Does the Wave Equation
Really Work?
Michael A. Karls
Ball State University
November 5, 2005
Modeling a Vibrating String
Donald C. Armstead
Michael A. Karls
The Harmonic Oscillator
Problem



A 20 g mass is attached to the bottom
of a vertical spring hanging from the
ceiling.
The spring’s force constant has been
measured to be 5 N/m.
If the mass is pulled down 10 cm and
released, find a model for the position
of the mass at any later time.
3
Newton’s Second Law

The (vector) sum of all forces acting on
a body is equal to the body’s mass
times it’s acceleration, i.e. F = ma.*

*The form F=ma was first given by Leonhard Euler in 1752,
sixty-five years after Newton published his Principia.
4
Hooke’s Law

A body on a smooth surface is connected to a
helical spring. If the spring is stretched a small
distance from its equilibrium position, the spring will
exert a force on the body given by F = -kx, where x
is the displacement from equilibrium. We call k the
force constant.

*This law is a special case of a more general relation, dealing
with the deformation of elastic bodies, discovered by Robert
Hooke (1678).
5
The Harmonic Oscillator
Model



Using Newton’s Second Law and Hooke’s Law, a model for a mass on a
spring with no external forces is given by the following initial value problem:
where proportionality constant 2 = k/m depends on the mass m and spring’s
force constant k, u0 is the initial displacement, v0 is the initial velocity, and u(t)
is the position of the mass at any time t.
We find that the solution to (1) - (3) is given by
6
Verifying the Harmonic Oscillator
Model Experimentally

Using a Texas Instruments
Calculator Based
Laboratory (CBL) with a
motion sensor, a TI-85
calculator, and a program
available from TI’s website
(http://education.ti.com/) ,
position data can be
collected, plotted, and
compared to solution (3).
7
The Vibrating String Problem



A physical phenomenon related to the
harmonic oscillator is the vibrating string.
Consider a perfectly flexible string with both
ends fixed at the same height.
Our goal is to find a model for the vertical
displacement at any point of the string at
any time after the string is set into motion.
8
The Vibrating String Model




Let u(x,t) be the vertical
displacement of the string at
any point of the string, at any
time.
Let x = 0 and x = a correspond
to the left and right end of the
string, respectively.
Assume that the only forces
on the string are due to gravity
and the string's internal
tension.
Assume that the initial position
and initial velocity at each
point of the string are given by
sectionally smooth functions
f(x) and g(x), respectively.
x=0
x=a
9
The Vibrating String Model
(cont.)

Applying Newton's Second Law to a small piece of the string,
we find that a model for the displacement u(x,t) is the following
initial value-boundary value problem:

Equation (5) is known as the one-dimensional wave equation
with proportionality constant c2 = T/ related to the string’s
linear density  and tension T.
Equations (6) - (8) specify boundary and initial conditions.

10
The Wave Equation


Solving the wave equation was one of
the major mathematical problems of
the 18th century.
First derived and studied by Jean
d’Alembert in 1746, it was also looked
at by Euler (1748), Daniel Bernoulli
(1753), and Joseph-Louis Lagrange
(1759).
11
The Vibrating String Model
(cont.)

Using separation of variables, we find

where
12
Checking the Vibrating String
Model Experimentally






To test our model, we stretch a
piece of string between two fixed
poles.
Tape is placed at seven positions
along the string so displacement
data can be collected at the same
x-location’s over time.
The center of the string is
displaced, released, and allowed
to move freely.
Using a stationary digital video
camera, we film the vibrating
string.
World-in-Motion software is used
to record string displacements at
each of the seven marked
positions.
Data is collected every 1/30 of a
second.
13
Assigning Values to
Coefficients in Our Model

We need to specify the
parameters in our model.




Length of string: a = 0.965 m.
String center: xm= 0.485 m.
Initial center displacement:
d = -0.126 m.
To find c, we use the fact that
in our solution, the period P in
time is related to coefficient c
by c = 2a/P.


From Figure 1, which shows
the displacement of the
center of the string over time,
we find that P is
approximately 0.165
seconds.
It follows that c should be
about 11.70 m/sec.
Figure 1
14
Initial String Displacement and
Velocity

For initial displacement we choose the piecewise
linear function:

For initial velocity, we take g(x)≡0.
15
Determining the Number of
Terms in Our Model


Using (10) and (11),
we can compute the
coefficients an and bn
of our solution (9).
Graphically comparing
the nth partial sum of
(9) at t = 0 to the initial
position function f(x),
we find that fifty terms
in (9) appear to be
enough.
16
Model vs. Experimental
Results






Figure 2 compares model and
actual center displacement over
time.
Clearly, the model and actual data
appear to have the same period.
However, our model does not
attain the same amplitude as the
measured data over time.
In fact, the measured amplitude
decreases over time.
This physical phenomenon is
known as damping.
The next slide compares our
model and experiment at each of
the seven points on the string over
time!
Figure 2
Model: -----Actual: ------
17
Model vs. Experimental
Results (cont.)
String
Model
0.1
u
0
-0.1
0.6
0.12
0.225
Model: -----Actual: - - - -
0.35
0.485
0.6
x
0.725
Figure 3
0.4
t
0.2
0.83
0
18
Model vs. Experimental
Results (cont.)



