Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 4, pp. 973-992, Warsaw 2008

THE PROPERTIES OF COUPLED WAVES PROPAGATING

IN LONG SUSPENDED CABLES

Jacek Snamina

Cracow University of Technology, Institute of Applied Mechanics, Cracow, Poland

e-mail: js@mech.pk.edu.pl

In this paper, the properties of coupled waves travelling along a long
cable are analysed. Since the tension and curvature in the equilibrium
position of the cable are slowly varying functions of the arc co-ordinate,
the problems concerning the travelling waves can be solved using the
Wentzel-Kramers-Brillouin (WKB) method. The waves propagating in
the plane of the equilibrium curve are coupled. The wave associated
with displacements perpendicular to the plane is uncoupled from the
remaining waves. Applying the WKB method, the dispersion relation
and equations describing the amplitudes of waves are determined. For
a longitudinal-dominated pair of waves, there exist two cut-off frequ-
encies depending on the arc co-ordinate. The results of calculations of
wavelengths and amplitudes are presented in the form of plots.

Key words: long cables, coupledwaves, dispersion relation, cut-off frequ-
encies

1. Introduction

Cables are used in many engineering applications. For instance, they are ap-
plied in ship equipment, cable railway, bridge suspensions and lift devices.
Above all, cables are used in the overhead transmission lines.

Unfortunately, cables transmit waves induced by relatively small distur-
bances. Oscillations of the overhead transmission line can cause considerable
damage in suspension towers and lead to failures of the conductor and electri-
cal ormechanical subsystems due tomaterial fatigue. There aremany reasons
that can induce waves in cables – for example: gusts of wind, interruptions
associated with installation works and various failures of devices mounted to
the cable.



974 J. Snamina

A wide range of cable applications leads to formulation of many research
strategies and equations describing motion of the cable. The equations have
been derived using various techniques and coordinate systems.

Many papers are devoted to the analysis of vibrations of cables. Motion of
the cables is usually analysedas a superpositionofmodes (IrvineandCaughey,
1974;PerkinsandMote, 1986;Burgess andTriantafyllou, 1988).Thisapproach
is very efficient for describing the response of short cables. For very long cables
and high frequencies, the wavelength is small relative to the cable length, and
the time in which the waves pass along the cable is relatively long. In this
case, motion of the cable (especially the transient response) is better and
more naturally described using the technique of superposition of travelling
waves.

In several papers, the overhead cable is modelled as a taut string. In these
papers, the effect of equilibrium curvature and non-constant tension along the
cable are ignored. However, the majority of papers takes into account the
equilibrium curvature in cable dynamics.

Many interesting problems in cable motion are associated with waves tra-
velling along the cables. Perkins in works (Perkins and Mote, 1987; Perkins
and Behbahani-Nejad, 1995, 1996) considered the wave propagation in elastic
cableswith a small curvature.The equations ofmotion derived in (Perkins and
Mote, 1987) were simplified to the form of linear equations with a constant
coefficient, neglecting variability of the tension and curvature along the cable.
The coupled longitudinal and transverse waves were determined.

Large-amplitude free vibrations of a suspended cable were investigated
e.g. in Luongo et al. (1984), Rega et al. (1984), Srinil et al. (2004). Wave
interactions in a non-linear elastic string were considered in Young (2002).
The aim of the present work is an investigation of the properties of travel-

ling waves in cables taking into account effects associated with variability of
cable tension and curvature along the cable. Linear equations of motion with
variable coefficients are considered.The cable is treated as a non-uniformwave
medium.Thedispersion relationswill beanalysed.The lengthsandamplitudes
of waves as functions of the arc co-ordinate will be derived.

2. General linear equations of motion

Observers usually describe motion of the cable in relation to its equilibrium
position using reference axes associated with the line of the cable equilibrium.
In this co-ordinate system, the points of the cable are identified by the arc-



The properties of coupled waves propagating... 975

coordinate s, and the base unit vectors eτ, en, eb have the tangent, normal
and binormal direction, as shown in Fig.1.

