Graphical Representations of One-Dimensional Wave Pulses and Their
Interactions with Boundaries
MASSACHUSETTS INSTIitfTE
OF TECHNOLOGY
by
JUL 3 0 20
Ayman S. AbuShirbi
Submitted to the
Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of
LIBRARIES
Bachelor of Science in Mechanical Engineering
at the
Massachusetts Institute of Technology
February 2014
2013 Massachusetts Institute of Technology. All rights reserved.
Signature redacted
Signature of A
r:
Department of Mechanical Engineering
February 1, 2014
Certified by:
Signature redacted
L)
James H. Williams, Jr.
Professor of Applied Mechanics and Wfting and Humanistic Studies
Thesis Supervisor
Signature redacted
Accepted by:
Anette Hosoi
Professor of Mechanical Engineering
Undergraduate Officer
1
Graphical Representations of One-Dimensional Wave Pulses and Their
Interactions with Boundaries
by
Ayman S. AbuShirbi
Submitted to the Department of Mechanical Engineering
on February 1, 2014 in Partial Fulfillment of the
Requirements for the Degree of
Bachelor of Science in Mechanical Engineering
ABSTRACT
Wave propagation is the transfer of energy through a medium without bulk motion of the matter
of the medium. The teaching of elementary wave propagation is a worthy educational goal.
Graphical models are presented to assist the physical understanding and interpretation of wave
propagation in one-dimensional mechanical elements. Wave models for square, triangular, and
sinusoidal wave pulses are illustrated for transverse and longitudinal particle displacements,
reflection at clamped and free boundaries, and transmission and reflection at interfaces between
media. These models should enhance the understanding of these elementary concepts.
Thesis Supervisor: James H. Williams, Jr.
Tile: Professor of Applied Mechanics and Writing and Humanistic Studies
2
Acknowledgments
I would like to thank the people who have made this journey possible. Professor James H.
Williams, Jr. (who is supported by the DDG-100 Program Manager/NAV SEA PMS 500 the
DDG-1000 Ship Design Manager/NAVSEA05D) for his support, patience, and guidance
throughout the process of completing this thesis. I would also like to thank Professor Sanjay
Sarma for his valuable advice, and for being the man. I would also like to thank Brandy Baker
who has helped me out so many times during my career at MIT.
3
Table of Contents
Abstract ............................................................................................................................................2
A cknow ledgm ents ...........................................................................................................................
3
List of Figures ..........................................................
5
1. Introduction..........................................................................................................................6
2. W ave Theory .......................................................................................................................
2.1 W ave Interference ...................................................................................................
7
10
3. Graphical M ethod ................................................................................................................
13
4. W ave M otion in Sem i-Infinite M edia ............................................................................
14
5. W aves at Fixed End Boundary Condition .....................................................................
27
5.1 W ave Pulse Interactions with a Clamped End ......................................................
27
5.2 W ave Pulse Interactions with a Free End ..............................................................
33
7. Conclusion..........................................................................................................................37
References.....................................................................................................................................38
4
List of Figures
Figure 1: Longitudinal wave propagation, the light gray lines relate to the depression of particles
and the darker grey lines relate to the compression of particles as the wave propagates through
7
the m edium ......................................................................................................................................
Figure 2: Particle movement in respect to wave propagation, the green dot monitors the vibration
of the particles in the media as the transverse wave pulse propagates through the medium..... 8
Figure 3: A general sinusoidal wave example to aid in the following sections.......................... 9
Figure 4: Constructive interference displayed for two triangular waves traveling in opposite
11
direction . .......................................................................................................................................
Figure 5: Destructive interference displayed for two triangular waves traveling in opposite
12
direction . .......................................................................................................................................
Figure 6: A junction at x = 0 joining a two rod system, containing an incoming wave in the
15
positive x direction ........................................................................................................................
Figure 7: Displacements of a longitudinal triangular wave, the reflection and transmission
process in a rod system, wave pulse shown moving towards the junction towards the positive x
19
direction . .......................................................................................................................................
Figure 8: A two rod system with a wave pulse traveling from rod 1 to rod 2 in the negative x
20
direction . .......................................................................................................................................
