7th Asia-Pacific Workshop on Structural Health Monitoring

November 12-15, 2018 Hong Kong SAR, P.R. China

Rooting strategy to determine dispersion characteristics of longitudinal

wave motion in a solid rod

T. Jothi Saravanan

Department of Civil Engineering, Yokohama National University, Japan.

Email: thiyagarajan-jothi-gh@ynu.ac.jp

KEYWORDS: Guided wave motion; Pochhammer frequency equation; Padé approximation;

Dispersion curve; Longitudinal modes; Solid rod.

ABSTRACT

The present research is based on the classical elastic theory of mechanics, to derive the simplified

solution for Pochhammer frequency equation for guided wave motion in an infinitely long solid

cylindrical rod with circular cross section. The concept of dispersion characteristics of longitudinal

mode is analysed. A simple theoretical procedure is employed and incorporated in an efficient manner

to study the dispersion curves of longitudinal mode. In this study, a novel rooting strategy is utilized

through Padé approximation which is a rational fraction. Based on the Padé approximants, the solution

strategy for the original Pochhammer frequency equation is improved. The calculation results show that

this method is stable and efficient under the condition of guaranteeing a certain wavenumber calculation

accuracy, which can avoid the leakage of its close root in the curve. The solution strategy is described

as follows: the frequency solution and the group velocity solution obtained from the previous

wavenumber point (i.e., wavenumber-frequency curve slope at this point) are used to predict the

frequency solution of the next wavenumber point by the Padé approximation algorithm. The root finding

strategy is performed again within the predicted range, and the obtained frequency solution can be used

to solve the group velocity value at this point. Compared with the algorithm used in the previous research,

the advantage is that the stability of solving the transcendental equation is improved and the

wavenumber-frequency curve can also be solved.

1. Introduction

The propagation of waves in an isotropic, elastic, solid media is a classic philosophy among many elastic

theories. Wave propagation in elastic rods and derivation of the Pochhammer frequency equation have been

comprehensively studied by many researchers [1-4]. In Pochhammer frequency equation, the different forms

of Bessel functions are utilized in the entire wavenumber-frequency domain as different intervals. Hence it

is mathematically a piecewise function, which adds difficulty to the solution of the equation [5-6]. In fact, for

theoretical level of frequency equation, only one Bessel function (Jn) can be used. However, this makes the

substituted variable into a complex number in a certain region, causing serious numerical instability, which

is not favorable for calculation. Some research scholars have contributed to numerically solving the

theoretical frequency equations of different waveguide media, such as the dispersion curve calculation

based on the global matrix algorithm – DISPERSE software [7-8], and the PCDISP software method [9-10].

The objects they target include more complex multilayer heterogeneous waveguide media.

In this research paper, an attempt to derive the simplified solution for Pochhammer frequency equation for

guided wave motion in an infinitely long solid cylindrical rod with circular cross section is presented. The

Padé approximation has been proposed for identification of dispersion curves. The basic theory of the Padé

approximation and its coefficients calculation are briefly introduced. The specific steps for solving the

Pochhammer frequency equation are also discussed. The solution for rooting strategy based on Padé

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approximants for solving the Pochhammer frequency equation is also well described. Finally, the roots

predicted by the Padé approximation and the theoretical solution obtained for longitudinal modes are

compared and it is in good agreement.

2. Wave motion in isotropic solid media

The solid medium discussed in this research is a homogeneous, continuous and isotropic. By using

Newton's second law and generalized Hooke's law, and ignoring the influence of body force, the

Navier’s governing stress equations of motion for the media is given as,

, , , 1,2,3j ij i jj iu u u i j (1)

where iu indicate the components of displacement vector, when 1,2,3i in the Cartesian coordinate

system , ,x y z respectively; , are the Lame constant; is the density of the solid medium.

A simpler way to express the above governing equations is to use the derivative of the potential energy

function to express the displacement components in three directions. These potential energy functions

can decouple the wave function.