Fig. 1. Equilibrium curve and the dynamic configuration of the cable

The displacement vector u has three components u1(s,t), u2(s,t), u3(s,t)
that describe displacements of cable points in the tangent, normal and binor-
mal directions. In order to simplify the equations, one has assumed that the
arc-coordinate s is equal to zero in the lowest point of the cable.
The general equations of cable motion are non-linear. They have been

derived in Perkins and Mote (1987). In this paper, it is assumed that the
components of displacementu are small. Thus the non-linear terms in general
equations can be neglected, and finally the linear equations can be written in
the form

µ
∂2u1
∂t2
= EA0

∂2u1
∂s2
− (N +EA0)κ

∂u2
∂s
−Nκ2u1−EA0

dκ

ds
u2

µ
∂2u2
∂t2
= N

∂2u2
∂s2
+(N +EA0)κ

∂u1
∂s
+
dN

ds

∂u2
∂s
+
d(Nκ)

ds
u1−EA0κ2u2

(2.1)

µ
∂2u3
∂t2
= N

∂2u3
∂s2
+
dN

ds

∂u3
∂s

where: µ is themass per unit length of the cable, N is the cable tension, κ is
the curvature of the equilibriumcurve, A0 is the area of the cable cross-section
and E is the Youngmodulus.
Using the equilibrium equations, one can determine the cable tension

N and the curvature κ of the equilibrium curve as functions of the arc-
coordinate s. The equations describing the cable tension and the curvature are
presented e.g. in Perkins and Behbahani-Nejad (1996). They can be written
in the following forms

N(s)=
√
N20 +(µgs)

2 κ(s)=
N0µg

N20 +(µgs)
2

(2.2)



976 J. Snamina

where: g is the gravitational acceleration and N0 is the horizontal component
of cable tension. Functions N(s), κ(s) are even. This property is associated
with symmetry of the cable. The waves propagating in both directions from
the lowest point are symmetric. For this reason, onlywaves propagating in the
positive direction are investigated in the next sections.

3. Wave propagation in a long cable

The typicalwavelength λ appearing in long cables is considerably smaller than
the cable length L. For convenience, we introduce a small coefficient εdefined
as the ratio of λ to L. The changes of parameters N(s), κ(s) are small if the
change of s is comparable with λ.
The calculation canbegeneralised by introducing an infinite cable inwhich

the horizontal component of the equilibrium tension is constant and it has the
samevalue N0 as in the considered cable. In this case, the cable can be treated
as an infinite non-uniformmediumwith slowly varyingparameters N(s),κ(s).
The distance L can be interpreted as such a distance that causes significant
changes of parameters N(s), κ(s).
The problems concerning the travelling waves in the cable can be solved

using the well-known Wentzel-Kramers-Brillouin (WKB) approximation me-
thod. In this method, presented e.g. in Yang (1990), the slowly varying co-
ordinate η and slowly varying time τ are introduced. The relation between
the arc co-ordinate s and the slowly varying co-ordinate η is as follows

η = εs (3.1)

By analogy, the relationship between time t and the slowly varying time τ is

τ = εt (3.2)

In the above expressions, the small parameter ε is used.
We assume that a wave is called the normal wave, if it develops only

normal displacements of cable points. By analogy, the tangent wave results
in tangent displacements only and the binormal wave develops only binormal
displacements, as shown in Fig.1.
From Eqs. (2.1) it is apparent that the tangent and normal waves are

coupled (the first and second equation) whereas the binormal wave is not
coupled to the remainingwaves (the third equation).Theseproperties ofwaves
propagating in long cables have been described in the literature. Thus, the



The properties of coupled waves propagating... 977

binormal wave can appear by itself, without tangent and binormal waves. In
accordancewith theWKBmethod, the binormalwave can be expressed in the
form

u3 = U3exp
[
iθ(η,τ)

1

ε

]
(3.3)

where U3 is the wave amplitude and θ(η,τ) is a slowly varying phase of the
wave.
In the proposed description of waves, it is convenient to define, as inWhi-

tham (1999), a local wave number k and a local frequency ω

k =
∂θ

∂η
ω =−

∂θ

∂τ
(3.4)

From equations (3.4), the following relationship, known as the eikonal equ-
ation, can be derived

∂k

∂τ
+
∂ω

∂η
=0 (3.5)

