Faculty of Civil Engineering and Geosciences

Section of Structural Mechanics

Dynamics of Structures CT4140

Part 2

Wave Dynamics

A.V. Metrikine

A.C.W.M. Vrouwenvelder

Delft University of Technology

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iii

It is not sufficient to have clear and true ideas. To impart them to the others,

one must be able to express them clearly.

Claude Adrien Helvtius

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Preface

The purpose of these lecture notes is to present basic ideas of the theory of elastic wave propagation in structural elements. Only one-dimensional elements such as cables, rods and beams are considered in this development. The intent to consider these elements has been twofold. On the one hand, it is hard to imagine a structural element that is used in Civil Engineering more often than a cable or a beam. On the other hand, the mathematical description of the wave dynamics of cables and rods is relatively simple and can be presented by employing basic mathematical methods only.

The notes are organised into five lectures and exercises. The first two lectures are concerned with wave motion of strings (cables). Such a motion is most easy to imagine for everyone who has ever seen a guitar or the wires that supply trains and trams with electric energy. Making use of the string model, we introduce the wave equation, discuss the wave propagation in non-dispersive media and study the wave reflection at different boundaries. The third and the fourth lectures cover some aspects of longitudinal wave motion in rods. Along with theoretical study of propagation and reflection of compressional waves in thin rods, two models for pile-soil interaction are introduced in these lectures. On the hand of one of these models that accounts for a distributed pile-soil interaction, the waves in dispersive media are discussed and basic properties of the harmonic waves are demonstrated. In the last lecture bending waves in a beam on elastic foundation are analysed. The lecture material is supplemented by exercises. A number of these exercises are given with solutions, whereas the other ones are meant to challenge the students and contain no solutions. The lecture notes are concluded with a list of notations and the bibliography that may be used for extensive studies of wave phenomena in solids.

These lecture notes are based on the course of lectures "Dynamica van Constructies, Deel 2" that, during many years, was delivered by Dr.ir. H.A. Dieterman who has untimely passed away in 1998. We gratefully acknowledge his work, which helped us significantly in preparing the present development.

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Contents

1. Lecture 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Wave phenomena in solids and structures (general) . . . . . . . . . . . . . . . . . . . . 1 1.1.1. Waves around us (sound, water waves, radio waves, etc.) . . . . . . . . . . 1 1.1.2. Applications of the wave theory in civil engineering . . . . . . . . . . . . . . 2 1.1.3. Historical background of the studies on wave

and vibration phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2. Transverse waves in long strings (guitar strings, electric cables, elevator

cables, etc.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1. Wave equation for transverse motion of a string. . . . . . . . . . . . . . . . . 6 1.2.2. General (DAlembert) solution to the wave equation

in the form of two counter-propagating waves. . . . . . . . . . . . . . . . . . . 8 1.2.3. Excitation of waves in the strings by impact

and by initial displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Lecture 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1. Transverse waves in long strings (second part) . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1. Geometrical representation of wave propagation

on characteristic plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2. Wave reflection at boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1. Main types of the boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2. Reflection from a fixed boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

The method of images Representation of reflection in the characteristic plane Wave particle analogy

2.2.3. Reflection from a free end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.4. Reflection from an elastic boundary . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.5. Reflection from a viscous boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Non-reflective boundary element

3. Lecture 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1. Longitudinal waves in thin long rods (piles) . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.1. Governing equation for longitudinal motion of a rod . . . . . . . . . . . . . 31 3.1.2. DAlembert solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.3. Particle velocity and the wave speed . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2. Reflection of waves at boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.1. Reflection from a fixed end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.2. Reflection from a free end . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3. Waves in a finite-length rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.1. Stress pulse in a free-fixed rod (impact against a pile) . . . . . . . . . . . 38 3.3.2. Stress pulse in a free-free rod; how do we move things? . . . . . . . . . 40

3.4. Transmission and reflection at interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.1. Transmission and reflection at a junction of two rods . . . . . . . . . . . . 42

The wave impedance

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4. Lecture 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1. Pile-soil interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.1. Local model for pile-soil interaction . . . . . . . . . . . . . . . . . . . . . . . . . 45

Boundary conditions Non-reflective boundary element

4.1.2. Distributed model for pile-soil interaction . . . . . . . . . . . . . . . . . . . . . 48 Equation of motion Failure of the DAlemberts solution

4.2. Harmonic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2.1. Amplitude, frequency and wave number of a harmonic wave . . . . . 49 4.2.2. Complex representation of a harmonic waves . . . . . . . . . . . . . . . . . . 50 4.2.3. Wave dispersion, dispersion equation, dispersion plane

and dispersion curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.4. Phase and group velocity of harmonic waves . . . . . . . . . . . . . . . . . . 52

4.3. Harmonic waves in distributed model for pile-soil interaction . . . . . . . . . . . 56 4.3.1. Harmonic excitation of a semi-infinite rod . . . . . . . . . . . . . . . . . . . . 56

Cut-off frequency 4.3.2. Reflection of harmonic waves at boundaries . . . . . . . . . . . . . . . . . . . 59

Frequency dependent non-reflective boundary element 4.3.3. Effect of distributed damping on the forced motion of the pile . . . . . 63

5. Lecture 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.1. Flexural waves in a railway track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1.1. Equation of motion for a beam on elastic foundation . . . . . . . . . . . . 68 5.1.2. Excitation of waves in the beam by a harmonic load . . . . . . . . . . . . . 69

Dispersion equation and dispersion curve Wavenumbers of excited waves Condition of resonance

5.2. Dynamic response of the beam to a harmonic load . . . . . . . . . . . . . . . . . . . 73

6. Problems with solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7. Problems for self-study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8. Notations (English and Dutch) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9. Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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LECTURE 1

1.1. Wave phenomena in solids and structures

Waves around us (sound, water waves, radio waves, etc.) - Applications of the wave theory in civil engineering - Historical background of the studies on

wave and vibration phenomena

1.1.1. Waves around us.

The effect of a sharply applied, localised disturbance in a medium soon transmits or spreads to other parts of the medium. This simple fact forms a basis for study of the fascinating subject known as wave propagation. The manifestations of this phenomenon are familiar to everyone in forms such as the transmission of sound in air, the spreading of ripples on a pond of water, the transmission of seismic tremors in the earth, or the transmission of radio waves. These and many other examples could be cited to illustrate the propagation of waves through gaseous, liquid, and solid media and vacuum.

Waves of any physical nature propagate with a finite speed so that no signal can be transmitted from one point to a different point instantaneously. This transmission always takes a non-zero time, although this time crucially depends on the type of wave that carries the signal. This can be seen from the table below that present some of the wave speeds

Wave Type Wave Speed (m s-1) Sound waves in the air 330 Elastic waves in steel 5000 Elastic waves in concrete 3000 Radio waves (the light speed) 300 000 000 Surface water waves 0.025 0.5 Waves in power lines for trains 100 Elastic waves in soil 60 300

This table shows, for example, that ripples perturbed by a stone dropping into water would need about two seconds to travel one meter along the water surface. The same time would be more than sufficient for the light to travel from the Earth to the Moon. There is quite a big difference between 1m and the Earth-Moon distance, isnt it?

Despite a great deal different physical mechanisms that govern propagation of the acoustic, elastic and electromagnetic waves, all these waves have a lot in common. They all are defined by the interdisciplinary characteristics such as the wave speed, frequency, wavelength, wave amplitude, etc. Waves are always reflected and transmitted at interfaces between two media, they can propagate or form a standing pattern, decay or be amplified in space and in time, interact or ignore each other while passing and so on. Such an impressive amount of common features of waves of different physical nature is explained by the fact that all waves are governed by similar mathematical equations. The simplest but the most important one among these

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equations is the wave equation, which will be studied in this course that is dedicated to waves in elastic structures.