Using vector notation, Equation (1) can be expressed as:

2 u u u (2)

The displacement vector u will be represented by Helmholtz decomposition [7],

u (3)

where and are the scalar and vector potentials, respectively. Substituting the above equation into

the Equation (2) is,

2

2

2t

(4)

Also 2, and the Equation (4) can be rewritten as,

2 22 0 (5)

The Equation (5) will be satisfied, if each term vanishes and its alternative form of scalar and vector

potential equations are established as,

2

2

2 0

0

(6)

Therefore, the Navier’s governing equation is decomposed into two simple wave equations. Although the components of the scalar, and vector, potential functions are universally coupled by boundary

conditions, the usage of displacement decomposition simplifies the analysis. In order to solve the

problem, the specific solution of Navier’s governing equation can be selected according to the integral

of an arbitrary function or function itself. If these selected equations satisfy the boundary and initial

conditions, then the solutions can be achieved. According to the principle of uniqueness, the obtained

solution for the equation is unique. It is worth noting that using Helmholtz decomposition the three

components of the displacement vector are related to the potentials namely, scalar potential function and

three components of the vector potential function. This indicates that there is a redundant constraint.

2.1 Dilatational and distortional waves

The two elementary wave types are dilatational and distortional, which can propagate individually with

a specific velocity in an infinite medium. Moreover, they are independent or uncoupled from one another.

If the vector operation of the divergence is implemented on the Equation (2),

2

2

1

LC (7)

where the propagation velocity, 2 /LC . Thus, it is determined that a change in volume, or

dilatational disturbance, will propagate at the velocity LC . The physical meaning is that the elastic

medium will only undergo volume expansion or contraction, and shear deformation will not occur. Such

waves are called expansion or dilatational waves.

Assuming that the curl of the gradient of a scalar is 0, then:

2

2

1

TC (8)

where the propagation velocity, /TC . The elastic medium will only produce shear deformation

without volume expansion or contraction. Such waves are called rotational or distortional waves. In

practice, the research object is often a finite medium, which cannot reach an infinite ideal scale in size,

thickness or scale direction, and the fluctuation will occur multiple times at the boundary, accompanied

by the dilatational and distortional wave. Therefore, in the actual finite boundary wave propagation, the

different modes are superimposed on each other and propagated in the form of guided waves.

3. Guided wave motion in a cylindrical rod

The Navier’s governing equation can be decoupled into two sets of equations by Helmholtz decomposition, which are the scalar and vector potential functions, respectively.

Scalar potential function is given as,

22

2 2

1

LC t

(9)

Vector potential function is available as,

22

2 2 2 2

22

2 2 2 2

22

2 2

2 1

2 1

1

r rr

T

r

T

zz

T

r r C t

r r C t

C t

(10)

where 2 is the Laplacian operator in the cylindrical coordinate system,

2 2 22

2 2 2 2

1 1

r r r r z

(11)

In the cylindrical coordinate system, the displacement components in three directions are , ,r zu u u ,

representing the radial, circumferential and axial displacement respectively.

1

1

1 1

zr

r z

rz

ur r z

ur z r

ru

z r r r

(12)

In wave motion of solid cylindrical rod with circular cross-section, the stress components , , rr r rz of

the outer surface of the rod provides the boundary conditions for the subsequent theoretical solution of

the wave equation. In the case of free vibration of rod, the values of these three stress components should

be 0. The relationship between these stress components and displacement is listed below.

2

1

rrr

r

r

r z

rz

u

r

uuu

r r

u u

z r

(13)

where is the volume invariant, defined as

1r zr

u u uu

r r z

(14)

3.1 Frequency equation for guided waves

Consider an infinitely long solid, circular, cylindrical rod of radius a , as shown in Figure 1; if the

surface of the cylinder is not stressed, i.e., r a ,

0 rr r rz (15)

Figure 1. Schematic diagram for a solid, cylindrical rod with circular cross-section

By considering the propagation of harmonic waves in the cylinder [11], the scalar potential function

, , ,r z t , and the wave motion along the z axis is in the following form,

expr i kz t (16)

Since the cross-sectional dimensions of a rod is limited, it can be assumed that the solution has a form

in which r and are decoupled, and the above formula is brought into the wave equation with respect

to r and as,

2 2 22

2 2 2

22

2

10

0

L

d d nk

dr r dr C r

dn

d

(17)