The described non-uniform medium depends on the spatial co-ordinate s
and it does not depend on time t because tension N associated with the
equilibrium configuration depends on the arc co-ordinate s alone. Therefore,
the dispersion relation can be written in the form

ω = ω(k,N) (3.6)

Substituting relationship (3.6) into Eq. (3.5), one can obtain a partial
differential equation describing the wave number k

∂k

∂τ
+vg

∂k

∂η
=−∂ω

∂N

dN

dη
(3.7)

where vg stands for the group velocity defined as

vg =
∂ω

∂k
(3.8)

Differentiating relationship (3.6) with respect to the slowly varying time τ
and using eikonal equation (3.5), one can obtain an equation describing the
frequency ω

∂ω

∂τ
+vg

∂ω

∂η
=0 (3.9)

The left-hand sides of Eqs. (3.7) and (3.9) are analogous. Hence, both
equations have the same characteristic curves which can be described by the
following relationship

η−
∫
vg dτ = const (3.10)



978 J. Snamina

Using Eqs. (3.8), (3.9) and (3.10), one can determine time derivatives of
the wave number and frequency along the characteristic curve. They have the
forms (dk

dτ

)

ch
=−∂ω

∂N

dN

dη

(dω
dτ

)

ch
=0 (3.11)

The above equations show that thewave number k varies and the frequen-
cy ω is conserved along the characteristic curve. If additionally the frequency
wave source is invariable, the wave frequency is constant at each point of the
cable. Thus the binormal wave can be expressed in a more convenient form

u3 = U3(η)exp
[
i
(
ωt− ψ3(η)

ε

)]
(3.12)

where U3(η) is the wave amplitude and ψ3(η) is a slowly varying component
of the wave phase depending on the spatial coordinate.
It is apparent that the amplitude U3(η) and the local wave number k3 are

slowly varying functions of the spatial co-ordinate η, but the displacement u3
varies due to fast oscillations aswell. In order to incorporate this requirement,
the phase of the wave is written in the form ωt− ε−1ψ3(η). The first term
provides the fast oscillation depending on time and the second one describes
the changes depending on the spatial co-ordinate. Using Eq. (3.4) and taking
into account the form of the phase function, the local wave number can be
calculated as follows

k3 =
dψ3
dη

(3.13)

The phase dependence on the spatial co-ordinate can also be expressed using
the function ϕ3(s). The following relation holds

ψ3(η) = εϕ3(s) (3.14)

It is easy to prove that the first derivative of ϕ3(s) with respect to s is equal
to the local wave number k3.
A great advantage of theWKBmethod in practical calculations is an easy

way of arranging terms in the expressions according to their rate of change.
Equation (3.12) describing thebinormalwave canbe substituted into the third
of Eqs. (2.1), in which the derivative of slowly varying cable tension N with
respect to s should be transformed in the following way

dN

ds
=
dN

dη

dη

ds
= N ′ε (3.15)

In theabove expressionand in the followingdescription, the sign (·)′ represents
the first derivative of the slowly varying function with respect to η.



The properties of coupled waves propagating... 979

By analogy, the first and second derivative of the function u3(s,t) with
respect to s should bewritten in terms of derivatives with respect to η. After
rearranging, they take the form

∂u3
∂s
=(U ′3ε− iU3k3)exp

[
i
(
ωt−

ψ3(η)

ε

)]

(3.16)

∂2u3
∂s2
= [U ′′3ε

2−U3k23 − i(2U ′3k3+U3k′3)ε]exp
[
i
(
ωt− ψ3(η)

ε

)]

After substituting Eqs. (3.12), (3.15) and (3.16) into the third of Eqs. (2.1),
the expression obtained can bewritten as a polynomial of ε. Equating the free
term and the coefficient at the first power of ε to zero, one obtains

µω2−Nk23 =0 2Nk3U ′3+U3(Nk3)′ =0 (3.17)

The above equations can be expressed in non-dimensional forms

ω̃2− Ñk̃23 =0 2Ñk̃3Ũ ′3+ Ũ3(Ñk̃3)′ =0 (3.18)

where the following dimensionless quantities

ω̃ = ω

√
L

g
k̃3 = k3L s̃ =

s

L

Ñ =
N

µgL
η̃ =

η

L
Ũ3 =

U3
L

(3.19)

are used.
Equation (3.18)1 is the dispersion relation for the binormalwave. Equation