Figure 9: Displacements of a longitudinal triangular wave, the reflection and transmission
process in a rod system, wave pulse shown moving towards the junction towards the negative x
22
direction . .......................................................................................................................................
Figure 10: A system of two semi-infinite strings joined at a junction at x = 0, with an incoming
23
wave pulse traveling in the positive x direction........................................................................
Figure 11: Displacements of a triangular wave, the reflection and transmission process in a string
system, wave pulse shown moving towards the junction towards the positive x direction.......... 25
26
Figure 12: Semi-infinite rod system with a clamped end. .........................................................
Figure 13: Displacement in a semi-infinite rod with a clamped end at x=O, shown at different
times for a triangular wave pulse propagating in the positive x direction, the resultant wave due
29
to the superposition represented by a smaller wave pulse.. ......................................................
Figure 14: Displacement in a semi-infinite rod with a clamped end at x=0, shown at different
times for a sine wave pulse propagating in the positive x direction, the resultant wave due to the
30
superposition represented by a smaller wave pulse...................................................................
Figure 15: Displacement in a semi-infinite rod with a clamped end at x=O, shown at different
times for a square wave pulse propagating in the positive x direction, the resultant wave due to
31
the superposition represented by a smaller wave pulse. ...........................................................
end
represents
a
free
green
connection
free
end,
the
a
rod
system
with
Figure 16: Semi-infinite
simulation, which disconnects the wall from the semi-infinite rod, and holds the boundary
32
conditions established...................................................................................................................
Figure 17: Displacement in a semi-infinite rod with a free end at x=0, shown at different times
34
for a triangular wave pulse propagating in the positive x direction...........................................
Figure 18: Displacement in a semi-infinite rod with a free end at x=O, shown at different times
35
for a sine wave pulse propagating in the positive x direction....................................................
Figure 19: Displacement in a semi-infinite rod with a free end at x=0, shown at different times
36
for a square wave pulse propagating in the positive x direction...............................................
5
1. Introduction
A system of particles connected together is by definition a medium. Since these particles in this
medium are connected, it is possible to send information through this medium. This information
can be in the form of energy. Energy is carried through media the same way sound is transmitted
through thin air. Waves function to carry this energy between different media.
Physical material in nature is not uniform; waves have to travel between different media.
Graphical representation of wave pulses traveling between different media is introduced. The
graphical method used is explained in section 3. The speed of the wave depends on the
characteristics defined by the medium to which the wave is traveling in. The wave speed will
increase or decrease depending on a relationship between these characteristics. When the
incoming wave reaches the junction between two media, a small part of the wave reflects, much
as our image is reflected when looking into a mirror. Under the same premise, a small portion of
the wave reflects when it is transmitted through a boundary. Section 4 studies these phenomena
of reflection and transmission, and attempts to display the reflection and transmission
coefficients.
When wave pulses collide with a boundary it is very much like a ball hitting a wall. The ball
carries momentum and energy with it. The wall in this case is a fixed boundary, and it will reflect
the ball's energy. Assuming that the ball and the wall are inelastic and incompressible, there will
be no energy exchange between the wall and the ball. In the case of waves, different phenomena
happen depending on the nature of the end condition. As a special case of a junction, the fixed
end boundary and the free end boundary are introduced. Section 5 graphically represents the
behavior of the waves while interacting with these ends.
6
2. Wave Theory
Any assembly of particles is by definition a medium. The interaction between these particles is
possible. This interaction creates waves. The forces of interaction between particles in different
mediums allows for the transfer of energy. The direction in which the energy is traveling in
defines the direction of the wave, or the propagation of the wave. Energy transfer happens in two
main types of waves, longitudinal waves and transverse waves.
Longitudinal waves exhibit a vibration of the particles in the medium by being parallel to the
propagation of energy in the media, shown in figure 1. The gray lines resemble vibrating waves.
As the wave propagates the particles are compressed and depressed. In figure 1 below, the red
arrows represent a function generator. In this case the function generator is creating a
longitudinal wave.