Studying waves in elastic structures, we deal with waves in solids. The physical basis for the wave propagation through a solid ultimately lies in the interaction of the discrete atoms. Solid mechanics, however, does not look that deep into the material structure, but considers the medium as continuous, so that properties such as density or elastic constants are assumed to be continuous functions representing averages of microscopic quantities. Nevertheless, in envisaging the basis for propagation of a mechanical disturbance it is helpful to first consider a model composed of discrete masses and springs, see Figure 1.1. If a disturbance is imparted (by a hammer) to a mass particle, it is transmitted to the next mass by the intervening

t = 0

t = t1 > 0

Undisturbed chain of masses and springs

Impact

Undisturbed partDisturbed part

t = t2 > t1

Disturbed chain

M

K d

Figure 1.1. Chain of masses and springs perturbed by an impact

spring. In this manner the disturbance is soon transmitted to a remote point, although any given particle of the system will have moved only slightly. The role of the mass and stiffness parameters in affecting the speed of the disturbance propagation is quite clear in such a model. If the stiffness of the connecting springs is increased or the particle masses decreased, or both, the speed of propagation would be expected to increase. So it is in the case of a continuous medium. The mass and elastic parameters are now expressed in terms of the mass density and the elastic moduli and the disturbance is passed through the system because of mutual interaction of differential elements of the continuum.

In this development, attention will be focused on propagation of waves in structural elements such as cables, piles and rails. These elements are often used in civil and mechanical engineering and, therefore, studying their dynamic behaviour is of practical importance. On the other hand, mathematical models for these elements are relatively simple and allow to gain insight into the fundamental wave phenomena without involving sophisticated mathematical treatments.

1.1.2. Applications of the wave theory in civil engineering.

The practical applications of wave phenomena surely go back to the early history of man. The shaping of stone implements, for example, consists of striking sharp, carefully placed blows along the edges of a flint. The resulting stress waves in the

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cone of percussion break out fragments of rock in very specific patterns. Starting at this early time, it may be safely said that interest in wave phenomena has been increasing ever since.

Nowadays, there exist an impressive number of principal applications of the wave theory in science and industry. The applications in civil engineering are, probably, the most lively among the others. Indeed, civil engineers employ the wave theory to

predict structural response to earthquakes, see Figure 1.2:

Figure 1.2. Propagation of tremors generated by earthquake

estimate and reduce the level of vibrations that are produced by construction works, see Figure 1.3:

Figure 1.3. Propagation of ground waves perturbed by piling towards a block of flats

improve sound isolation in buildings and acoustics of concert halls;

reduce the level of vibrations generated by high-speed trains in the ground and in the catenary system; ensure that the generated waves do not destabilise vibrations of the train and the current collector, see Figure 1.4:

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Figure 1.4. Waves in rails, ground and catenary generated by a high-speed train.

perform ultrasonic inspection of materials (NDT), see Figure 1.5:

Steel plate Crack

Radiator Receiver

Figure 1.5. Inspection of a crack by ultrasonic wave.

There are many more applications of the elastic waves. One can recall here the ultrasonic delay lines that are used in electronics; waves in a rock that are strongly effecting the percussive drilling, the acoustic emission that enables an easy inspection of pipelines, etc., etc. It is possible to mention plenty of other applications of the wave theory that are related to visco-elastic and plastic waves that occur as soon as any peace of material is formed. This is a very interesting subject that lies, however, far beyond the scope of this development.

1.1.3. Historical background of the studies on wave and vibration phenomena.

First studies on wave and vibration phenomena go back hundreds of years. Most early studies were naturally more observational than quantitative and frequently were concerned with musical tones or water waves, two of the most common associations with wave motion. From the time of Galileo onward, the science of vibrations and waves progressed rapidly in association with developments in the theory of solids. Some of the major developments in the area over the years are chronologically ordered in the following list (after Karl F. Graaf Wave motion in elastic solids).

Sixth Century BC: Pythagoras studied the origin of musical sounds and the vibrations of strings.

1636: Mersenne presented the first correct published account on the vibrations of strings.

1638: Galileo described the vibrations of pendulums, the phenomenon of resonance, and the factors influencing the vibrations of strings.

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1678: Robert Hooke formulated the law of proportionality between stress and strain for elastic bodies. This law forms the basis for the static and dynamic theory of elasticity.

1686: Newton investigated the speed of water waves and the speed of sound in air. 1700: Sauveur calculated vibrational frequency of a stretched string. 1713: Taylor worked out a complete, dynamical solution for the vibrations of a string. 1744: Leonard Euler (1744) and Daniel Bernoulli (1751) developed the equation for

the vibrations of beams and obtained the normal modes for various boundary conditions.

1747: DAlembert derived the equation of motion of the string and solved the initial-value problem.

1755: Daniel Bernoulli developed the principle of superposition and applied it to the vibrations of strings.

1759: Lagrange analysed the string as a system of discrete mass particles. 1766: Euler attempted to analyse the vibrations of a bell on the basis of the behaviour

of curved bars. James Bernoulli (1789) also attempted analysis of this problem

1802: E.F.F. Chladni reported experimental investigations on the vibrations of beams and on the longitudinal and torsional vibrations of rods.

1815: Madame Sophie Germain developed the equation for the vibrations of a plate. 1821: Navier investigated the general equations of equilibrium and vibration of

elastic solids. Although not all of the developments of the work met with complete acceptance, it represented one of the most important developments in mechanics.

1822: Cauchy developed most of the aspects of the pure theory of elasticity including the dynamical equations of motion for a solid. Poisson (1829) also investigated the general equations.

1828: Poisson investigated the propagation of waves through an elastic solid. He found that two wave types, longitudinal and transverse, could exist. Cauchy (1830) obtained a similar result.

1828: Poisson developed approximate theories for the vibrations of rods. 1862: Glebsch founded the general theory for the free vibrations of solid bodies using

normal modes. 1872: J.Hopkinson performed the first experiments on plastic wave propagation in

wires. 1876: Pochhammer obtained the frequency equation for the propagation of waves in

rods according to the exact equations of elasticity. Chree (1889) carried out similar studies.

1880: Jaerisch analysed the general problem of the vibrations of a sphere. The result was obtained independently by Lamb (1882).

1882: Hertz developed the first successful theory for impact. 1883: St. Venant summarised the work on impact of earlier investigators and

presented his results on transverse impact. 1887: Rayleigh investigated the propagation of surface waves in a solid. 1888: Rayleigh and Lamb (1889) developed the frequency equation for waves in a

plate according to exact elasticity theory. 1904: Lamb made the first investigation of pulse propagation in a semi-infinite solid. 1911: Love developed the theory of waves in a thin layer overlying a solid and

showed that such waves accounted for certain anomalies in seismogram records.

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1914: B. Hopkinson performed experiments on the propagation of elastic pulses in bars.

1921: Timoshenko developed a theory for beams that accounted for the shear deformations.

1930: Donnell studied the effect of a non-linear stress-strain law on the propagation of stress waves in a bar.

1942: von Karman, Taylor and Rakhmatulin developed a one-dimensional finite-amplitude plastic wave theory.

1949: Davies published an extensive theoretical and experimental study on waves in bars.

1951: Mindlin presented an approximate theory for waves in a plate that provided a general basis for development of higher-order plate and rod theories.

1951: Malvern developed a rate-dependent theory for plastic wave propagation. 1955: Perkeris presented the solution to Lambs problem of pulse propagation in a

semi-infinite solid.

Developments in elastic wave theory did not, of course, cease in 1955. The date only represents the desire not to offend more recent significant contributions to the field through inadvertent omission from a mere listing.

1.2. Transverse waves in long strings

Wave equation for a string - General (DAlembert) solution to the wave equation in the form of two counter-propagating waves - Excitation of waves

in the strings by impact and by initial displacement

Transverse motion of strings is one of the easiest examples of wave motion to visualise both in reality and in interpretation of various solutions to the governing equation. While it is sufficient to rest the case of analysing the taut strings on mathematical grounds alone, it should be appreciated that practical motivations also exist. The characteristics of many musical instruments are based on the vibrations of strings. The dynamics of electric transmission lines as well as catenaries (overhead wires supplying trains with electricity) may be modelled using the strings. Not to forget is a wide application of the string model for predicting the thread manufacture process.

1.2.1. Wave equation for a string motion.

Let us obtain a governing equation for the taut string. Since boundaries inevitably introduce complications in wave propagation due to the reflection phenomenon, the first considerations will involve long strings, that is, infinite strings.