(3.18)2 is thedifferential equation containing theunknownwaveamplitudeasa
function of slowly varying co-ordinate η̃. Using dispersion relation (3.18)1, one
can determine the dimensionless phase velocity ṽf3 (the frequency ω̃ divided

by the wave number k̃3) and the dimensionless group velocity ṽg3 (the first

derivative of ω̃ with respect to k̃3)

ṽf3 = ṽg3 =

√
Ñ (3.20)

These velocities are functions of the arc co-ordinate. Their plots are presented
in Fig.2 for two values of the horizontal component of the equilibrium cable
tensions: Ñ0 =0.3, Ñ0 =0.6.
Differential equation (3.18)2 is equivalent to the condition that the energy

of the binormal wave is conserved during propagation along the cable. Assu-
ming the boundary condition Ũ3(0)= Ũ30 (the amplitude in the lowest point



980 J. Snamina

Fig. 2. Phase velocity and group velocity versus non-dimensional arc co-ordinate s̃
for: Ñ0 =0.3 and Ñ0 =0.6

of the cable is equal to Ũ30), Eq. (3.18)2 can be solved with respect to the
amplitude of the wave as a function of the dimensionless arc co-ordinate s̃

Ũ3(s̃)= Ũ30
4

√√√√ Ñ0
Ñ(s̃)

(3.21)

Thewave number k̃3(s̃) and the amplitude Ũ3(s̃) are plotted inFig.3 (for the
amplitude in the lowest point Ũ30 =10

−4).

Fig. 3.Wave number and amplitude versus non-dimensional arc co-ordinate s̃ for:
ω̃ =100, Ñ0 =0.3 and Ñ0 =0.6

Thewave number and the amplitude become smaller when thewavemoves
away fromthe lowest point of the cable.Thesephenomenaaremore explicit for
lower values of the horizontal component of the cable tension, see the plots for
Ñ0 =0.3. The change of the wave number is much greater than the change of
the wave amplitude. Thus, the varying cable tension N has a bigger influence
on the wave number than on the amplitude of the binormal wave.



The properties of coupled waves propagating... 981

Using Eqs. (3.1), (3.12), (3.13) and (3.14), the binormal wave can be writ-
ten in the form

ũ3(s̃, t̃)= Ũ3(s̃)exp
[
i
(
ω̃t̃−

s̃∫

0

k̃3(s̃1) ds̃1
)]

(3.22)

Figure 4 shows the shape of thewave for Ñ0 =0.3 and ω̃ =100. It is easy
to notice the increase of the wavelength and the decrease of the amplitude.
Taking into account that the frequency of thewave is constant, the increase of
the wavelength results in an increase of the phase velocity, as shown in Fig.2.

Fig. 4. Plot of the travelling wave for: Ñ0 =0.3 and ω̃ =100

In accordance with previous considerations, the tangent- and the normal
waves can be assumed in the following forms

u1 = U1(η)exp
[
i
(
ωt−

ψ(η)

ε

)]
u2 = U2(η)exp

[
i
(
ωt−

ψ(η)

ε

)]
(3.23)

The waves have different amplitudes. Since the waves are coupled, the pha-
ses are described by the same expressions, but taking into account that the
functions U1(η), U2(η) describing the amplitudes are complex functions the
physical phases of the waves cannot be the same. Their difference can be con-
stant or a slowly varying function.
Relations (3.23) have to satisfy the first and the second of Eqs. (2.1).

Derivatives of displacements u1 and u2 with respect to time andwith respect
to the arc co-ordinate should be determined similarly to derivatives of the
displacement u3 presented in Eqs. (3.16). The derivatives of slowly varying
functions in Eqs. (2.1) should be replaced by the following derivatives with
respect to the co-ordinate η

dN

ds
= N ′ε

dκ

ds
= κ′ε

d(Nκ)

ds
=(Nκ)′ε (3.24)



982 J. Snamina

Substituting Eqs. (3.23) and (3.24) into the first and second equation of Eqs.
(2.1) and comparing the free terms on both sides and the expressions at ε1,
one obtains two systems of equations.Thefirst systemconsists of the following
algebraic equations