III II I I II II
II
II
I
II
III II II II
II
II
II II
I Il
II II
II
Figure 1: Longitudinal wave propagation, the light gray lines relate to the depression of particles and the darker grey
lines relate to the compression of particles as the wave propagates through the medium.
7
The second type of waves is transverse waves. These waves exhibit a vibration of the particles in
the medium perpendicular to the direction of travel or direction of propagation. If a transverse
wave is introduced as a single disturbance it can be called a pulse. If a pulse or a transverse
wave is introduced into a medium the particles in the medium will vibrate in the vertical manner
shown in figure 2. The green dot in figure 2 monitors the movement of one particle in the
medium.
Propagation
Figure 2: Particle movement in respect to wave propagation, the green dot monitors the vibration
media as the transverse wave pulse propagates through the medium.
8
of the particles in the
Since waves travel over a distance or a spatial x variable, they also have speeds. The speed of the
wave depends on the medium which the wave is traveling in. The speed of the wave can also be
expressed as a function between the wave length, shown in figure 3, and the frequency of the
wave. Figure 3 shows the most basic form of a wave; a sinusoidal wave propagating along a set
path. Figure 3 also shows the amplitude of the wave. The amplitude is the change over a period
of time. It allows for a relevant measure of the particles vibration.
V = fL
(2-1)
Displacement
wavelength X
Amplitude
Distance
Figure 3: A general sinusoidal wave example to aid in the following sections.
Assuming that the wave is traveling in the same medium, if the frequency is increased the wave
length will decrease to maintain the speed of the wave. If the frequency is decreased the wave
length will increase to satisfy the condition set by equation (2-1). If the wave travels between
different media it is possible for the speed to change, where wave length of the wave will
change, but it will not affect the frequency of the wave.
9
2.1 Wave Interference
When a wave pulse travels down a medium, it is possible to have a different wave pulse traveling
in the opposite direction. Wave interference is the theory that describes the interaction between
these wave pulses. There are mainly two types of interferences, constructive interference, and
destructive interference. To further study this phenomenon, two wave pulses are generated
through a string using a virtual pulse generator. Figure 4 shows a case of constructive
interference.
In this example two wave pulses travel in opposite direction, and are meant to interfere with each
other. Once the wave pulses start to interfere, their energies will add up. The amplitude of the
resulting wave is the algebraic some of the original amplitudes. The wave pulses will continue to
propagate, to their destination with no change to their original nature.
Figure 5 shows a case of destructive interference. Two wave pulses are simulated; but, with
being 1800 out of phase from each other. In the example shown, the wave pulses are traveling in
opposite direction, and in different domains. Once the wave pulses' energies contact each other,
the resulting wave has the algebraic sum of their amplitudes. Thus the wave pulses will
momentarily cancel each other. The waves will continue to propagate with no amplitude change.
10
Red Wave Propagation
Blue Wave Propagation
A
t=t,
A
Red Wave Propagation
Blue Wave Propagation
Red Wave Propagation
Blue Wave Propagation
Blue Wave Propagation
Red Wave Propagation
t=t,
1=,
t=t3
t%4
Blue Wave Propagation
Red Wave Propagation
t=15
4
00-
Figure 4: Constructive interference displayed for two triangular waves traveling in opposite direction.
11
Red Wave Propagation
I=tj
Blue Wave Propagation
A
Red Wave Propagation
Blue Wave Propagation
Red Wave Propagation
Blue Wave Propagation
Blue Wave Propagation
Red Wave Propagation
1=t,
t=t3
hhh,
t=t1
A
Blu WvoaainRdWvPrpgtn
Figure 5: Destructive interference displayed for two triangular waves traveling in opposite direction.
12
3. Graphical Method
The examples in the following sections are generated using two programs. The first program is
Excel. The sinusoidal functions were generated numerically, and the propagation of the function
was modeled through a series of graphs inside the program. A selection of these graphs was
exported into Inkscape for further study. Inkscape being a graphical tool used to generate a
rendered version of these graphs. Inkscape not only being a great graphical tool, but was also
used to generate the quasi-static interaction for the triangular wave pulses and the square wave
pulses.
