoÞ is always of the form:

BðoÞ ¼o2M ð15Þ

for wave problems in elasticity because of the velocity-squaredterms in the kinetic energy.

The eigenvalue problem (2) has the following special formnow

KðkÞw ¼ lMw, ð16Þ

where l¼o2 is the eigenvalue and w is the correspondingeigenvector. We will call K as the stiffness matrix that dependson the wave number k; and M is the mass matrix. The frequencydependence in (16) is explicitly o2 since the only time derivativeappears from the acceleration terms. On the other hand, KðkÞ is al-matrix [28] which is a polynomial in the variable k; the degreeof the polynomial depends on the order of spatial derivatives in L.For example, for the string problem, KðkÞ is quadratic in k whereasfor beam bending problems, it is quartic. Because K is Hermitianand positive semi-definite, and M is real, symmetric and positivedefinite, the eigenvalues l are all real for prescribed positive k.

For every prescribed value of the wavenumber k, one can solvethe eigenproblem (16) for o2 thus giving us the dispersionrelation for any desired value of k. In this way, we can expressthe eigenvalues and the eigenvectors l and w as functions of k:

l¼ lðkÞ, w ¼ wðkÞ: ð17Þ

Now consider a perturbed problem in the spirit of Rayleigh [1]when the wavenumber has changed to a new value k-ðkþDkÞ.The analysis that follows here is on the lines of the perturbationanalysis presented in [24] which was carried out for gyroscopicsystems in a design context. We look for the correspondingchange in the frequency o-ðoþDoÞ that is admissible with

A. Bhaskar / International Journal of Mechanical Sciences 55 (2012) 42–4946

assumed propagation (12) and satisfies (16) non-trivially. Thismeans that we are looking for changes along the dispersion curvewhen a small change in the wavenumber is brought about. Theadmissibility requires Eq. (13) to change according to

DðkþDk,oþDoÞ ¼ 0: ð18Þ

This is equivalent to making the following changes to theeigenproblem (16)

w-ðwþDwÞ, K-ðKþDKÞ, l-ðlþDlÞ, ð19Þ

where KðkþDkÞ ¼KðkÞþDKðk,DkÞ. Substituting these into (16)and neglecting second order terms after expanding the productsof the terms within the parentheses, we have

KDwþDKw ¼ lMDwþDlMw: ð20Þ

The change in the i-th eigenvector Dwi can be expanded interms of the eigenvectors of the original unperturbed problem wj

as

Dw i ¼X

j

aijwj, ð21Þ

where aij is the weight given to the j-th modal contribution.Writing (20) for the i-th eigenvalue-eigenvector pair li, w i andsubstituting from Eq. (21), we haveX

j

aijKwjþDKw i ¼ liMX

j

aijwjþDliMw i: ð22Þ

Reorganizing terms after recognizing that Kwj ¼ ljMwj and pre-multiplying throughout by w

H

i we have

wH

i DKwi ¼X

j

ðli�ljÞaijwH

i MwjþDliwH

i Mwi: ð23Þ

The first term in the above equation is always zero since

ðli�ljÞ ¼ 0 for i¼ j and wH

i Mwj ¼ 0 when ia j due to the orthogon-

ality of eigenvectors with respect to M. Recall that both K and Mare symmetric and M is at least positive semi-definite due to thepreviously assumed adjointedness and definiteness of L and

MFthis is a requirement for the orthogonality.After dividing both sides by Dk and taking the limit Dk-0, we

obtain the rate of change of the i-th eigenvalue with respect to theeigenvalue as

@li

@k¼

wH

i ðkÞf@KðkÞ=@kgw iðkÞ

wH

i ðkÞMwiðkÞ: ð24Þ

Since li ¼o2i and that the group velocity is given by cg,i ¼ @oi=@k

for the i-th waveguide mode, we have

cg,i ¼1

2oi

� �w

H

i ðkÞð@KðkÞ=kÞw iðkÞ

wH

i ðkÞMw iðkÞ: ð25Þ

This expression is exact despite the second order terms beingignored in Eq. (20) which is usual while establishing differentials.Note that Eq. (24) is the Rayleigh’s quotient associated with thematrix pencil ð@K=@k,MÞ when w iðkÞ is taken as the trial vector.