(EA0k
2+Nκ2−µω2)U1− i(N +EA0)κkU2 =0

(3.25)

i(N +EA0)κkU1+(EA0κ
2+Nk2−µω2)U2 =0

The second system includes two differential equations

(N +EA0)κU
′

1+(Nκ)
′U1− i[2NkU ′2+(Nk)′U2] = 0

(3.26)

(N +EA0)κU
′

2+EA0κ
′U2+i(2EA0kU

′

1+EA0k
′U1)= 0

Introducing the following non-dimensional quantities

k̃ = kL κ̃ = κL εg =
µgL

EA0
(3.27)

and using appropriate dimensionless quantities taken from (3.19), system
(3.25) can be written in the form

(k̃2+ Ñεgκ̃
2−εgω̃2)Ũ1− i(εgÑ +1)κ̃k̃Ũ2 =0

(3.28)

i(εgÑ +1)κ̃k̃Ũ1+(κ̃
2+ Ñεgk̃

2−εgω̃2)Ũ2 =0

The calculations were done for εg =0.00018.
The non-trivial solution to equations (3.28) exists if and only if the de-

terminant of the system is equal to zero. Two wave numbers k̃1, k̃2 can be
calculated from this condition. They have the following forms

k̃1 =

√√√√
κ̃2+

1

2
(εgÑ +1)

ω̃2

Ñ
− ω̃√

Ñ

√

2(εgÑ +1)κ̃2+
1

4
(1−εgÑ)2

ω̃2

Ñ
(3.29)

k̃2 =

√√√√
κ̃2+

1

2
(εgÑ +1)

ω̃2

Ñ
+

ω̃
√
Ñ

√

2(εgÑ +1)κ̃2+
1

4
(1−εgÑ)2

ω̃2

Ñ

The above expressions describe the dispersion properties of two coupledwaves
travelling along the cable in thepositive direction of the arc co-ordinate (which
defines the positive direction of wave propagation). It is apparent that the



The properties of coupled waves propagating... 983

second wave number k̃2 is a positive real number for each frequency of the
wave, whereas the first wave number k̃1 can be a positive real number or an
imaginary number depending on the frequency ω̃ and arc co-ordinate s̃. The
results of the wave number calculation are presented in Figs. 5, 6, 7 and 8.
Figures 5, 6 show the wave numbers as functions of the frequency (for s̃ =0).
Figures 7, 8 illustrate the wave numbers as functions of the arc co-ordinate s̃.

Fig. 5. The wave number k̃1 versus frequency (s̃ =0 and Ñ0 =0.3, Ñ0 =0.6)

Fig. 6. The wave number k̃2 versus frequency (s̃ =0 and Ñ0 =0.3, Ñ0 =0.6)

When the curvature of the cable approaches zero, the coupling between
waves vanishes and the wave number k̃1 approaches the wave number of lon-
gitudinal waves in a slender rod whereas the wave number k̃2 approaches the
wave number of the transverse wave in a taut string. Hence, it can be conclu-
ded that the wave number k̃1 is associated with the longitudinal-dominant
pair of waves and k̃2 is associated with the transverse-dominant pair of
waves.

The results of calculations show that the waves belonging to the
transverse-dominant pair are dispersive, whereas the waves belonging to the
longitudinal-dominantpair canbedispersive in thepassband frequencyranges



984 J. Snamina

Fig. 7. The wave number k̃1 versus non-dimensional arc co-ordinate (ω̃ =100 and

Ñ0 =0.3, Ñ0 =0.6)

Fig. 8. The wave number k̃2 versus non-dimensional arc co-ordinate (ω̃ =100 and

Ñ0 =0.3, Ñ0 =0.6)

ω̃ ∈ (0; ω̃g1)∪(ω̃g2;∞) ornon-propagating (exponentiallydecayingwithdistan-
ce) in the stop band frequency range ω̃ ∈ (ω̃g1; ω̃g2). The cut-off frequencies
ω̃g1, ω̃g2 can be calculated using Eq. (3.29)1. Their non-dimensional form is
given by the following formulas

ω̃g1 = κ̃

√
Ñ ω̃g2 = κ̃

1
√
εg

(3.30)

Figure 9 shows the graphs of the cut-off frequencies as functions of the arc
co-ordinate.