The choice of using Inkscape was obtained after testing with different static image manipulation
programs. The aim was to find a graphical tool with an acceptable learning curve. Adobe
Photoshop was the first tool tested. It is a very strong tool which provided too many
customizable functions. With proper practice it could have provided a great venue for creating
the needed examples; but, during the course of this creating this thesis it was found to be too
complicated to be used.
Adobe Illustrator was the second tool tested. With its huge library of useful functions, it provided
a great medium for creating the examples. Although Adobe Illustrator provided a great
environment it was not easy to learn and master. The third program tested was Inkscape which
provided the important functionalities found in Adobe Illustrator, and was very easy to use.
Inkscape also has a function generator tool. For those reasons Inkscape was chosen to be the
graphical generation tool for this thesis.
13
4. Wave Motion in Semi-Infinite Media
It is purely a theoretical argument to assume that any medium is infinite. Real systems have
finite dimensions. In the course of our study we are interested in one-dimensional boundaries. In
general when a wave encounters a boundary, some part of it will be reflected. As in real systems
with three-dimensional boundaries there will be multiple reflections. To simplify the discussion
we will assume that the media boundaries we are dealing with result in one reflection. To follow
the argument that media are semi-infinite, there will be studying one-dimensional media.
To further study the characteristics of the transmitted wave pulses, we need to establish
relationships between the original wave pulse and the daughter wave pulse. These relationships
will serve as a mathematical model, which will aid in the graphical representation of the waves'
interaction with boundaries.
In the following discussions the wave traveling towards the junction will be termed the incoming
wave, and the portion reflected back will be termed the reflected wave. The portion of the wave
that passes through the junction will be named the transmitted wave.'
Figure 6 exhibits the characteristics of a two-rod system joined with a junction at x=O. Rod l's
density, cross-sectional area, and modulus of elasticity are po, A,, El, and rod 2's characteristics
are p2, A 2 ,E2 . Further study shows that the speeds in rods 1 and 2, respectively, are cl = V E1/pl
and c2 = VE2/p2. This shows that the speed of the wave is a function of the characteristics of
the medium through which it is traveling, in this case the density and the modulus of elasticity.
14
As section 2 discusses, a wave's speed is determined only by the characteristics of the medium in
which it is traveling in.
Figure 6 shows a longitudinal incoming wave i =
fi (x
- c1 t). In this case the subscript i refers
to the wave being incoming. When the wave gets to the junction at x= 0, it separates into two
-
waves. The reflected and transmitted waves can be denoted respectively as 'r (x, t) = fr (x
c1 t) , and ft (x, t)
=
ft(x
-
c 2t). The subscript r refers to the reflected wave, and the subscript t
refers to the transmitted wave.
Cl
4i =AiX-Cy t)
I
-7
pl, A 1,EI
X=O
Rod 1
Nox
P2, A 2 ,E2
Rod 2
E2 A25
-P.I &r 1Figure 6: A junction at x = 0 joining a two rod system, containing an incoming wave in the positive x direction.
To be able to solve this case some boundary conditions are required. The first needed boundary
condition is a geometric requirement. This requirement is generally called the continuity of
15
displacement at x = 0. This represents the argument that the media are in contact at all times. The
second boundary condition that we will introduce is a natural requirement. This requirement
denotes that the equality of the axial force at x=0. This boundary condition is important because
if there is a finite force jump across the junction on an infinitesimal element, that element will
have an infinite acceleration. Infinite acceleration is impossible; hence this condition is needed.
The displacement boundary condition can be expressed as'
(4-1)
(Opt) + fr(Olt) = ft(Olt)
or
fh(-cit) +
Yr(Cit)
= ft(-c 2 t)
(4-2)
The force boundary condition can be expressed as
E1A1
(0, t) + E 1A 1 Lfr (0, t) = E2 A 2 f- (O, t)
(4-3)
Upon attempting to solve these equations, it is clear that is impossible to solve them directly.
This is because we have the functions fi (0, t), f, (0, t) and ft (0, t) . However, if we attempt to
integrate the boundary condition (4-3) in respect to time, we will be able to arrive to a solution.