In the Lagrangian associated with the elastic wave propaga-tion, the potential energy usually involves quadratic functions ofterms that are independent of, gradient of, or curvatures of thebasic field variables. This results in a general mathematicalstructure of the stiffness matrix as a polynomial in wavenumberk. Further, because the elasticity operators are self-adjoint, thegoverning field equations involve differentiation of the variablesthat are of even order. Therefore, the stiffness matrix in suchsituations can be written as KðkÞ ¼K0þk2K2þk4K4þ � � �. Opera-tors that are matrices of differential operators (e.g. when the fieldequations are coupled partial differential equations), the odd

order partial derivatives are still possible leading to a generalform of Eq. (14). An example of this is taken later. Suchpolynomial matrices are known as l-matrices as mentionedbefore and have been studied extensively by Lancaster [27,28].For KðkÞ which is a real symmetric l-matrix in k, Eqs. (24)and (25) can be simplified by replacing the Hermitian by a transposeeverywhere.

The eigensensitivity relationship which enables analyticaldifferentiation of the relevant coefficient matrices relies on thestructure of the Lagrangian that is specific to classical mechanics.In the absence of gyroscopic and ‘centrifugal’ forces, the kineticenergy function is quadratic in velocities, i.e. T ¼ T2, where thesubscript denotes the dependence with respect to the generalisedvelocities. The strain energy, however, can have terms such asU ¼U0þ

~U1þU2þ~U3þU4þ � � �, where a tilde indicates that the

corresponding stiffness coefficients are skew-symmetric as in Eq.(14); the subscript denotes the order of differentiation withrespect to the spatial co-ordinate. This means that the Lagrangianin the double Fourier domain involving frequency and wavenum-ber takes the following form:

Lðo,kÞ ¼o2T2�U0�ik ~U1�k2U2�ik3 ~U3�k4U4� � � � : ð26Þ

A possible generalisation of this can be made to wave problemsthat involve gyroscopic effects (as in problems in a rotating frameof reference or convecting elastic media, for example) or centri-fugal effects. The kinetic energy term then takes the form:T ¼ T0þ

~T 1þT2. The second term is attributed to gyroscopiceffects and it leads to a skew symmetric coefficient matrix. Wewill not be concerned with these generalisations here.

Eq. (25) provides an exact expression for the group velocity interms of the eigenvectors of Eq. (16) and the derivative of KðkÞwith respect to k. Since KðkÞ is often a polynomial in k withconstant matrix coefficients, differentiation is a trivial operation.We do need to have the eigenvectors w i available, which is alwaysthe case after having calculated the dispersion relation.

3.2. Eigensensitivity and the stationarity of Rayleigh’s quotient

The i-th eigenvalue of (16) can be expressed as a Rayleigh’squotient associated with the matrix pencil ðKðkÞ,MÞ:

li ¼o2i ¼

wH

i ðkÞKðkÞw iðkÞ

wH

i ðkÞMw iðkÞ: ð27Þ

When we compare Eqs. (24) and (27) we note that Eq. (24) is adifferential of Eq. (27) with respect to the wavenumber k. Notethat although the eigenfunction depends on the wavenumber,i.e. wi ¼ w iðkÞ, it appears that the differentiation of (27) has takenbeen carried out while w i were held constant! In the following weshow that this observation is consistent with the stationarity ofRayleigh’s quotient.

If we let k-kþDk, the right side of (27) is a Rayleigh’s

quotient associated with the matrix pencil ðKðkþDkÞ,MÞ with

wiðkþDkÞ as the trial vector; the left side equals liþDli. We could

increment the wavenumber by an amount Dk so that

liðkþDkÞ ¼ liþDli. The perturbed eigenfunction w iðkþDkÞ differs

from the unperturbed eigenfunction wðkÞ by an amount Dwi.

Writing liþDli as a Rayleigh’s quotient and asserting the statio-

narity of the Rayleigh’s quotient, liþDli differs from a Rayleigh’s

quotient associated with the operator pencil ðKðkþDkÞ,MÞ withthe trial vector corresponding to the unperturbed propagation

mode by an amount which is of the order of 9Dwi92¼ E29w i9

because of the well known result that Rayleigh’s quotients arerelatively insensitive to changes in the trial vector (the difference

is of the order E2). Hence after replacing wi ðkþDkÞ by w iðkÞ,

60

80

100

120

Freq

uenc

y ω

A. Bhaskar / International Journal of Mechanical Sciences 55 (2012) 42–49 47

we obtain the change in the eigenvalue as a quadratic form in

terms of the change in the stiffness matrix DK, the mass matrix M

and the trial vector w iðkÞ which is the unperturbed eigenvector.The result (24) then follows. This argument still does not formally

tell us why the trial vector in (24) ought to be exactly w iðkÞ,namely the i-th eigenvector of (16). It, however, shows that thestationarity property of the Rayleigh’s quotient is consistent withthe observation that while differentiating (27) the result is such

as if w iðkÞ is independent of k. The formal proof comes from Eqs.