Consider a differential (small) element of taut string under tension T as shown in Figure 1.6. It is assumed that any variation in the tension due to the displacement of the string as well as the gravity force is negligible. The mass density of the string material is and the cross-sectional area of the string is A. Suppose that the string element is subjected to a distributed vertical load f x t,b g, which is small enough to cause displacement of the string w x t,b g satisfying the following inequality:

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FHGIKJ

8

A w x tt

Tdx

w x dx tx

w x t

xf x t

=

+

FHG

IKJ +

2

2

, , ,

,

b g b g b g b g . (1.4)

By definition,

lim, , ,

dx dxw x dx t

x

w x t

x

w x t

x

+

FHG

IKJ =

0

2

21 b g b g b g

.

Therefore, equation (1.4) for the string transverse motion takes the form

A w x tt

Tw x t

xf x t

=

+2

2

2

2

, ,

,

b g b g b g. (1.5)

Of particular interest is the homogeneous form of equation (1.5) obtained by setting f x t,b g = 0. This form reads

=

=

2

22

2

2

w x t

tc

w x t

xc T A

, ,

, .

b g b g b gwhere (1.6)

Equation (1.6) governs transverse motion of the string in the case that no external forces are applied and is known as the wave equation. The constant c is usually referred to as the wave speed of transverse waves in the string.

1.2.2. General (DAlembert) solution to the wave equation in the form of two counter-propagating waves

It is possible to derive a general solution to the wave equation (1.6) in several ways. At this stage, however, it is most suitable to study the classical solution by DAlembert (1747) that provides considerable insight into wave propagation phenomena.

To obtain this solution, we introduce the new variables

,x ct x ct = = + (1.7)

Using the chain differentiation rule, the following equalities are obtained:

,

.

w w w w w

x x x

w w w w w w wc c c

t t t

= + = +

= + = + = (1.8)

Analogously, the second derivatives can be expressed as

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2

2

2 2 2

2 2

2

2

2 ,

w w w w w w

x x x x x x

w w w w w w w

w w w w w wc c c

t t t t t t

= = + = + =

+ + + = + +

= = = =

2 2 22 2 2

2 22 .w w w w w w w

c c c

= +

(1.9)

Substitution of (1.9) into the wave equation (1.6) yields

( )2 , 0w

=

(1.10)

This may be integrated directly to give

( ) ( ),w F

=

and then

( ) ( ) ( ), ,w f f + = + (1.11)

where f + and f are arbitrary functions. Finally, returning to the original variables x and t , we obtain the classical

DAlembert solution to the wave equation,

( ) ( ) ( ), .w x t f x ct f x ct+ = + + (1.12)

Considering solution (1.12), one may note that

Functions f + and f represent propagating waves ( f + is a wave propagating in the positive x direction and f is a wave propagating in the negative x direction).

Both f + and f propagate with the same wave speed c . f + and f are arbitrary functions of their arguments; a particular shape of these

functions is defined by the initial conditions or forcing function of a given problem.

Whatever the shape of the waves f x ct+ b g and f x ct +b g initially is, that shape is maintained during the propagation. Thus, the waves propagate without distortion.

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1.2.3. Excitation of waves in the strings by impact and by initial displacement

We now wish to determine the form of the functions arising in the general solution (1.12) under prescribed initial conditions.

Impact excitation. Consider an infinitely long string that undergoes an impact as it is shown in Figure 1.7a (the initial velocity of the string is depicted in Figure 1.7.b).

Figure 1.7a. String under impact excitation.

Figure 1.7b. Initial velocity of the string.

The term impact excitation means that initially (at 0t = ) the string displacement is zero but the vertical velocity of a certain part of the string has a non-zero value. Accordingly, the initial conditions (the vertical displacement and the vertical velocity of the string at 0t = ) for the case at hand can be written as

( )( ) ( )

0

00

, 0,

0,,1,

t

t

w x t

x xw x tx v

t x x

=

=

=

> = = <

(1.13)

To find the string shape as a function of the time t and spatial co-ordinate x , we substitute the DAlemberts solution (1.12) into the initial conditions (1.13). This yields

f x ct f x ct

tf x ct f x ct x

t t

t

+

=

=

+

=

+ + =

+ + =

b g b gb g b gc h b g

0 0

0

0,

. (1.14)

Taking into account that

w x t,0b g

x x=0

2x

v0

T T x

w(x,t) 2x

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+ + =

+ +

FHG

IKJ =

=

+

FHG

IKJ =

+

+

=

+

=

= = +

+

=

+

tf x ct f x ct f x ct

t

f x ctt

cf x ct

cf x ct

cf x

xc

f xx

t t

x ct x ct

t

b g b gc h b g b g

b g b g b g b g0 0

0

,

,

(1.15)

the system of equations (1.14) can be rewritten as

f x f x

cf x

xc

f xx

x

+

+

+ =

+

=

b g b gb g b g b g

0,

. (1.16)

From the first equation of (1.16) it follows that f x f x += b g b g. Inserting this relation into the second equation of (1.16), we obtain the following ordinary differential equation:

=

=

+ +

22

cf x

xx

f xx

x

c

b g b g b g b g (1.17)

Since the string is assumed to be infinitely long, the solution to equation (1.17) may be written in the following form:

f xc

z dz xx

+

= =zb g b g b g12 . (1.18)

If a particular shape of the initial velocity of the string were not known, we would have to stop here and write the string response to an impact in the following general form

w x t f x ct f x ct x ct x ct,b g b g b g b g b g= + + = ++ . (1.19)

For the particular form of the initial velocity that is given by expression (1.13), we can specify ( )x by evaluating the integral in (1.18). The result of this evaluation reads

f x x vc

x x

x x x x x

x x x

+= =

<

+ <

RS|T|

b g b g

0

2

0

2

,

,

,

(1.20)

Therefore, the string response to the impact that is depicted in Figure 1.7, is governed by equation (1.19) with ( )x given by expression (1.20). Corresponding patterns of the string are shown in Figure 1.8 for five consecutive time instants (perturbed domain is shaded).

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Figure 1.8. String patterns in the case of impact excitation of the rectangular form shown in Figure 1.7b.

Excitation by initial displacement. Consider a string whose initial shape is shown in Figure 1.9.

Figure 1.9. Initial displacement of the string.

Assuming that at 0t = the string is not given any initial velocity but displaced in the manner that is shown in Figure 1.9, the initial conditions to the problem may be written as

w x,0b g

x x = 0

2x w0

t = 0

t = x/2c

t = x/c

t = 3x/2c

t = 2x/c

00.5w v c=

0w v c=

0w v c=

x

w (x,t)

x -x

0w v c=

13

( ) ( )( )

00

0

0,, ,

1,

,

0.

t

t

x xw x t x w

x x

w x t

t

=

=

> = =

<

=

(1.21)

Substitution of the DAlemberts solution (1.12) into the initial conditions (1.21) yields

f x ct f x ct x

tf x ct f x ct

t t

t

+

=

=

+

=

+ + =

+ + =

b g b g b gb g b gc h

0 0

00

,

.

(1.22)

Using equation (1.15) which reads

+ + =

+

+

=

+

tf x ct f x ct c f x

xc

f xxt

b g b gc h b g b g0

,

the following relation between f + and f can be obtained from the second initial condition of the set (1.22):

( ) ( ) ( ) ( )( ) ( ) ( )0 0 ,f x f x f x f x f x f x Bx x x

+ + +

+ = + = + =

(1.23)

where B is a constant. Combining the first equation of (1.22) and the last equation in (1.23), the following

system of two algebraic equations is obtained:

( ) ( ) ( )( ) ( ) ,

f x f x xf x f x B+

+

+ =

+ =

which, being solved with respect to f + and f , yields

( ) ( )( ) ( ) ( )( )1 1, .2 2f x x B f x x B+ = = +

Substituting these expressions into the DAlemberts solution, we finally come to the following general expression for the string displacement in the case that the string is given an initial displacement ( )x :

w x t f x ct f x ct x ct x ct,b g b g b g b g b g= + + = + ++ 12

12

(1.24)

14

For the particular form of ( )x that is defined by Eq.(1.21), solution (1.24) represents a sum of two identical rectangular pulses propagating in the opposite directions. The resulting string patterns in this case are depicted in Figure 1.10.

Figure 1.10. Propagation of initial rectangular displacement in the string.