The solution to this case can presented as
,
= E1 A 1 C2 -E 2 A 2 c1
Er(cit)=
2 -Ac 1
f(-cit)
(4-4)
and
ft(-c 2 t)
2E 1 cA 1
2
E1 A 1 C2 +E 2
16
A 2 Cl
(
CIt
(4-5)
Now that we have the results (4-4) and (4-5), we can derive physical quantities like the velocity,
forces, stresses and energies in transmitted and reflected wave pulses. For further discussion the
coefficient displacement in the transmitted wave pulse and the coefficient of displacement in the
reflected wave pulse are, respectively, shown in (4-6) and (4-7)
r(0,)0
gr (ci) 0 E1A 1cE2 - E2 A2 c1
1A 1c 2 + E2 A 2 c1
= Rjfi(-cit) = Rf 4 (0, t)
ft (0, 0)
fr (- C2) =
(4-6)
2E A c
1 1 2
AC fi (- Ct)
,,C+
E1 A 1 c2 + E2 A 2 clf ct
= Tjfh(-cit) = Tfi(0, t)
(4-7)
Figure 7 shows the representation of the interaction between the longitudinal wave pulse
described in figure 6, and the junction at x-O. The red triangle represents the longitudinal wave
pulse
fi (x -
c1 t) traveling left in the positive x direction towards the junction. The junction is
represented by the solid black vertical axis. Once the wave pulse reaches the junction at t = t2 a
small portion of the wave will be reflected back in the opposite direction. The small portion of
the wave being reflected is presented by the solid green line emerging from the junction traveling
in the negative x direction. The transmitted wave continuing to travel in the positive direction is
now traveling at a speed c2. The speeds of the wave pulses in this case are defined as c1 =
E1/p1 andC2
=
E2/P2-
17
In this example rod 1 and rod 2 share the same densities; however, the modulus of elasticity is
not the same. For representation purposes it will be assumed that the modulus of elasticity of rod
2 is twice as much as the modulus of elasticity of rod 1.
.
At t = t3 the transmitted wave pulse has cleared the junction and is now traveling at a speed c 2
The change in speed of the wave is graphically shown in Figure 7. In the same time frame
shown, the portion of the wave pulse that is reflected is traveling back into rod 1 at the same
speed at which it came in. The other thing to note is the change in the spatial variable x's change
in the transmitted wave. Since the properties of the material did not change during the interaction
in rod 1, the wave pulse traveling in the rod will have the same spatial extent, the spatial width of
the incoming wave is a, hence the reflected wave will maintain that spatial width'. The
transmitted wave's space variable is scaled accordingly to comply with the change of speed of
the wave upon entering the second medium. The new spatial width for the transmitted wave
pulse is -a. The change in the spatial width is shown in Figure 7.
C1
Upon reflection and transmission the wave pulses' displacement is scaled by the factors derived
in the previous section. The reflected wave is scaled by the magnitude of the R4. The transmitted
wave displacement is scaled in its magnitude by a factor of T4. Figure 7 also has provided insight
to the events of the wave pulses propagation at t =
18
4,
once it cleared the junction completely.
~(xt1
)
I
'K
Incoming Wave
(a) t~
Boundary Interaction
x
0
(b) t =t2
)
{(x,t3
Transmitted Wave
Reflected Wave
4-4
A
Tz
x
(c)t
t3
C2
{(x,t4)
Reflect
4
Transmitted Wave
T4
A R4
(d) t = t4
- -w
-
L4r
Figure 7: Displacements of a longitudinal triangular wave, the reflection and transmission process in a rod system, wave
pulse shown moving towards the junction towards the positive x direction.
19
In order to further study the phenomenon of wave pulses traveling between two media through a
junction the following example is presented. Figure 8 presents the parameters which we are
working with in this example.
CI
i=g(x-c 1 t)
5V
P2, A 2,E 2
X
X=o
Rod 2
pi, AI,EI
Rod 1
Figure 8: A two rod system with a wave pulse traveling from rod 1 to rod 2 in the negative x direction.