(16) through to (24).

40

4. Application to elastic waveguides

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

Wavenumber k

Fig. 4. Dispersion curves for the Timoshenko waveguide. Both branches are

dispersive and show a dominant quadratic behaviour for long waves. The values

of the non-dimensional parameter used with Eqs. (30) and (31) are E¼ 0:1, s¼ 0:4.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

Wavenumber k

Gro

up v

eloc

ity c

g

Fig. 5. Group velocity branches corresponding to the dispersion curves in Fig. 4.

Example 2 (Timoshenko waveguide–a case of complex Hermitian K).Consider the problem of wave propagation in Timoshenko beamswhich exhibits bending-shear coupling. The Lagrangian associatedwith this problem is given by

L¼1

2

Z l

0½frI

_c2þrA _w

2g�fEIc

2

,xþkGAðw ,x�cÞ2g�dx, ð28Þ

where the first term within the braces represents the kinetic energydensity whereas the second term represents the potential energydensity. Here E is the Young’s modulus, r is the density,A the cross-sectional area, G the shear modulus, I the second momentof the cross-section, and k the Timoshenko shear coefficient which isa property of the cross-sectional shape and the Poisson’s ratio. w andc are the transverse deflection and the shear degree-of-freedom ofthe cross-section. A subscript following a comma indicates differen-tiation with respect to the relevant independent variable. ApplyingHamilton’s variational principle d

R 21 Ldt¼ 0 we obtain the following

set of coupled field equations of motion in the non-dimensional form:

Ew,xx�c,x ¼ E3sw,tt , Ew,x�cþE2sc,xx ¼ E4sc,tt , ð29Þ

where s and E are two non-dimensional groups. The reader can referto Eq. (2.1) of [30] for the non-dimensionalization used.

To obtain the form of the matrices as in Eq. (16), we multiply

the first of the above two equations by E and seek travelling wave

solution for the field variables w and c to obtain

KðkÞ ¼ k2 E2 0

0 E2s

" #þ ik

0 E�E 0

� �þ

0 0

0 1

� �, M¼

E4s 0

0 E4s

" #:

ð30Þ

Note that the K-matrix is complex Hermitian in this example. The

gradient of this matrix with respect to the wavenumber k can be

analytically calculated as

@KðkÞ

@k¼ 2k

E2 0

0 E2s

" #þ i

0 E�E 0

� �: ð31Þ

The dispersion relation oiðkÞ is numerically calculated by solving

KðkÞ/¼o2M/ for different values of specified k. The two branches

thus obtained are shown in Fig. 4. The values of the non-dimensional

parameters E and s are taken as 0.1 and 4.0 respectively. The branch

passing through the origin represents the bending wave with shear

correction whereas the branch with a cut-on frequency corresponds

to the so-called second spectrum of Timoshenko equations [29,30].

Both branches have dominant quadratic behaviour in the long

wavelength limit which should result in group velocity that varies

linearly with wave number for long waves, if we differentiated along

the two curves. Instead we use Eq. (25) at every step of wave

number to calculate the group velocity for the two wave types

involved. The results are plotted in Fig. 5 which is in agreement with

our expectation.

Example 3 (Bending-torsion coupled waveguide–a case of real

symmetric K). As a second example, consider the problem ofwaves freely propagating in a rod when the transverse motionand the torsional motion are coupled [31]. The dynamics aregoverned by the following pair of coupled partial differentialequations:

lZw0000

þ €wþcy€f ¼ 0,

�lpf00þcy €wþcx

€f ¼ 0, ð32Þ

where it is assumed that the cross-section possesses an axis ofsymmetry, thus decoupling the other transverse motion v fromthe above two field equations and also that the cross-sectionalwarping effects are negligible. Here w and f are the bending andtorsional degrees-of-freedom of the cross-section, a prime meansdifferentiation with respect to the axial co-ordinate, lZ, cy, cx, lp

are constants associated with the cross-sectional properties [31].

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavenumber k

Gro

up v

eloc

ity c

g

Fig. 7. Group velocity corresponding to the dispersion curves in Fig. 6 for the

bending-torsion coupled waveguide.