Since the cut-off frequencies depend on the arc co-ordinate, there exists
a neighbourhood of the lowest point of the cable where the longitudinal-
dominant pair consists of two waves decaying exponentially. At points laying
outside this neighbourhood, the same longitudinal-dominant pair consists of
two dispersive waves.



The properties of coupled waves propagating... 985

Fig. 9. Cut-of frequencies versus non-dimensional arc co-ordinate (Ñ0 =0.3,

Ñ0 =0.6)

Using Eqs. (3.29), one can calculate the phase and group velocity for each
pair of waves. The results of such calculations performed for Ñ0 = 0.3 are
presented inFigs. 10 and 11. For the longitudinal-dominant pair, the velocities
have been calculated assuming the wave frequency greater than the cut-off
frequency ω̃g2.

Fig. 10. The phase and group velocity of longitudinal-dominant waves versus
frequency (s̃ =0, Ñ0 =0.3)

Analysing the graphs, it is apparent that the group velocity is lower than
the phase velocity for the longitudinal-dominant pair (normal dispersion) and
higher than the phase velocity for the transversal-dominant pair (anomalous
dispersion).

In the case of shortwaves (high frequencies), the groupvelocity approaches
the phase velocity for both transversal-dominant and longitudinal-dominant
pairs, see Figs. 10 and 11.Therefore, the dispersionphenomenon can bebetter
observed for lower than for higher frequencies. The difference between the
group- and phase frequencies diminishes when the waves move away from the
lowest point of the cable, as shown in Figs. 12 and 13. The velocities of the



986 J. Snamina

Fig. 11. The phase and group velocity of transverse-dominant waves versus
frequency (s̃ =0, Ñ0 =0.3)

Fig. 12. The phase and group velocity of longitudinal-dominant waves versus
non-dimensional arc co-ordinate (Ñ0 =0.3, ω̃ =400)

Fig. 13. The phase and group velocity of transversal-dominantwaves versus
non-dimensional arc co-ordinate (Ñ0 =0.3, ω̃ =15)



The properties of coupled waves propagating... 987

longitudinal-dominant pair are significantly greater than the corresponding
velocities of the transversal-dominant pair.
Taking into account Eqs. (3.23) and the results of the wave number calcu-

lation (Eqs. (3.29)), the formulas describing the displacements of cable points
in the tangent and normal directions can be written in the form

u1 = U11exp
[
i
(
ωt−

s∫

0

k1(s1) ds1
)]
+U22δ12exp

[
i
(
ωt−

s∫

0

k2(s1) ds1
)]

(3.31)

u2 = U11δ21 exp
[
i
(
ωt−

s∫

0

k1(s1) ds1
)]
+U22exp

[
i
(
ωt−

s∫

0

k2(s1) ds1
)]

The displacements are linear combinations of two waves travelling along
the cable in the positive direction. The waves have the same frequencies and
a different length.
InEqs. (3.31),wehave taken intoaccount theprevious conclusion that each

tangentwave (with anamplitude equal to U11) is coupled to the corresponding
normal wave (with an amplitude equal to δ21U11). Both waves have the same
frequencyandwavenumber.Byanalogy, eachnormalwave (with anamplitude
equal to U22) is coupled to the corresponding tangentwave (with an amplitude
equal to δ12U22).
The coefficient δ21 is the amplitude ratio in the longitudinal-dominant

pair, and δ12 in the transversal-dominant pair. They can be determined from
any of Eqs. (3.28). Their final expressions are as follows

δ12 = i
κ̃2+εgÑk̃

2
2 −εgω̃2

(εgÑ +1)k̃2κ̃
δ21 =−i

k̃21 +εgÑκ̃
2−εgω̃2

(εgÑ +1)k̃1κ̃
(3.32)

Taking into account that both the above coefficients are imaginary, the diffe-
rences between wave phases in each pair are equal to π/2.
Graphs of modules of the ratios δ21 and δ12 are shown in Figs. 14 and 15.