Since the system describes the same situation described in figure 6 with the wave traveling in the
negative x direction, we can assume the mathematical model will hold for this case. In this
example we will assume that the initial parameters produce a case where the ratio between
T4 / R4 = 3, and the scaling in the speed of the transmitted wave is -= 2.
C1
Figure 9 below shows the results of such parameters. The first time frame at t = t, shows the
wave pulse that traveling towards the junction. The wave reaches the junction in the time frame
at t = t2 . The interaction between the wave pulse and the boundary is shown. The original wave
shown in red, starts to move through the junction with the same boundary conditions established
in the mathematical model presented in the previous section. At the same time the green reflected
20
portion of the wave. It is required to note that the representation omits the parts of the wave that
have not gone through the transmission or reflection process. That is to allow for a better
representation of the interaction at the junction. The time frame t = t3 shows the change in the
spatial variable. The green wave travels back at the same spatial displacement it came in. the
graph also depicts the change in speeds of the waves as the wave pulses propagate through the
junction.
21
4(1
Incoming Wave
CI
(a) t =
t,-
a
4(X,t2)
Boundary Interaction
0
(b)
t =t
((x,13)
Reflected Wave
Transmitted Wave
4
-
-4
(c)t
=,
Reflected Wave
Transmitted Wave
4R4
0r-r
0
(d)
t= 1,
--ool a [**-
Figure 9: Displacements of a longitudinal triangular wave, the reflection and transmission process in a rod system, wave
pulse shown moving towards the junction towards the negative x direction.
22
To further explore the behavior of wave pulses in media, we will present the refection and
transmission coefficients in a system of strings.
Presented in Figure 10 below a system of strings, which are taut semi-infinite. The density and
cross-sectional area of the strings are p, and A 1 for string 1, and P2 and A2 , for string 2. The
tension in the strings is equal since there is no gross acceleration in the x direction, and will be
denoted by P.
'I(x,t)
C1
l1i=f j(x-cft
pi, Ai
x=O
String1
x
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII~
-
I
p2, A 2
String 2
Figure 10: A system of two semi-infinite strings joined at a junction at x = 0, with an incoming wave pulse traveling in the
positive x direction.
As developed when creating the model for the system of rods, it is important to set boundary
conditions for this system. There is one continuity condition for the transverse displacement, and
on condition for the equality of transverse force.
The displacement boundary condition can be expressed as'
t)
+
1i(0,
77,.(0, 0) = rnt(0, 0)
or
23
(4-8)
A (-c 1 t) + g,.(clt) = ft(-c 2 t)
(4-9)
Now the transverse force boundary condition can be expressed as'
t) + P 8x
L(0, t) = P 6x7 (0, t)
P Z(0,
6x
(4-10)
Upon integrating the presented equation in respect to time, and solving the equations, one would
get the following result for the coefficient of reflection and the coefficient of transmission
R
cl-c 2
(4-11)
c1+c 2
T = 2c 2
(4-12)
c 1 +c 2
In the example presented figure 11 c1 =
c 2. This result the wave speeds in a string are defined
as cl = JP/Ap and c 2 = J PA2 2. The generated figure shows the change in speed of the
waves pulses. The wave changes speed once it passes through the junction. The different color
represent to the two different strings. The weight of the colors is not correlated to the parameters
of the string material, but only a tool of representation.
24
11(x t)
I
Incoming Wave
a-
(a)
t'= t,
1(x, t0
Boundary Interaction
0
(b)
= 12
)
(x,t3
Transmitted Wave
Reflect
Cl,
R
X
a(c)t =0c
)
11(x,t4
Reflec
Transmitted Wave
CI
R
(d) t =t4
-P-[C
'
-- a
-
R
Figure 11: Displacements of a triangular wave, the reflection and transmission process in a string system, wave pulse
shown moving towards the junction towards the positive x direction.
25
5. Wave Pulse Interaction with a Clamped End and a Free End
In this section we are studying the reflection of waves on both at a free end, and a clamped or
fixed end. As discussed in section 4, we are interested in semi-infinite media. These cases that
we will be studying refer to the junction interaction limiting the reflection and transmission
between two media. It is only logical when there is a free end, or a clamped end there will be no
wave pulse transmission.