A. Bhaskar / International Journal of Mechanical Sciences 55 (2012) 42–4948

Indeed Eq. (32) is obtained by specialising equation (2.3) of [31]by setting cz ¼ 0 and lw ¼ 0. The above pair of field equations canbe cast in terms of two non-dimensional numbers E and p1 as

w ,xxxxþw ,t t þEf,t t ¼ 0,

�p1f,xxþEw ,t t þð1þE2Þf,t t ¼ 0: ð33Þ

Considering spatially and temporally propagating wave solu-

tions (space and time co-ordinates being x and t respectively) of

the form:

wðx,tÞ ¼W exp½iðkx�otÞ�,

fðx,tÞ ¼F exp½iðkx�otÞ�, ð34Þ

we have the following set of algebraic equations arranged as

follows:

k4 1 0

0 0

� �W

F

� �þk2 0 0

0 p1

" #W

F

� �¼o2

1 EE ð1þE2Þ

" #W

F

� �:

ð35Þ

This equation has the same generic form as (16). In this example,

the left side of the equation depends on two terms that have the

explicit dependence on the wavenumber as � k4 and � k2

respectively. Therefore,

KðkÞ ¼ k4 1 0

0 0

� �þk2 0 0

0 p1

" #;

@KðkÞ

@k¼ 4k3 1 0

0 0

� �þ2k

0 0

0 p1

" #:

ð36Þ

The two branches of the dispersion curve are shown in Fig. 6. The

branch that shows dispersive behaviour for long waves becomes

non-dispersive for short waves and vice-versa. It is straightfor-

ward here to analytically calculate the group velocity; however,

for many practical problems involving waveguides, this needs to

be carried out numerically by differentiating the curves. The

problem is worsened when these curves intersect or come close

together (veer). This is because the order in which the eigenvalues

are computed is not consistent along the branches of the disper-

sion curves (hence the well known difficulty while producing

continuously joined curves as in Fig. 6; most often they are left as

a series of closely spaced dots). The values of the non-dimensional

parameters p1 and E are taken as 0.05 and 0.1 respectively for this

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Wavenumber k

Freq

uenc

y ω

Fig. 6. Dispersion curves for the bending-torsion coupled waveguide discussed in

Example 3. The two branches come very close but do not cross.

numerical example. The group velocity branches are plotted in

Fig. 7 by making use of Eq. (25). The roughly horizontal inclina-

tion of different parts of these curve indicate non-dispersive

behaviour where they should.

We chose a rather simple problem for illustrating the basic idea

in order to use the eigensensitivity expression (15). In many

practical waveguide problems, the general mathematical struc-

ture remains the same as here but the details are likely to be more

complex. Indeed the analysis outlined in this work was used to

calculate the group velocity of a structurally complex waveguide

for the problem of wave propagation in rails [32,33], however,

this aspect has so far remained unpublished.

5. Conclusions

The problem of calculating group velocity was presented aseigensensitivity of a parameter dependent non-linear eigenvalueproblem which appear in a host of wave propagation problems inphysics. First, an expression for the eigenderivative was obtainedin the form of a quotient whose trial vectors are the same asthe eigenvectors of the general non-linear eigenvalue problem.This quotient exhibits a Rayleigh-quotient-like behaviour andbecomes the Rayleigh quotient divided by the eigenvalue whenspecialised to linear eigenvalue problems. Examples are given toillustrate the results. An expression for group velocity in terms ofan eigensensitivity is derived when the field equations involvesecond order derivatives with respect to time–a feature specific toelastic waves. In this special case, the wavenumber dependentmatrix in the eigenproblem is a l-matrix and not a general non-linear function, which can be easily differentiated and used incalculations. Two applications of this result to—(i) Timoshenkowaveguide and (ii) bending-torsion coupled elastic waveguide–are presented. The approach is general and can be applied to morecomplex situations.

Acknowledgements

I thank Professor Jim Woodhouse, Cambridge University who,in 1994, suggested to me that the sensitivity Eq. (24) and thestationarity of the Rayleigh’s quotient may have a connection.Thanks are also due to my colleague Professor Neil Stephen and

A. Bhaskar / International Journal of Mechanical Sciences 55 (2012) 42–49 49

Professor Aleksei Kiselev of Steklov Mathematical Institute foruseful comments.

References

[1] Rayleigh L. The theory of sound. NY: Dover; 1945. (reprint).[2] Adelman HM, Haftka RT. Sensitivity analysis of discrete structural systems.

AIAA J 1986;24:823–32.[3] Rudisill CS, Bhatia KG. Complex structures to satisfy flutter requirements.