The solution of , is the sine or cosine function whose argument is n . The solution should behave

as a continuous function of and n can only be 0 or an integer to have continuous derivatives. The

solution of r is a Bessel function. For a solid cylindrical rod, the potential function should be finite

at the centre of the section, so the Bessel function of the first kind is preserved and the Bessel function

of the second kind is discarded. In summary, the scalar potential function is given as,

1 2 1cos sin expnA n A n W r i kz t (18)

where 1 is expressed as,

2

2 2

1 2

L

kC

(19)

when 2 0 , 1nW r is the

thn order Bessel function of the first kind 1nJ r and when

2 0 ,

1nW r is the th

n order modified Bessel function 1nI r ; n is the circumferential order number;

is angular frequency expressed as, 2 f ; k is wave number expressed as, 2 / lk , where l is

the wavelength. Similarly, the vector potential function component z can be expressed as,

1 2 1cos sin expz nB n B n W r i kz t (20)

where 1 is defined as,

2

2 2

1 2

T

kC

(21)

when 20 , 1nW r is the

thn order Bessel function of the first kind 1nJ r and when 2

0 ,

1nW r is the th

n order modified Bessel function 1nI r .

Now consider the equations governing r and , which are relatively more difficult to solve because

they are coupled. It is, however, evident that r and also comprise trigonometric function .

Furthermore, the form of coupling in Equation (10) indicates that a sine-dependence on in r is

consistent with a cosine-dependence on in and vice-versa. Accordingly, the potentials are given

as,

sin exp

cos exp

r r r n i kz t

r n i kz t

(22)

Substituting into the Equation (10),

2 22 2

2 2 2

2 22 2

2 2 2

1 12 0

1 12 0

r rr r r r

T

r

T

d dn n k

dr r dr r C

d dn n k

dr r dr r C

(23)

Here, it is more convenient to solve the r and by subtracting the above formula to obtain the

equation for r , given as,

2 1 12r nC W r (24)

Adding the expressions in Equation (23) to get,

1 1 12r nC W r (25)

The corresponding expressions of r and are

1 1 1 2 1 1

1 1 1 2 1 1

r n n

n n

C W r C W r

C W r C W r

(26)

The scalar potential and three components of the vector potential functions are obtained in terms of four

arbitrary functions. Still, the displacement vector is indicated in terms of three constants and only three

boundary conditions are available which provide only three homogenous equations. Thus, the other

essential condition is assumed by setting an unknown coefficient, 1 0C , which results in r .

Accordingly, the modified set of potentials are given as,

3

1

1

cos exp

sin exp

sin exp

cos exp

z

r

f r n i kz t

g r n i kz t

g r n i kz t

g r n i kz t

(27)

where,

1 1

3 1 1

1 2 1 1

n

n

n

f r AW r

g r B W r

g r C W r

(28)

Equations (27-28) can be employed to calculate the stresses in terms of the potentials. The boundary

condition in Equation (15), yield three homogenous equations for the three constants 1 1 2, ,A B C

respectively. It can be obtained as,

0 , 1,2,3ij jc X i j (29)

where jX represents the unknown coefficient vector, 1 2 1jX A C B . The frequency equation is

obtained from the resulting determinant of the coefficients,

0ijc (30)

The wavenumber-frequency ( k ) curve can be obtained by solving the countless pair of ,k values

satisfying the determinant condition. The obtained classical equation is the well-known Pochhammer

frequency equation [4]. Due to its complexity, an effective and stable numerical calculation method is

required. After solving the k curve, the phase velocity and group velocity corresponding to each

pair of ,k solutions can be obtained. This is obvious for the phase velocity, which is obtained by

multiplying the angular frequency by the reciprocal of the wavenumber, /pc k . However, for the

group velocity, it can be directly obtained by numerically solving the derivative at each point of the

k curve, i.e., g

ck

. Still, this method requires that the points of the calculated group velocity be

as close as possible to each other, and theoretically very rigorous.