5.1 Wave Pulse Interactions with a Clamped End
4(x, t)
Cq
4iJ=f(X'Cqt)
IF
1b
Ir
I
p ,A ,E
x
x=
Figure 12: Semi-infinite rod system with a clamped end.
The first case we will be looking at is the clamped end case. Figure 12 shows a semi-infinite rod
system with a clamped end. The wave traveling down the rod system in the positive x direction is
has the form i (x, t) =
fi (x -
cq t). The figure shows a triangular wave for simplicity. The rod
26
system has density, cross sectional area, and modulus of elasticity p, A, and E. Figure 12 shows a
triangular wave for simplicity. To better understand the interaction of between the wave pulse
and the clamped end, we need to express the reflection of the wave pulse in terms of the
incoming wave.
The boundary conditions at a clamped end are defined by the fact that the clamped end cannot
move. If we denote the reflected wave displacement as 'r (x, t) = g, (x - cq t) boundary
condition can be defined asi
fi((Ot) + fr(Ot) = 0
(5-1)
fi(-cqt) +gr(cqt) = 0
(5-2)
or
R= fj(ct)
,gr(cqt)
-
r(Ot) -
M04~t
1
(5-3)
Using equation (5-3) it can be shown that the reflected displacement wave function is
fr(X,
0
= gr(x -
cqt) =
Rffi(x
- cqt) = -f(X
- cqt)
(5-4)
Figures 13, 14, and 15 investigate three different cases of wave pulses interacting with a clamped
or fixed end. The wave pulse retains its original speed once it exits the clamped end.' In all the
figures the red lines represent the reflected wave. It is required to note that the time frames for at
t2 ,
and t3 , are different from the duration segments chosen for t1 , and t4. This is to further study
the interaction at the boundary itself. Since the reflection coefficient = -1, the reflected wave
27
travels back in the negative F domain. The figures simulate the interference of the waves as a
wave pulse with less magnitude. If the waves are to be added up, there will be destructive
interference between the wave pulse and itself. In the duration of the interference the wave does
not hold the shape it originated in, it can be displayed to a better extent using dynamic
simulations of the phenomenon.
28
4x i)}
Cq
x
1
Superposition of wave components
4(X, t2)
Conponents of wave pulse
'
11
z
0
Superposition of wave components
4(x,
Conponents of wave pulse
3)
4(x, t3
)
0
Ix
0
4(x, 04)
Cq
g-
.
1
It
V
i
I
AF
P
0
Figure 13: Displacement in a semi-infinite rod with a clamped end at x=O, shown at different times for a triangular wave
pulse propagating in the positive x direction, the resultant wave due to the superposition represented by a smaller wave
pulse.
29
P(xt,)
p Cq
I1
x
x
0
4(Xt 2
1
V
0- rr
.
)
I
-)
41%
V
((x t 3
Components
)
Superposition of wave components
4(Xt 2
Components of wave pulse
)
Superposition of wave components
of wave pulse
11 __O_ 1~r
0
(X t3)
I
so
x
L
.3
Y
x
-
-i
0
0
)
4(xt
Cq O
0
Figure 14: Displacement in a semi-infinite rod with a clamped end at x=O, shown at different times for a sine wave pulse
propagating in the positive x direction, the resultant wave due to the superposition represented by a smaller wave pulse.
30
4(x,ti)
-~Cq
a
TC
2
Superposition of wave components
-
I
10- x
x
Components of wave pulse
4(Xt2)
4(X, t,)
F
0
0
I
4(x, t 3
)
Components of wave pulse
4(x,t 3
)
Superposition of wave components
I
0
0
4
)
(X,
x
Cq .-
Li
0
Figure 15: Displacement in a semi-infinite rod with a clamped end at x=O, shown at different times for a square wave
pulse propagating in the positive x direction, the resultant wave due to the superposition represented by a smaller wave
pulse.