They are goodmeasures of thewave coupling in each pair. Using thesemeasu-
res, it can be asserted that waves are not coupled, if the ratios |δ21|, |δ12| are
less than the assumed limit value δm. The discussion about δm is associated
with the aim and accuracy of calculations. Taking into account Eqs (3.32) and
(3.29), the limit values for the horizontal component of the cable tension N0
or curvature κ can be determined. The calculations were done for Ñ0 = 0.3
and Ñ0 =0.6. These values of the horizontal component of the cable tension
correspond to the cable curvature (at s̃ =0)which is equal to κ̃(0)∼=3.3 and
κ̃(0)∼=1.7.



988 J. Snamina

Fig. 14. The modulus of amplitude ratio in longitudinal-dominant pair of waves
(Ñ0 =0.3, Ñ0 =0.6), (a) for s̃ =0, (b) for ω̃ =400

Fig. 15. The modulus of amplitude ratio in transversal-dominant pair of waves
(Ñ0 =0.3, Ñ0 =0.6), (a) for s̃ =0, (b) for ω̃ =15

The system of differential equations (3.26) has an integral that is associa-
ted with the energy of a pair of waves. In order to determine this integral, the
following transformations are proposed. The first equation is pre-multiplied
by (iU2), the second equation is pre-multiplied by (iU1) and the resultant
equations are added.The expression on the left-hand side of the obtained equ-
ation can be rearranged to give the derivative of the double energy of coupled
waves with respect to the arc co-ordinate. Since the energy is conserved, the
derivative is equal to zero. Taking into account the non-dimensional quanti-
ties defined so far (relationships (3.19), (3.27)) and denoting (iŨ1) as Ũ

∗

1 , the
integral of system (3.26) can be expressed in the following form

k̃(Ũ∗1)
2+ Ñεgk̃Ũ

2
2 + κ̃(1+ Ñεg)Ũ

∗

1Ũ2 = const (3.33)

Equation (3.33) is the basic equation used in calculations of wave amplitudes.
Additionally, in order to determine the amplitudes, Eqs. (3.29) and relation-
ships (3.32) should be taken into account.



The properties of coupled waves propagating... 989

The results of such calculations are shown in Figs.16 and 17. The am-
plitudes of waves in the longitudinal-dominated pair are presented in Fig.16,
whereas the amplitudes of waves in the transversal-dominated pair are shown
in Fig.17.

Fig. 16. Amplitudes of waves in the longitudinal-dominant pair versus
non-dimensional arc co-ordinate, for ω̃ =1000, Ñ0 =0.3 and Ñ0 =0.6

Fig. 17. Amplitudes of waves in the transversal-dominant pair versus
non-dimensional arc co-ordinate, for ω̃ =100, Ñ0 =0.3 and Ñ0 =0.6

The calculations for the longitudinal-dominated pair have been done for
ω̃ =1000, i.e. the frequency higher than the cut-off frequency ω̃g2.

It follows from the calculations that the amplitudes of both waves in each
pair diminish when the waves move away from the lowest point of the cable.
The amplitude of the dominant wave in each pair diminishes slower than the
amplitudeof thewave coupled to it.Thedifferences between thevelocity of the
normal wave in the transverse-dominant pair and the velocity of the binormal
wave are small as shown in Fig.2 and Fig.13.



990 J. Snamina

4. Conclusions

A suspended elastic cable is a non-uniform wave medium. The features of
this medium can be described accounting for the curvature and tension of the
sagged cable. They are slowly varying functions of the arc co-ordinate. In this
case, the WKB method is an effective tool to derive the dispersion relations
and amplitude equations. The wavelengths and the amplitude vary along the
cable.

The curvature causes that the waves travelling in the plane of the equili-
brium line are coupled. There are two distinct pairs of coupled waves propa-
gating along the cable – the longitudinal-dominant and transverse-dominant
pair. When the curvature tends to zero, the coupling gets smaller and the
model simplifies to well known models of lateral waves in a taut string and
longitudinal waves in an elastic rod. The coupling of waves is very weak when
the frequency tends to infinity or when the waves move away from the lowest
point of the cable.