3.2 Propagation of Longitudinal mode

Longitudinal waves are axially symmetric, which are determined by the existence of the radial and axial

directions displacement components. The reduction of the general frequency equation for the case 0n

in Equation (30), it reduces to the term 23c and its cofactor matrix.

0 0

11 12

0 0

22 23

0 0

31 32

0

0 0

0

n n

n n

n n

c c

c c

c c

(31)

The above equation is rewritten according to the Laplace expansion,

0 0

011 12

230 0

31 32

0

n n

n

n n

c cc

c c

(32)

The frequency equation for the longitudinal modes is given by the cofactor matrix which is,

0 0

11 12

0 0

31 32

0

n n

n n

c c

c c

(33)

2 2 2 2

1 1 1 1 0 1 1 0 1 1 1

2 2 2 2

1 1 1 1 1 1

2 2 20

2

aW a k a W a k a W a kaW a

k a W a k a W a

(34)

This expands to give the Pochhammer frequency equation for the longitudinal modes, 0,L m . Figure

2 shows the dispersion curves of dimensionless frequencies for longitudinal modes in a solid cylindrical

rod with circular cross-section.

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

L(0,7)L(0,6)

L(0,5)

L(0,4)

L(0,3) L(0,2)

2a

f/C

T

ka/

L(0,1)

Figure 2. Dimensionless frequencies for longitudinal modes, 0,L m

4. Simplified solution for the Pochhammer frequency equation

A simpler way to solve the Pochhammer frequency equation is to use the bisection method for the

real number field, so that the continuity of the function is only in the definite domain, and the frequency

equation is bound to meet this requirement. The ,k domain to be solved is, divided into several

intervals in the k and directions respectively. For each discrete k value, the bisection method can

be used to search for the roots in the corresponding intervals and vice versa. Each step of the solution

in the whole process will be independent of each other. However, the shortcoming of this method is, it

is essential to find roots in all areas, which is unnecessary. It losses the characteristics of the k

curve, and is not favourable for further studies. Since, it is solved independently on each discrete k (or

), the difficulty of root finding increases, and sometimes individual roots are lost. For this reason, a

more effective solution strategy is needed. The Padé approximation has been proposed for identification

of dispersion curves.

4.1 Padé approximation

The rooting strategy based on Padé approximation [12-14] will be described here. Consider that in the

,k plane, a certain k curve is being solved (i.e., a specific mode) and three sets of solutions

0 0,k , 1 1,k , and 2 2,k have been obtained through the previous steps. Subsequently, by using

the information of the previous points, the Maclaurin expansion [15] can be to extrapolate the approximant

of 3k at 3 and then construct the roots within its proximity. Suppose, the power series representing the

function f (k) is,

0

i

i

i

f k c z

(35)

where 𝑐𝑖 = 0, 1, 2, 3, … , for particular set of coefficients. Precisely, consider the first four items of this

power series.

2 3

2 2 2 2 2f k k f k f k k f k k f k k (36)

where,

0 2

1 2 2 2

2 2 2 2 1 1

3 2 2 2 0 0 1 12

1,

2

1, ,

2

1, , 2 ,

2

g

g g

g g g

c f k

c f k C k

c f k C k C kk

c f k C k C k C kk

(37)

It can be seen that extrapolation method based on Maclaurin expansion can make full use of the

information of the previous solution, including the group velocity of each point. However, Maclaurin

series expansion may cause large deviations in the far-predicted area, and the Padé approximation

minimizes this effect.

A Padé approximant is a rational fraction, where /L M is expressed in Equation (38), which has a

Maclaurin expansion that agree with Equation (35) as far as possible. This expansion is an important

preliminary step of any investigation using Padé approximants.

10 1

0 0 1

/L

i L MLi M

i M

a a z a zc z L M O z

b b z b z

(38)

Though, the numerator and denominator parts have a total of 2L M coefficients, there is relatively

an inappropriate common factor among them, and for certainty it is taken that 0 1b . This selection

tends to be a vital part of the accurate definition. So, there are 1L M independent coefficients in the

Padé approximation of the /L M . This indicates that generally /L M should fit in the power series

in Equation (35) through the orders 21, , , L M

z z z

.