31
5.2 Wave Pulse Interactions with a Free End
Cq
4i =fi(X-Cqt)
p , A ,E
Figure 16: Semi-infinite rod system with a free end, the green connection represents a free end simulation, which
disconnects the wall from the semi-infinite rod, and holds the boundary conditions established.
In this section we will be considering a free end. Figure 16 above shows a semi-infinite rod
system connected to a free end boundary. The free end is illustrated as green slab between the
semi-infinite rod system and the wall. The semi-infinite rod system has a density, cross sectional
area, and modulus of elasticity defined to be p, A, and E. For simplicity the figure shows an
incoming triangular wave traveling in the positive x direction. The incoming wave equation is
fi (x, t) =
fh (x -
Cq t) . As in the previous sections, appropriate boundary conditions are required
to be able to further study the interaction between the wave pulse and the free end. The needed
boundary condition describes the axial force, which is by definition the sum of the incoming
wave and reflected portion of the wave pulse at the free end. We will denote the reflected wave
pulse as fr (x, t) = gr (x - cq t). Components of wave pulse
32
The force boundary condition at the free end can be expressed as'
E1A 1 j(0, t) + EjAj L (0, t) = 0
(5-5)
To further understand the interaction at the free end, the reflection coefficient is defined. It can
be shown after integrating the boundary condition in respect to time, that the reflection
coefficient is
Rf
f i(-cqt)
(5-6)
(,t)
fi(0,t)
gr(Cqt)
Following the same argument in the previous section, we need establish in which domain the
wave pulse will be traveling in. It can be shown that
fr(X,t) = gr(x -
cqt) = Rffi(x
-
cqt) = fi(x - cqt)
(5-7)
Equation (5-7) shows that the wave will be traveling back in the same domain as it originated
from.
Figures 17, 18, and 19, show three different cases of the discussed phenomenon. The figures
show a triangular, sinusoidal, and square wave pulses traveling in the positive x direction. The
semi-infinite rod system shown in the examples presented has the same properties as the semiinfinite rod system presented in figure 16. When wave reaches a free end, it will be reflected
back in the same domain, and travel at the same speed to which it originated with. The
interaction at the boundary is presented as if the wave builds up on its self. The wave in fact
undergoes a constructive interference with itself at the boundary.
33
joCq
A
ft-'r
0
)
(xt2
I
.
-
IF
Y
I"x
4(X,t4)
i
Aq
l0
0
Figure 17: Displacement in a semi-infinite rod with a free end at x=O, shown at different times for a triangular wave pulse
propagating in the positive x direction.
34
-b-
">1-x
0
)
C(x t2
-x
)
(x, t3
0
Cq
~
- x
0
Figure 18: Displacement in a semi-infinite rod with a free end at x=O, shown at different times for a sine wave pulse
propagating in the positive x direction.
35
(x t,)
~Cq
-ox
0
)
4(x, t2
-x-
)
4(X,t 3
T_
-
x-
0
Cq
~
0
Figure 19: Displacement in a semi-infinite rod with a free end at x=O, shown at different times for a square wave pulse
propagating in the positive x direction.
36
6. Conclusion
Using Inkscape as both a graphical and a computational tool it was possible to represent the
behavior of wave pulses as they interacted with different conditions. Mathematical models for
the phenomena were presented in section 4. These mathematical models described the interaction
between the wave pulses and the junction between different media. Graphical representation of
wave pulses was depicted for the interaction between media. The forms and details shown
graphically are mechanically correct from a mathematical stand point. Graphical representation
of a triangular, square and sinusoidal wave was presented.
The subsequent sections studied the interaction of a wave pulse with a clamped end, and free
end. The appropriate mathematical model was presented, and as in the previous section the forms
and details graphically presented are in fact mechanically correct from a mathematical stand
point. In the future, if this study is to be continued, a way to graphically represent the destructive
interference the wave undergoes upon reaching the fixed end is needed. The graphs presented
can aid in the understanding and teaching of wave pulses behaviors as they interact with
boundary condition.
37
References
1. James H. Williams, Jr., Wave Propagation, Draft of Notes, MIT, 2013.
38