Thus, we have studied two ways of the string excitation, namely excitation by impact and by initial displacement. In the case of the impact excitation, in accordance with Eq.(1.19), the string response can be represented as the difference of two identical, counter-propagating pulses. In the case of an initially displaced string, this displacement propagates in the form of the sum of two identical, counter-propagating pulses, see Eq.(1.24). Taking into account that any initial disturbance of the string can be decomposed into a superposition of initial velocity and initial displacement, it is safe to say that combining solutions (1.19) and (1.24) we can describe the string response to any initial disturbance.

t = 0

t = x/2c

t = x/c

t = 3x/2c

t = 2x/c

00.5w w=

x

w (x,t)

x -x

0w w=

0w w=

15

LECTURE 2

2.1. Transverse waves in long strings (second part)

Geometrical representation of wave propagation in characteristic plane

2.1.1. Geometrical representation of wave propagation in characteristic plane

The objective of this section is to get acquainted with the so-called characteristic plane. This plane is useful, for example, for determining positions of wave fronts as they propagate along the string.

Recalling the new variables and that were introduced by Eq.(1.7), we note that x ct constant = = represents a straight line in the ( ),x t -plane, along which

( ) ( )f x ct f constant+ + = = . Similarly, x ct constant = + = is a straight line, opposite in slope to , along which ( ) ( )f x ct f =constant + = . The lines constant = and

constant = are called the characteristics of the solution of the wave equation. If an initial disturbance of the string is located in the interval x x< (as it was

assumed in the first lecture for both types of the string excitation), then the characteristics x = and x = play the most significant role in description of the string dynamics. These four characteristics are plotted in Figure 2.1.

Figure 2.1. Characteristic representation of propagation of a disturbance in the string.

Comparing Figure 2.1 to Figures 1.8 and 1.10 that represent the string response to the rectangular initial velocity and similar initial displacement, it is easy to see that

x x -x

tx

c

2FHGIKJ

= x = -x = x = -x

1

f +f

16

the characteristics x = and x = bound the perturbed domain of the string; at all four characteristics x = and x = either the string velocity or the

string displacement experience a discontinuity, therefore, by definition, these characteristics mark positions for the fronts of waves that propagate along the string;

the time instant x c , which is geometrically determined as the crossing point of the time axis with characteristics x = and x = , serves as the critical time after which the string displacement between these characteristics is either zero or constant.

Combining aforementioned observations, one may say that the energy that was initially given to the string in the interval x x< , later travels leftward and rightward being kept between characteristics x = (for the wave f travelling leftward) and

x = (for the wave f + travelling rightward). Comparing the characteristic plane (Figure 2.1) to the representation of the wave

motion depicted in Figures 1.8 and 1.10, it becomes apparent that the characteristic representation lacks the amplitude information contained in the latter figures. Thus, for the simple examples at hand, there is no particular advantage of the characteristic plane. However, in more complicated problems involving wave reflection and transmission at boundaries, this will not be the case and the characteristic representation will prove most helpful.

Concluding this section, it is worthwhile to mention that the characteristics are not a special feature of the wave propagation in a string but arise, in fact, in the general theory of hyperbolic partial differential equations, of which the wave equation is a fairly simple example. In general, the characteristics are curved lines and the quantities that are constant along these lines may be fairly involved. Methods of analysis employing characteristics find many applications in fluid and solid mechanics and acoustics as well as in problems involving shock waves.

2.2. Wave reflection at boundaries

Main types of boundaries - Reflection from a fixed boundary - The method of the images (representation of the reflection in the characteristic plane; wave particle

analogy) - Reflection from a free end - Reflection from an elastic boundary - Reflection from a viscous boundary (non-reflective boundary element)

So far we have dealt with infinitely long strings. In reality, all structures have finite dimensions and, therefore, some boundaries. If a wave falls on a boundary, it reflects in a certain manner, which is defined by the type of the boundary or, in mathematical terms, by the boundary conditions. In this section we will discuss the wave reflection from main types of the boundaries assuming that the string occupies the region 0x < and the boundary is located at x = 0.

17

2.2.1. Wave reflection at boundaries

To study a bounded string with an arbitrary boundary element, it is necessary to know a mathematical expression for the force with which the string acts on this element. Let us find this expression.

Consider a differential element of a string that is located just to the left from the boundary, see Figure 2.2.

Figure 2.2. String element located next to the boundary.

Figure 2.2 shows that the vertical component of the force that acts on the boundary (x = 0) due to the tension T in the string can be expressed as

F t TVert 0, sinb g b g= . (2.1)

If vibrations of the string are assumed to be small (see condition (1.1)), then the following equalities hold:

sin , tan ,, , , , , x t x t w dx t w t

dxw t w dx t

dxw x t

xx

b gc h b gc h b g b gb gb g b g b g

=

=

=

00

0

0

Therefore, Eq.(2.1) can be rewritten as

F t Tw x t

xVert

x

00

,

,b g b g= =

(2.2)

to give the expression for F tVert 0,b g , which is valid if the string performs small vibrations.

Employing expression (2.2), one can write a boundary condition for any kind of the boundary element, which is connected to the string at 0x = . If the end of the string is not fixed, then the boundary condition represents the balance of vertical forces at 0x = and normally referred to as the dynamic (or natural) boundary condition. If a certain displacement of the string end is prescribed, then it is said that a kinematic boundary condition is given.

Let us specify the main types of boundary elements for the string and write down the corresponding boundary conditions. These conditions are shown in the table below.

w (x,t)

x

T

w 0b g

w dxb g

x = 0 x = -dx

18

Type Diagram Boundary condition

Fixed ( )0, 0w t =

Free ( )0, 0w tx

=

Mass ( ) ( )2

2

0, 0,w t w tm T

t x

=

Spring ( ) ( )0,0, w tkw t Tx

=

Dashpot ( ) ( )0, 0,dp w t w tc Tt x

=

More complicated boundary elements can be composed by combining the spring-mass-dashpot elements.

2.2.2. Reflection from a fixed end

Reflection of an incident wave from a fixed end of the string represents the simplest type of the wave-boundary interaction and is governed by the following boundary condition:

w t0 0,b g = . (2.3)

This condition permits somewhat intuitive an approach to the problem, which is called the method of the images. This method allows to study reflection from the free end and some other boundaries by making no use of laborious mathematical treatments.

Let us demonstrate how this method works. Consider a displacement pulse f x ct x ct+ = b g b g propagating in the positive x direction towards the fixation point as it is depicted in Figure 2.3. Imagine now that the boundary at x = 0 is removed and the string is extended towards positive infinity. Let us consider this infinite string and in addition to the incident pulse ( )f x ct+ introduce an image pulse so that the superposition of these two pulses would satisfy the boundary

x=0 x

w

x=0 x

w

x=0 x

w

m

x=0 x

w

x=0 x

w

dpc

19

condition (2.3). It is easy to understand that to reach this goal, the image pulse should be introduced symmetrically (with respect to x = 0) to the incident pulse, be opposite in sign to the incident pulse and propagate in the negative x direction with the wave speed c as shown in Figure 2.3.

Figure 2.3. Reflection of a displacement pulse from a fixed end.

t = 3x/c

w (x,t)

t = 0

t = x/2c

t = x/c

t = 3x/2c

t = 2x/c

t = 5x/2c

c

2x x=0

c

c

c

x

x

x

x

x

x

x

20

As long as the image pulse travels in the domain 0x > ( ( )2t x c< ), the deflection of the real string that occupies the interval 0x < is fully determined by the incident pulse. At ( )2t x c= both pulses reach the position of the fixation point and the string displacement starts to be determined by the sum of these pulses. Since the pulses are opposite, the string deflection at 0x = remains always zero thereby satisfying the boundary condition. At ( )5 2t x c= the incident pulse disappears from the domain occupied by the real string and the string deflection starts to be governed by the image pulse alone.

Thus, having been reflected by a fixed end of the string, the pulse reverses keeping, however, its original shape. This is one of the main characteristics of the wave reflection from a fixed end.

Let us consider the pulse reflection in the characteristic plane shown in Figure 2.4.

Figure 2.4. Pulse reflection in the characteristic plane

Figure 2.4 contains characteristics 2.5 x = and 0.5 x = that bound the incident pulse and characteristics 2.5 x = and 0.5 x = that bound the reflected pulse. Looking at the figure, it is easy to distinguish three following time intervals:

( ) ( )1 2 2t x c t x c < < , when the incident pulse has not yet touched the fixation point;

( ) ( )1 2 5 2 2 5 2t x c x c t x c< < < < , when there is an interference between the incident and the reflected pulses;

( ) 5 2 5 2t x c t x c > > , when only the reflected pulse exists in the string.

xt

c

1

x x

2

3

-1 -2 -3

21

x=0

c t=0

t=3x/2c

c t=5x/2c

All these intervals can be, of course, found form Figure 2.3, but in the characteristic plane this is done easier.