AIAA J 1971;8:1487–91.[4] Bhaskar A, Sahu SS, Nakra BC. Approximations and reanalysis over a

parameter interval for dynamic design. J Sound Vib 2001;248:178–86.[5] Brandon JA. Strategies for structural dynamics modification. NY: John Wiley;

1990.[6] Brillouin L. Wave propagation and group velocity. New York: Academic Press

Inc.; 1960.[7] Graff KF. Wave motion in elastic solids. New York: Dover; 1975.[8] Fox RL, Kapoor MP. Rate of change of eigenvalues and eigenvectors. AIAA J

1968;6:2426–9.[9] Wang BP, Pilkey WD. Eigenvalue reanalysis of locally modified structures

using a generalised Rayleigh’s methods. AIAA J 1986;24(6):983–90.[10] Nagaraj VT. Changes in the vibration characteristics due to changes in the

structure. J Spacecr 1970;7:1499–501.[11] Rogers LC. Derivatives of eigenvalues and eigenvectors. AIAA J 1970;8:943–4.[12] Nelson RB. Simplified calculation of eigenvector derivatives. AIAA J

1976;14:1202–5.[13] Jung H. Structural dynamic model updating using eigensensitivity analysis,

PhD thesis, Imperial College, London; 1992.[14] Jankovic MS. Analytical solutions of the nth derivatives of eigenvalues and

eigenvectors for a non-linear eigenvalue problem. AIAA J 1988;26:204–5.[15] Jankovic MS. Exact nth derivatives of eigenvalues and eigenvectors. J

Guidance, Control Dyn 1994;17:136–44.[16] Rudisill CS. Derivatives of eigenvalues and eigenvectors of a general matrix.

AIAA J 1974;12:721–2.

[17] Garg S. Derivatives of eigensolutions of a general matrix. AIAA J1973;11:1191–4.

[18] Murthy DV, Haftka RT. Derivatives of eigenvalues and eigenvectors of ageneral complex matrix. Int J Num Methods Eng 1988;26:293–311.

[19] Mills-Curran WC. Calculation of derivatives of structures with repeatedeigenvalues. AIAA J 1988;26:867–71.

[20] Ojalvo IU. Efficient computation of modal sensitivities for systems withrepeated frequencies. AIAA J 1988;26:361–6.

[21] Choi K-M, Cho S-W, Ko M-G, Lee I-W. Higher order eigensensitivity analysisof damped systems with repeated roots. Comput Struct 2004;82:63–9.

[22] Plaut RH, Huseyin K. Derivative of eigenvalues and eigenvectors in non-selfadjoint systems. AIAA J 1973;11:250–1.

[23] Adhikari S, Friswell MI. Eigenderivative analysis of asymmetric non-con-servative systems. Int J Num Methods Eng 2001;51:709–33.

[24] Bhaskar A. Rayleigh’s classical perturbation analysis extended to damped andgyroscopic systems, In: Proceedings of the IUTAM-IITD international winterschool on optimum dynamic design; December 15–19, 1997. p. 417–30.

[25] Bhaskar A. Group velocity as an eigensensitivity problem, In: Proceedingsof 7th EUROMECH solid mechanics conference, Lisbon; September 2009.p. 629–30.

[26] Meirovitch L. Computational methods is structural dynamics. Alphen aan denRijn, The Netherlands: Sitjhoff-Noordhoff; 1980.

[27] Lancaster P. On eigenvalues of matrices dependent on a parameter. NumerMath 1964;6:377–87.

[28] Lancaster P. Lambda-matrices and Vibrating Systems. New York: PergamonPress; 1966.

[29] Chan KT, Wang XQ, So RMC, Reid SR. Superposed standing waves inTimoshenko beams. Proc R Soc London A 2002;458:83–108.

[30] Bhaskar A. Elastic waves in Timoshenko beams: the ‘lost & found’ of aneigenmode. Proc R Soc London A 2009;465:239–55.

[31] Bhaskar A. Waveguide modes in elastic rods. Proc R Soc London A2003;459:175–94.

[32] Bhaskar A, Johnson KL, Wood GD, Woodhouse J. Wheel-rail dynamics withclosely conformal contact. Part 1. Proc Inst Mech Eng F 1997;211:11–26.

[33] Bhaskar A, Johnson KL, Wood GD, Woodhouse J. Wheel-rail dynamics withclosely conformal contact. Part2 1. Proc Inst Mech Eng F 1997;211:27–40.