The basic idea to find the coefficients of the Padé approximants is to cross-multiple, and make

corresponding terms to have equal powers. For the specific derivation, please refer to [14], only the /1L

is listed here. The formula for calculating the coefficients of the Padé approximants is as follows,

1 1

0 0

1 1 1 0

2 2 1 1 2 0

min ,

1

/L L

L M

L L i L i

i

b c c

a c

a c b c

a c b c b c

a c b c

(39)

Thus, the Padé numerator and denominator are determined from the Padé equations. Figure 3 shows the

progression of a function, 1 / 2 1 2f z z z obtained using the 2nd order Maclaurin series

expansion and 1/1 Padé approximant. The advantage of the Padé approximation relative to truncated

Maclaurin series expansion can be observed. The Padé approximants exploits the differences of the

coefficients to do its long-range extrapolation, and so the differences must all be accurate.

Figure 3. Comparison of Padé approximation with Maclaurin series expansion

4.2 Rooting strategy based on Padé approximation

The basic theory of the Padé approximation and its coefficients calculation are briefly introduced. The

specific steps for solving the Pochhammer frequency equation are discussed:

(1) The initial wavenumber and cutoff frequency are utilized as a limit to define the solution area to

be determined;

(2) In order to solve some of the initial roots, bisection method is used (interval solution), where the

cutoff frequency can be solved at the initial wavenumber or vice versa. However, considering

that the solutions of several frequencies may be very similar in the wavenumber region (for

example, the cutoff frequencies for 0,2L and 0,3L , 0,5L and 0,6L for longitudinal

modes, as shown in Figure 2). It is necessary to determine the control interval which in turn

define the number of roots. This reduces the stability of the solution by attaining several solutions

for wavenumber at cutoff frequency. If the number of roots are large, the distance between them

is also large, and the interval solution is stable;

(3) After obtaining a number of initial roots, the solution for the frequency on the upper and lower

discrete wavenumber values of the respective curves is searched. As described above, the process

proceeds to the low wavenumber region. According to the Padé approximation algorithm, each

discrete wavenumber values should be at equidistant. Steps (2) and (3), which begin in this

process, can use the 0 /1 and 1/1 Padé approximant respectively. Starting from step (4), only

the 2 /1 Padé approximant can be used. Use the previous three steps, and then advance further;

(4) After the approximation of the next point is obtained by Padé approximant, 1 or 2 is

fluctuated up and down around the point to form a root-seeking interval, and the bisection method

is used to solve the interval. If the function value is the same at both ends, then gradually expand

the root-seeking interval until it is satisfied for the interval of bisection method.

The solution diagram for rooting strategy based on Padé approximation for solving the Pochhammer

frequency equation is shown in the Figure 4.

Figure 4. Rooting strategy based on Padé approximation

In Figure 5, it shows the comparison of the root predicted by the Padé approximation and the theoretical

solution obtained for longitudinal modes. It can be seen that in the region where the slope does not

change much, they both have good agreement, which is favourable for the rapid convergence of the

theoretical solution. When the slope changes sharply, the results of the Padé approximation prediction

are also satisfactory. Similarly, the comparison study for theoretical solution of torsional and flexural

modes with Padé approximation are carried out and it has good agreement.

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Exact curves

predicted points

ka/

2af/C

T

0.2 0.4 0.6 0.81.0

1.2

1.4

1.6 3.0 3.5 4.02.5

3.0

3.5

4.0

Figure 5. Comparison between Padé approximation and theoretical solution of longitudinal modes

5. Conclusions

In this research work, the simplified solution for the Pochhammer frequency equation is derived for

guided wave motion in an infinitely long solid cylindrical rod with circular cross section. The Padé

approximation has been proposed for identification of dispersion curves. The dispersion curves obtained

for longitudinal modes in a solid rod resulting from the root finding method thorough Padé

approximation are compared with the analytical ones obtained by solving the corresponding

Pochhammer frequency equation. The accuracy of the proposed methodology has been confirmed and

illustrated. Very good agreement has always been encountered. This research can be extended to the

solution of the wavenumber-frequency curve of the complex wave number domain and the frequency-

group velocity curve can be solved simultaneously.

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