Before turning our attention to more complex boundaries of the string, it is worthwhile to briefly discuss the so-called wave-particle analogy. This analogy is not widely used in mechanics of solids in contrast to, for example, quantum mechanics where this analogy is considered as one of the fundamental concepts. Anyhow, this analogy can sometimes serve as a tool to easily understand wave phenomena in solids.

The idea of the wave-particle analogy is fairly simple and can be formulated as follows. Suppose that there is a displacement pulse that propagates along a structure with a given speed. Then, under certain conditions, this pulse can be considered as a moving discrete particle, which is characterised by the energy and the momentum only. What would such a particle (imagine a tennis ball) do once it meets a fixation (a wall)? Obviously, it will be reflected by the wall and then move backwards as it is depicted in Figure 2.5.

Figure 2.5. Reflection of a ball from a wall as an analogy to reflection of a pulse from a fixed end of the string (compare to Figure 2.4).

Comparing Figures 2.4 and 2.5, it is easily seen that the ball (particle) reflection from the wall is very similar to the pulse reflection that is depicted in Figure 2.5. It would be especially evident if the momentum and the energy conservation laws were compared. This, however, is beyond the scope of this development.

Concluding this section, let us come back to mathematics and write down an expression that describes reflection of a pulse from a fixed boundary. In accordance with the method of images, it can be done very easily by just subtracting the image pulse from the incident one. Thus, if the incident pulse is given as ( )t x c , then the displacement of the semi-infinite string that is fixed at 0x = reads

( ) ( ) ( ),w x t t x c t x c= + . (2.4)

2.2.3. Reflection from a free end

Suppose now that the end of the string is free. It implies that no external vertical force is applied to the string at x = 0, i.e.

=

w t

x

00

,b g. (2.5)

22

Consider a displacement pulse f x ct x ct+ = b g b g propagating in the positive x direction towards the end of the string as it is depicted in Figure 2.6.

Figure 2.6. Reflection of a rectangular displacement pulse from a free end of the string

Like in the previous case, the solution of the problem at hand may be represented as the superposition of the incident pulse and the image pulse. The latter should be again placed symmetrically with respect to x = 0, be propagating in the negative x direction but, in order to satisfy the boundary condition (2.5), have the same sign as the incident pulse, see Figure 2.6.

Mathematically, the displacement of the string in this case reads

w (x,t)

t = 0 2x x=0

c

x

t = 3x/2c x

t = x/c x

t = 2x/c x

t = x/2c

c

x

t = 5x/2c

c

x

23

t = x/c

t = 2x/c

w (x,t)

t = 0

+1

x=0

c

2x

x

x

+1

+2

x

+1

x

+1

t = 3x/c

-1

-2

( ) ( ) ( ),w x t t x c t x c= + + . (2.6)

2.2.3. Reflection from an elastic boundary

In both cases considered above (the fixed and free ends), the reflected pulse exactly repeated the shape of the incident pulse. This was, in fact, the reason why the solutions (2.4) and (2.6) could be obtained so easily. In general, if a more sophisticated boundary is considered, the reflection process becomes more complex. This will be exemplified is this section considering reflection of a pulse from an elastic boundary.

Figure 2.7. Reflection of a rectangular pulse from an elastic boundary ( = 33. ).

24

Suppose that the end of the string is attached to a spring as it is shown in Figure 2.7. In this case, the boundary condition at x = 0, which represents the balance of vertical forces at this point, reads

=Tw t

xk w t0 0, ,b g b g , (2.7)

where k is the stiffness of the spring. Consider a displacement pulse f t x c t x c+ = b g b g propagating in the positive

x direction towards the end of the string. In accordance with the DAlemberts solution (written in a modified form, which can be easily validated by substituting it into the wave equation), the string displacement may be represented as

w x t f t x c f t x c t x c f t x c,b g b g b g b g b g= + + = + ++ , (2.8)

where ( )t x c is the incident wave (known) and ( )f t x c + is a reflected wave propagating in the negative x direction (unknown). Substitution of the solution (2.8) into the boundary condition (2.7) yields

+ + = + +

=

=

Tx

t x c f t x c k t x c f t x cx

x b g b gc h b g b gc h

00 (2.9)

The left-hand side of equation (2.9) can be evaluated as follows

( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )

,

0 0

0

.

t x ct x c

xx

x

t x c f t x cT t x c f t x c T

x x x

t x c f t x c t f tT Tc c t t

+

=

+ = +

+ =

=

+

=

+ + + = +

+ = =

(2.10) Substitution of this expression into Eq. (2.9) gives

Tc

t

t

f tt

k t f t

FHG

IKJ = +

b g b g b g b gc h . (2.11)

Collecting now the unknowns on the left-hand side of Eq.(2.11), the following ordinary differential equation of the first order with respect to f t b g is obtained:

+ =

f tt

f t tt

tb g b g b g b g , (2.12)

with = kc T . The general solution to equation (2.12) can be written in the following form (this

can be checked by substitution):

25

( ) ( ) ( ) ( )( ) ( ) ( )0

exp 0 expt

f t t d f t

= + . (2.13)

The integral in (2.13) can be rewritten as

( ) ( )( ) ( ) ( )( ) ( ) ( )( )0

0 0

exp exp expt t

tt d t t d

=

=

=

to give the following expression for ( )f t :

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )0

exp 0 0 2 exp expt

f t t t f t d = + (2.14)

Assuming that the end of the string (x = 0) is not disturbed at t = 0 we set f = =0 0 0b g b g . This yields the final form for f t b g

( ) ( ) ( ) ( ) ( )0

2 exp expt

f t t t d = . (2.15)

As an example we consider an incident wave of the rectangular form (see Figure 2.7):

( ) ( )( )0, 3 2

1, 3 2

t x x c x ct x c

t x x c x c

+ > = + <

(2.16)

Substituting x = 0 into representation (2.16), and then inserting the obtained expression into formula (2.15) and accomplishing the integration, we obtain

( ) ( )( )

( ) ( )( )( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )

0

0, 0 2

2 1 exp exp 2 , 2 5 2

2exp exp 5 2 exp 2 , 5 2x

t x c

f t t x c t x c x c t x ct x c x c t x c

=

< = < < =

>

( )( ) ( )( ) ( ) ( )

( ) ( )( ) ( )( )( ) ( )

0, 0 2

1 2exp exp 2 , 2 5 2

2exp exp 5 2 exp 2 , 5 2

t x c

t x c x c t x c

t x c x c t x c

< + < < >

(2.17)

Thus, in accordance with (2.8), reflection of the rectangular wave (2.16) from the elastic boundary may be represented as

26

( ) ( ) ( )

( )( )

( )( ) ( )( ) ( ) ( )( )( ) ( )( ) ( )( )( ) ( )

,

0, 0 2

1 2exp exp 2 , 2 5 2

2exp exp 5 2 exp 2 , 5 2

w x t t x c f t x ct x c x c

t x c t x c x c x c t x c x c

t x c x c x c t x c x c

= + + =

+ < + + < + < + + >

(2.18)

The string patterns plotted in accordance with (2.18) are shown in Figure 2.7. It is obvious that during the reflection the incident rectangular pulse undergoes a considerable distortion. Such a distortion takes place at any boundary but not if the string end is fixed, free or attached to a dashpot. The latter case is considered below.