The transverse-dominant pair of waves is dispersive, whereas the
longitudinal-dominant pair can be dispersive in two pass band frequency ran-
ges or exponentially decaying in the stop band frequency range. Two cut-off
frequencies depend on the curvature and the arc co-ordinate. Since the cut-off
frequencies depend on the arc co-ordinate, the longitudinal-dominant pair of
waves can exponentially decay in the neighbourhood of the lowest point of the
cable and, for the same frequency, can be dispersive at points with a longer
distance from the lowest point.

The curvature does not influence the waves associated with the displace-
ment perpendicular to the plane of the equilibrium line. These waves are not
coupled with the in-plane waves.

The amplitudes of waves in each pair diminishwhen the waves move away
from the lowest point of the cable. The amplitude of the dominant wave in
each pair diminishes slower than the amplitude of the wave coupled with it.

The method applied in the present calculations can also be used in the
analysis of wave motion appearing in other continuous systems.

References

1. Burgess J.J., Triantafyllou M.S., 1998, The elastic frequencies of cables,
Journal of Sound and Vibration, 120, 153-165



The properties of coupled waves propagating... 991

2. Irvine H.M., Caughey T.K., 1974, The linear theory of free vibrations of a
suspended cable,Proceedings of the Royal Society, London,A341, 299-315

3. Luongo A., Rega G., Vestroni F., 1984, Planar non-linear free vibrations
of an elastic cable, International Journal of Non-Linear Mechanics, 19, 39-52

4. Perkins N.C., Behbahani-Nejad M., 1995, Forced wave propagation in
elastic cableswith small curvature,ASME, Design Engineering Technical Con-
ferences, 3 – Part B, 1457-1464

5. Perkins N.C., Behbahani-Nejad M., 1996, Freely propagating waves in
elastic cables, Journal of Sound and Vibration, 2, 189-202

6. Perkins N.C., Mote C.D., 1986, Comments on curve veering in eigenvalue
problems, Journal of Sound and Vibration, 106, 451-463

7. Perkins N.C., Mote C.D., 1987 Three-dimensional vibration of travelling
elastic cables, Journal of Sound and Vibration, 114, 325-340

8. Rega G., Vestroni F., Benedittini F., 1984, Parametric analysis of large
amplitude free vibrations of a suspended cable, International Journal of Solids
and Structures, 20, 95-105

9. Srinili N., Rega G., Chucheepsakul S., 2004, Three-dimensional non-
linear coupling and dynamic tension in the large-amplitude free vibrations of
arbitrarily sagged cables, Journal of Sound and Vibration, 269, 823-852

10. Yang H., 1990, Wave Packets and their Bifurcations in Geophysical Fluid
Dynamics, Springer Verlag

11. YoungR., 2002,Wave interactions in nonlinear elastic strings,Arch. Rational
Mech. Anal., 161, 65-92

12. Whitham G.B., 1999, Linear and Nonlinear Waves, John Wiley and Sons
Inc., NewYork

Własności sprzężonych fal rozprzestrzeniających się w linach o znacznej

długości

Streszczenie

W pracy przedstawiono analizę własności fal mechanicznych rozprzestrzeniają-
cych się w linach o znacznej długości.W liniowej teorii fale rozchodzące się w płasz-
czyźnie zwisu są ze sobą sprzężone, a fala wywołująca przemieszczenia punktów liny
w kierunku prostopadłym do płaszczyzny zwisu rozprzestrzenia się niezależnie od
pozostałych fal.W rozważaniach uwzględniono wolno-zmienną zależność siły osiowej
i krzywizny od współrzędnej łukowej, w położeniu równowagi statycznej liny. Za-
gadnienia dotyczące przemieszczających się wzdłuż przewodu fal rozwiązanometodą



992 J. Snamina

WKB(Wentzel-Kramers-Brillouin).Wyznaczonozwiązkidyspersyjne charakteryzują-
ce ruch falowyw linach, wyprowadzonoprędkości fazowe i grupowe z uwzględnieniem
sprzężenia fal rozchodzących się w płaszczyźnie zwisu. Analizie poddano zależność
liczb falowych oraz amplitud od współrzędnej łukowej.W rozważaniachwykorzysta-
no symetrię ruchu fal.Wyniki obliczeń zostały zobrazowane na szeregu wykresach.

Manuscript received March 28, 2008; accepted for print August 6, 2008