2.2.5. Reflection from a viscous boundary.

Suppose that the end of the string is attached to a dashpot as it is shown in Figures 2.8 and 2.9. The boundary condition at x = 0 in this case reads

( ) ( )0, 0,dp

w t w tT c

x t

=

, (2.19)

where dpc is the damping coefficient of the dashpot. Consider a displacement pulse f t x c t x c+ = b g b g propagating in the positive

x direction towards the end of the string. The solution to the problem, as in the previous case, is sought in the modified DAlemberts form:

w x t f t x c f t x c t x c f t x c,b g b g b g b g b g= + + = + ++ . (2.20)

Substituting this solution into the boundary condition (2.19), taking into account expression (2.10) for w t x0,b g and evaluating w t t0,b g as

( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )

,

0 0

0

,

t x ct x c

dp dpx

x

dp dp

x

t x c f t x cc t x c f t x c c

t t t

t x c f t x c t f tc c

t t

+

=

+ = +

+ =

=

+

=

+ + + = + +

= + = +

(2.21) we obtain

( ) ( ) ( ) ( )dp

t f t t f tTc

c t t t t

= +

. (2.22)

Expression (2.22) may me rewritten as

27

+ + =t

f t tb gb g b gb gc h 1 1 0 , (2.23)

with ( )dpc c T = . Evidently, the general solution of Eq.(2.23) reads

f t t A = +

+b g b g b gb g1

1

,

where A is an independent of time constant. If the string end is undisturbed at t = 0, then this constant should equal zero to give

f t t = +

b g b g b gb g1

1

. (2.24)

Thus, reflection of the incident wave t x cb g from a viscous boundary (a dashpot) may be described by the following expression:

w x t t x c f t x c t x c t x c,b g b g b g b g b g b gb g= + + = + +

+

11

(2.25)

with ( )dpc c T = . Expression (2.24) shows that the wave reflected by a dashpot keeps the shape of

the incident wave, like in the case of wave reflection from the free and fixed ends. In contrast to these two cases, however, the pulse reflected by the dashpot is smaller than the incident pulse. The ratio of the displacements in the reflected pulse and the incident pulse (the amplitude reflection factor) is given by

( )( )1

1R

=

+. (2.26)

Obviously, since is positive, the absolute value of the reflection factor is always smaller than one. This must be so, since the dashpot consumes a part of the energy of the incident pulse and, therefore, the reflected pulse has to be smaller than the incident one.

The reflection factor, as follows from (2.26), can be positive, negative or zero. If the reflection factor is positive ( )1 < , then the reflected pulse has the same sign as the incident pulse and in this respect the reflection is similar to the reflection from a free end of the string (see Figure 2.8 and compare to Figure 2.6). If the reflection factor is negative ( )1 > , then the reflected pulse and the incident pulse have opposite signs. Thus, in this case the reflection is similar to that from the fixed boundary (see Figure 2.8. and compare to Figure 2.3)

28

Figure 2.8. Reflection of a rectangular pulse from a viscous boundary, 1 3 0.5R = =

(It is worthwhile to mention that, since the reflected pulse keeps the shape of the incident pulse, the method of the images can be applied to analyse reflection of a

pulse from a dashpot. For the case at hand the image pulse should have the same sign as the incident pulse but be twice smaller, see the upper figure).

t = 5x/2c c

x

t = x/c x

t = 3x/2c x

t = 2x/c x

w (x,t)

t = 0 2x

x=0

c

x

+1/2

t = x/2c

c

x

29

Figure 2.9. Reflection of a rectangular pulse from a viscous boundary, 3 0.5R = =

(To apply the method of images to the case that is illustrated in this figure, the image pulse should again be twice smaller than the incident pulse but have the opposite sign, see the upper

figure).

t = x/2c

c

x

w (x,t)

t = 0 2x

x = 0

c

x

t = 5x/2c c

x

t = 3x/c c

x

t = x/c x

t = 3x/2c x

t = 2x/c x

30

The most interesting situation arises when the reflection factor is equal to zero, which implies that

0 1 nrdp dpTR c AT cc

= = = = = , (2.27)

In this case there is no reflected pulse and the incident pulse is fully absorbed by the boundary. This process is depicted in Figure 2.10.

Figure 2.10. Absorption of an incident pulse by the non-reflective viscous element ( 1 0R = = )

Thus, if the string end is attached to a dashpot with the damping coefficient given by Eq. (2.27), then whatever incident pulse would approach this boundary it will be fully absorbed.

This fact is of high significance for numerical modelling, since it allows to model the wave propagation in infinite structures by considering finite-extension parts of these structures. This can be achieved by introducing the non-reflective elements. Such a trick makes all incoming waves be absorbed by the boundaries (often such boundaries are called silent boundaries) and, thereby, enforces the part of the structure located within the boundaries to behave exactly as if it were infinitely long. Unfortunately, the non-reflective elements work perfectly for one-dimensional and non-dispersive structures only (the term dispersive will be explained in Lecture 4). In all other cases it becomes impossible to develop a boundary element that would fully absorb any incident pulse.

w (x,t)

t = 0 2x

x = 0

c

x

t = 5x/2c x

t = x/c x

t = 2x/c x

31

LECTURE 3

3.1. Longitudinal waves in thin long rods

Governing equation for a rod motion - DAlemberts solution - Particle velocity and the wave speed

Physically, longitudinal wave motion in a thin rod and the transverse wave motion of a taut string are different. Mathematically, however, these two motions are quite similar. It turns out that the wave equation, which governs the motion of the string, also governs the longitudinal motion of the rod, at least within a range of circumstances. It is to be expected, therefore, that many results obtained for the string, such as the DAlembert solution, will apply to the rod directly.

Despite this apparent similarity, there are several reasons to pay considerable attention to the longitudinal wave motion in rods. The most important reason for us is that the rod is a structural element, which plays a major role in Civil Engineering. One example is a pile (see Figure 3.1), which is driven into the ground by a series of impacts. These impacts excite longitudinal waves in the pile, which, propagating up and down the pile, make it move into the ground. The longitudinal waves in a pile will be the main objective of the current lecture.

Figure 3.1. Longitudinal wave in a pile.

3.1.1. Governing equation for a rod motion.

Consider a straight, prismatic rod as shown in Figure 3.2(a). The co-ordinate x refers to a cross-section of the rod, while the longitudinal displacement of this section is given by u x t,b g . We presume the rod to be under a dynamically varying stress field x t,b g , so that adjacent sections are subjected to varying stresses. A body force q x t,b g is also assumed present.

Wave in a pile

32

Figure 3.2. A thin rod (a) with co-ordinate x and displacement u of a section and (b) the stresses acting on element x of the rod.

With these assumptions the equation of the rod motion in the x direction reads

+ + + =

x A x x A qA x A x ut

b g b g 2

2 , (3.1)

where is the mass density of the rod material and A is the cross-sectional area of the rod . The latter is a constant in this development, since we are considering a prismatic rod. We note that tensile stress is assumed positive.

By taking into account the following Taylor expansion of x x+ b g

x x xx

x+ +

b g b g , (3.2)

Eq.(3.1) reduces to

+ =

x

q ut

2

2 . (3.3)

Material effects have not been introduced yet, so equation (3.3) is applicable to non-elastic as well as elastic problems. We now presume that the material behaves elastically and follows the simple Hookes law

= E , (3.4)

where E is the Youngs modulus and is the axial strain that for the case at hand is defined by

= u x . (3.5)

x u

x

(a)

(b) x

q xb g x x+ b g

33

Using (3.4) and (3.5) in the equation of motion, we obtain

FHGIKJ + =

x

E ux

q ut

2

2 . (3.6)

If the rod is homogeneous so that E does not vary with x , equation (3.6) may be written as

E ux

q ut

+ =

2

2

2

2 . (3.7)

One can simply recognise this equation to be identical in form to Eq.(1.6) derived for the taut string.

There are several assumptions implicit in the development of Eq.(3.7), some of which have been already mentioned, such as the prismatic shape and homogeneity. It is also assumed that plane, parallel cross-sections remain plane and parallel and that a uniform distribution of the stress over the rod cross section exists. Finally, we note that a very important assumption regarding lateral effects has been made. In fact, we have neglected the lateral inertia effects associated with lateral expansions and contractions arising from the axial stress (the Poissons effect). If these effects were taken into account, the equation of motion would assume the form

E ux

qt

uJ

Au

x

+ =

FHG

IKJ

2

2

2

2

2 2

2

, (3.8)

where is the Poissons ratio of the rod material and J is the polar moment of inertia. Let us focus, however, on the simple form of the equation of motion given by

Eq.(3.7). In the absence of body forces, this equation reduces to

=

=

2

2 2

2

21u

x c

u

tc

E, (3.9)

which is the familiar wave equation.

3.1.2. DAlemberts solution.

Solution to the wave equation, exactly like in the string case, may be written in the form

u f x ct f x ct= + ++ b g b g (3.10)

or, equivalently,

u f t x c f t x c= + ++ b g b g. (3.11)

34

3.1.3. Particle velocity and the wave speed.

In this section it will be shown that the wave speed with which disturbances travel in the rod is not equivalent to the velocity of particles in the rod.

By definition, the horizontal particle velocity in the rod is given by

v x t u t,b g = . (3.12)

To compare this velocity v x t,b g to the wave speed c we consider a wave of the form

u x t f x ct,b g b g= + . (3.13)

The particle velocity and the stress in this wave may be expressed as follows

v x tu

t

f x ctt

cf

x t E ux

Ef x ct

xE

f, , ,b g b g b g b g b g b g= =

=

=

=

=

+ + + +

.

Therefore, the relation between the particle velocity and the stress reads

v x tc

Ex t, , .b g b g= (3.14)

Under elastic conditions, the stress is always much smaller than the elastic modulus. Consequently, the particle velocity is much smaller than the wave propagation velocity. As an example, suppose that a stress pulse of magnitude = 108 N m2 propagates in steel with c 51 103. m s and E = 2 07 1011. N m2 . This gives a particle velocity of 2.4m sv which is about 0.05 % of the wave speed.

Thus, not only is the particle velocity smaller than the wave speed, but it is smaller by several orders of magnitude.

3.2. Reflection of waves at boundaries

Reflection from a fixed end - Reflection from a free end

In this section the wave reflection from the simplest boundaries, i.e. from a fixed end and a free end will be considered. Like in the string case, the reflection process in these cases can be easily analysed by applying the method of the images.

3.2.1. Reflection from a fixed end.

Consider an incident pulse travelling to a fixed end of a rod. The boundary condition at the end is given as

u t0 0,b g = . (3.15)

35

Any incident pulse propagating in the rod in the positive x direction can be described as

( )iu t x c .

Taking this into account and applying the method of the images (completely analogously to the string case by the anti-symmetric imaging), we easily find that the reflection process is governed by the following expression

u x t u t x c u t x ci i,b g b g b g= + . (3.16)

Evidently, the stress space-time dependence in this case reads

( ) ( ) ( ) ( ) ( )( )( ) ( )

( ) ( ) ( ) ( )

,

,

, ,

1 1

t x ct x c

i i

i i

i i i i

u x tx t E x t E E u t x c u t x c

x x

u t x c u t x cE

x x

u t x c u t x c u uEEc c c

=

= + = = = + =

+

= = +

= =

(3.17)

Since ( )ii uEc

=

is nothing else but the stress distribution in the incident pulse,

we may conclude that

x t t x c t x ci i,b g b g b g= + + . (3.18)

Thus, while the displacement in the reflected pulse is opposite to that in the incident pulse (see the minus sign in (3.16)), the stresses in the reflected incident pulses are of the same sign. This causes the stress doubling during reflection of a pulse from a fixed end, see Figure 3.3. This phenomenon is called stress multiplication.

Since the reflected stress pulse keeps the sign of the incident stress pulse, one may say that compression is reflected as compression and tension as tension. This also is a characteristic of reflection from a fixed end.

3.2.2. Reflection from a free end.

Consider an incident pulse travelling to a free end of a rod. The boundary condition in this case is given as

0 00

0,,

.tu t

xb g b g= = (3.19)

Introducing the incident pulse as

( )iu t x c ,

36

Figure 3.3. Reflection of (a) displacement pulse and (b) stress pulse from a fixed end.

and applying the method of the images, i.e. introducing the symmetric image pulse, the reflection process can be presented as

u x t u t x c u t x ci i,b g b g b g= + + . (3.20)

Evidently, the stress field in the rod now reads

x t t x c t x ci i,b g b g b g= + (3.21)

x

c

u

t = 0

t = 5x/2c

t = 3x/c

t = 4x/c

t = 5x/c

(a) (b)

37

Thus, at a free end, the displacement in the reflected pulse is identical to that in the incident pulse, while the stress in the reflected pulse is opposite to the stress in the incident pulse, see Figure 3.4.

So, in contrast to the case of a fixed end, reflection from a free end provides the doubling not of the stress field but of the displacement field.

Figure 3.4. Reflection of (a) displacement pulse and (b) stress pulse from a free end.

x

c

u

t = 0

t = 5x/2c

t = 3x/c

t = 4x/c

t = 5x/c

(a)

38

3.3. Waves in a finite-length rod

Stress pulse in a free-fixed rod (impact against a pile) - Stress pulse in a free-free rod - how do we move things?

So far we considered wave propagation in infinite or semi-infinite strings and rods. In this section these considerations will be extended to the case of a finite-length rod. Commonly, vibrations of such a rod are studied by considering natural frequencies and normal modes. We will do it differently, with the help of the method of the images, which enables the rod motion to be presented in the form of propagating waves.

3.3.1. Stress pulse in a free-fixed rod (impact against a pile).

Consider a finite-length rod that is fixed at the right end and free at the left end, see Figure 3.5 (a). Let the free end of the rod undergo an impact, which provides the stress ( )0, t that is depicted in Figure 3.5 (b).

Figure 3.5. Impact excitation of a free-fixed rod.

When a stress pulse is excited in a rod of a finite length, this pulse then travels along the rod until it meets one of the boundaries. At the boundary the pulse reflection takes place into the rod interior. This process continuously repeats itself. If the rod boundaries are free or fixed and no damping mechanisms are accounted for, the pulse will travel in the rod infinitely long, experiencing no distortion. As we already know, in this case, the method of the images can be applied as a suitable tool to describe the system motion.

In contrast to the semi-infinite string and rods, where only one image pulse should be introduced to satisfy the boundary conditions, in a finite rod we need infinitely many image pulses. For the rod depicted in Figure 3.5(a) the necessary series of the image pulses is shown in Figure 3.6.

Figure 3.6. Stress pulses that satisfy the boundary conditions of the free-fixed rod.

t

0

0

0 6T l c

T=

=

0, tb g

l0

(a) (b)

x l0

x

39

The stress pulses in Figure 3.6 are constructed in such a way that every pulse has a symmetric counterpart with respect to the fixed end and an anti-symmetric counterpart with respect to the free end.

Using Figure 3.6. the actual stress distribution in the rod as a function of time and spatial co-ordinate x can be found easily. Indeed, since the stress pulses propagate with the constant speed (the wave speed), it is quite straightforward to calculate the time intervals when one or another pulse in Figure 3.6. travels through the actual rod

[ ]( )00,x l . This pulse will be then the one that represents the stress distribution in the rod. This distribution, together with the corresponding displacement field, is shown in Figure 3.7.

Figure 3.7. Displacement (left) and corresponding stress (right) pulse traversing the free-fixed rod.

2 0

T + 02

T + 4 0

220T +

2 2 0T +

320T +

3 4 0T +

4 2 0T +

u

40

This figure shows that the motion of the finite-length rod that is initiated by an impact is a repetitive process of the pulse propagation and reflection.

3.3.2. Stress pulse in a free-free rod - how do we move things?

Imagine a rod that is suspended by two light strings (Figure 3.8.(a)), so that it is free to move horizontally under the action of applied loads. Assume that both ends of the rod are free and the left one experiences an impact that is shown in Figure 3.8(b).

Figure 3.8. Impact excitation of a free-free rod.

This example should serve to illustrate how the gross motion of a rigid body is related to wave propagation in this body. In other words, we should get an idea on how, actually, we move things by hitting them.

To solve this problem the method of the images is applied. The necessary set of the image pulses that satisfy the boundary conditions for the free-free rod is given in Figure 3.9.

Figure 3.9. Stress pulses that satisfy the boundary conditions of the free-free rod.

The image pulses shown in the figure are constructed in such a way that every pulse has the anti-symmetric counterpart with respect to both the right end and the left end of the rod.

Using Figure 3.9, the actual stress distribution in the rod can be found. For the case at hand, however, it will be most interesting to present not the stress distribution, but the displacement of certain points of the rod as functions of time. This is done in Figure 3.10 that shows the displacement of three points of the rod (the left and the right ends and the middle).

Figure 3.10 shows that point B (the middle point of the rod) moves to the right in a series of repetitive movements. Each movement consists of the rest phase (the horizontal lines) and the movement phase (inclined lines). The movement phase takes place as the stress pulse passes the point. Once the pulse is gone, the point does not move and the rest phase takes place. The end points A and C move similarly but

t

0

0

0 4T l c

=

=

0, tb g

l0

(a) (b)

x

A

B

C

x

l0

41

the period of their movements is twice larger relative to that of point B. This is so because the stress pulse comes to the ends of the rod twice more seldom than to the middle point.

Figure 3.10. The displacements of three points along the rod that is depicted in Figure 3.8.

Thus, Figure 3.10 shows that no point of the rod moves continuously. On the contrary, every point experiences a jerky motion, which consists of the rest phase and the movement phase. In reality, however, there are effects like material damping, for example, that will smooth down such jerky motion of the rod particles. Specifically, curve B will approach (as time increases) the dashed line shown in the figure. There will then be no vibrations of the rod. Instead, each particle of the rod will have a constant translational velocity given by the slope of the dashed line.

1

xl02

tlc

0

1 4 3 2 7 6 5 1/4

2

3

4

5

A

B

C

42

3.4. Transmission and reflection at interfaces

Transmission and reflection at a junction of two rods - The wave impedance

So far we considered reflection of waves from a boundary (termination) of a rod. The reflection, however, also may occur at a junction between two rods, see Figure 3.11. In this case the reflected wave will be always accompanied by a wave transmitted across the junction. These waves carry information about the rod and the junction properties, which can be extracted quite easily. Therefore, the effect of wave reflection and transmission at a junction finds practical application in experimental studies of dynamic loading of materials.

3.4.1. Transmission and reflection at a junction of two rods.

Consider an incident stress wave i x ctb g propagating towards a junction of two semi-infinite rods, where there is a discontinuity in the cross-section, material properties or both. The situation is shown in Figure 3.11.

Figure 3.11. Incident, reflected and transmitted stress waves at the junction of two rods Z Z2 1>b g.

As depicted in Figure 3.11, the incident and transmitted waves propagate in the positive x direction whereas the reflected wave travels in the negative x direction. Therefore, in accordance with the DAlemberts solution, the rod displacements associated with these waves can be described as

u f t x c u f t x c u f t x ci r t= = + = + +1 1 1 1 2 2b g b g b g, , . (3.22)

Accordingly, the stress distribution = E u xb g and the particle velocity v u t= b g associated with these waves are given as ( )t x c =

( ) ( )

( ) ( )

( ) ( )

1 11 1 1 1

1 1

2 12 2 1 1

2 1

1 21 1 2 2

1 2

, ,

, ,

, ,

i r

t i i

r r t t

E Ef t x c f t x cc c

E cf t x c v f t x cc E

c cv f t x c v f t x c

E E

+

+

+ +

+ +

+

+

= = +

= = =

= + = = =

(3.23)

i

A E1 1 1, , A E2 2 2, ,

r t

x

43

Suppose that the rods are free of external loading. Then, both the force field F A= b g and the particle velocity field must be continuous at the junction x = 0b g:

A A v v vi r t i r t1 2 + = + =b g , (3.24)

Substituting the stress-velocity relations from (3.23) into the second equation of (3.24) the following system of two algebraic equations is obtained

A Ac

Ec

Ec

E

i r t

i r t

1 2

1

1

1

1

2

2

+ =

+ =

b g.

(3.25)

Solving this system with respect to r and t we obtain

( )( )( )

1 1

2 2 2 1 1 1

1 1 1 2 2 21

1 1 1 2 2 21

2,

.

t i

r i

c Ec E A c A E

c E A c A Ec E A c A E

=

+

=

(3.26)

Problems involving wave transmission across junctions are often spoken of in terms of an impedance. This term and concept, borrowed from electric circuit theory, express the ratio of a driving force to the resulting velocity at a given point of the structure. For an elastic rod, the impedance is given by

Z Fv

Av

EAc

A E= = = = . (3.27)

Using the impedance, it is possible to rewrite (3.26) as

t i r iZ Z A A

Z ZZ Z

Z Z=

+=

+

21

11

2 1 1 2

2 1

2 1

2 1

b gb g, , (3.28)

where Z A E Z A E1 1 1 1 2 2 2 2= = , .

Expressions (3.28) for the transmitted and reflected stress fields show the following: If the impedance of the left rod equals to the impedance of the right rod, no

reflected wave occurs at the junction. The transmitted stress pulse always keeps the sign of the incident stress pulse, i.e.

compression is transmitted as compression and tension as tension. The sign of the reflected stress pulse depends on the ratio of Z2 and Z1 . When Z Z2 1> ,

the reflected stress pulse keeps the sign of the incident one (like in the case of the reflection from a fixed end). On the contrary, when Z Z1 2> the signs of the reflected and the incident stress pulses are opposite (similar to reflection from a free end).

These properties of wave reflection and transmission at a junction of a rod are widely used in experiments on dynamic loading of materials.

44

45

LECTURE 4

4.1. Pile-soil interaction

Local model for pile-soil interaction (boundary conditions, non-reflective boundary element) - Distributed model for pile-soil interaction (equation of

motion, failing of the DAlemberts solution)

In this lecture, the effect of soil reaction on waves in a pile is discussed. First, we assume that only the tip of the pile is subjected to the soil reaction. This model is referred to as a local model for pile-soil interaction. The second model is slightly more sophisticated and accounts for the soil reaction all over the pile length. The soil reaction is modelled with the help of continuously distributed along the pile (rod) springs and dashpots and, accordingly, the model is called either distributed or continuous. The equation of motion for this model consists of the wave equation part and two additional terms reflecting the distributed stiffness and distributed damping of the soil. These additional terms ruin the DAlembert solution and cause propagating waves to experience distortion. The physical phenomenon causing the wave distortion is known as dispersion and plays a major role in the wave theory and its practical applications.

4.1.1. Local model for pile-soil interaction

Let us first neglect the soil reaction that the pile experiences along its length and assume that the soil reacts at the pile tip only. In this case the pile can be modelled as a rod, one end of which is free and the other one is attached to a visco-elastic element. This rod is depicted in Figure 4.1.

Figure 4.1. Local model for pile-soil interaction.

As shown in the previous lecture, the longitudinal motion of the rod is described by the wave equation, which reads

=

=

2

2 2

2

21u

x c

u

tc

E, . (4.1)

To write a boundary condition at x = 0 we consider a differential (small) element of the rod located at x = 0 , see Figure 4.2.

f + k

cdp x=0 x u(x,t)

c

46

Figure 4.2. Differential end-element of the rod.

To provide equilibrium, three forces (imposed by the dashpot, the spring and the part of the rod, which is located to the left from the differential element) should be balanced to give

( )0, 0sp dpA t F F + + = , (4.2)

or, taking into account that 0 0, ,t E u t xb g b g= ,

( ) ( ) ( )0, 0,0, 0dpu t u tAE ku t cx t

=

. (4.3)

Reflection of an incident longitudinal wave from the spring-dashpot boundary element may be analysed in exactly the same manner as it has been done for the string (see Lecture 2). This is shown below.

Consider a displacement pulse f t x c t x c+ = b g b g propagating in the positive x direction towards the end of the rod x = 0 . In accordance with the DAlemberts solution, the longitudinal displacement of the rod may then be expressed as

u x t f t x c f t x c t x c f t x c,b g b g b g b g b g= + + = + ++ , (4.4)

where ( )f t x c + is the reflected pulse that appears after the incident pulse has reached the boundary x = 0 . Substitution of solution (4.4) into boundary condition (4.3) yields

( ) ( )( ) ( ) ( )( )( ) ( )( )

00

00.

xx

dpx

AE t x c f t x c k t x c f t x cx

c t x c f t x ct

=

=

=

+ + + +

+ + =

(4.5)

Employing relations (2.10) and (2.21) from Lecture 2, Eq.(4.5) is reduced to

( ) ( ) ( ) ( )( ) ( ) ( ) 0dpt f t t f tAE k t f t cc t t t t

+ + =

. (4.6)

Collecting now the unknowns in the left-hand side, the following ordinary differential equation of the first order with respect to f t b g is obtained:

x=0

Fsp

Fdp

(0)

47

( ) ( ) ( ) ( )dp dpf t tAE AEc k f t c k tc t c t

+ + = (4.7)

Combination of parameters AE c in (4.7) represents the rod impedance Z , see Eq.(3.27) in Lect