The next 21 slides show our results as
“snapshots” in time at 1/30 second
intervals.
Dots represent tape positions along
the string.
The solid curve represents the model.
19
Vibrating String
0
0.1
0.05
0
x
m
-0.05
-0.1
0.2
0.4
0.6
0.8
20
Vibrating String
21
Vibrating String
22
Vibrating String
23
Vibrating String
24
Vibrating String
25
Vibrating String
26
Vibrating String
27
Vibrating String
28
Vibrating String
29
Vibrating String
30
Vibrating String
31
Vibrating String
32
Vibrating String
33
Vibrating String
34
Vibrating String
35
Vibrating String
36
Vibrating String
37
Vibrating String
38
Vibrating String
39
Vibrating String
40
Vibrating String

That was the last frame!
41
Vibrating String Model Error



How far are we off?
One measure of the error is the mean of the
sum of the squares for error (MSSE) which
is the average of the sum of the squares of
the differences between the measured and
model data values.
We find that over four periods, the MSSE is
0.000890763 m2 or 0.0298457 m.
42
Revising Our Model



From our first experiment, it is clear that
there is some damping occurring.
As is done for the harmonic oscillator, we
can assume that the damping force at a
point on the string is proportional to the
velocity of the string at that point.
This leads to a new model with an extra
term in the wave equation.
43
Revised Model with Damping


Equation (12) is known as the one-dimensional wave equation
with damping, with damping factor .
Coefficient c, initial values, and boundary values are the same
as before.
44
Solution to the Revised Model

Again, using separation of variables, we find that the solution
to (12)-(15) is

where
45
Solution to the Revised Model
(cont.)

with

Note that if g(x)≡0, the RHS of (18) is zero for all n,
it follows that
46
Assigning Values to Coefficients in
the Revised Model


For our revised model, we keep the same values for a, c, and
initial position and velocity functions f(x) and g(x).
The only parameter we still need to find is the damping
coefficient .




Using the string center’s period in time of P = 0.165 seconds,
c = 11.70 m/sec, and the fact that 2 = P1, we guess that 
should be approximately 0.0127 sec/m2.
Once we know , the coefficients in our solution (16) can be
found with (17) - (19). Again we use fifty terms in (16).
Unfortunately, our choice of  does not produce enough
damping in our model.
Through trial and error, we find that  = 0.0253 sec/m2 gives
reasonable results!
47
Revised Model vs.
Experimental Results



Figure 4 compares model
and actual center
displacement over time.
With damping included,
there appears to be much
better agreement between
model and experiment!
The next slide compares
our model and experiment
at each of the seven points
on the string over time!
Figure 4
Model: -----Actual: -----48
Revised Model vs.
Experimental Results (cont.)
Model: -----Actual: - - - Figure 5
49
Revised Model vs.
Experimental Results (cont.)



The next 21 slides show our results as
“snapshots” in time at 1/30 second
intervals.
Dots represent tape positions along
the string.
The solid curve represents the model.
50
Damped Vibrating String
51
Damped Vibrating String
52
Damped Vibrating String
53
Damped Vibrating String
54
Damped Vibrating String
55
Damped Vibrating String
56
Damped Vibrating String
57
Damped Vibrating String
58
Damped Vibrating String
59
Damped Vibrating String
60
Damped Vibrating String
61
Damped Vibrating String
62
Damped Vibrating String
63
Damped Vibrating String
64
Damped Vibrating String
65
Damped Vibrating String
66
Damped Vibrating String
67
Damped Vibrating String
68
Damped Vibrating String
69
Damped Vibrating String
70
Damped Vibrating String
71
Damped Vibrating String

That was the last frame!
72
Damped Model Error

We find that over approximately four
periods, the MSSE is 0.000296651 m2
or 0.0172236 m.
73
Modeling a Vibrating Spring



In order to see if there is
any other way to reduce
the amount of error we are
seeing in our models, we
repeat our experiment with
a long thin spring in place
of our string.
Since the spring is “hollow”,
we assume damping due to
air resistance is negligable.
Therefore, the classic wave
equation IVBVP may be a
reasonable model.
74
Assigning Values to Coefficients in
the Spring Model



For our spring model, we choose the same
initial position and initial velocity functions
f(x) and g(x).
For this experiment, a = 1 m, xm = 0.5 m, d =
-0.135 m.
The spring’s period in time is about 0.263
seconds, so using the relationship c = 2 a/P,
we find c = 7.6 m/sec.
75
Spring Model vs. Experimental
Results



Figure 6 compares model
and actual center
displacement over time,
after shifting our model in
time by -0.02 seconds.
There appears to be even
better agreement than in
the damped case!
The next slide compares
our model and experiment
at each of the seven
points on the string over
time!
Figure 6
Model: -----Actual: ------
76
Spring Model vs. Experimental
Results (cont.)
Model: -----Actual: -----Figure 7
77
Spring Model vs. Experimental
Results (cont.)



The next 26 slides show our results as
“snapshots” in time at 1/30 second
intervals.
Dots represent tape positions along
the string.
The solid curve represents the model.
78
Vibrating Spring
79
Vibrating Spring
80
Vibrating Spring
81
Vibrating Spring
82
Vibrating Spring
83
Vibrating Spring
84
Vibrating Spring
85
Vibrating Spring
86
Vibrating Spring
87
Vibrating Spring
88
Vibrating Spring
89
Vibrating Spring
90
Vibrating Spring
91
Vibrating Spring
92
Vibrating Spring
93
Vibrating Spring
94
Vibrating Spring
95
Vibrating Spring
96
Vibrating Spring
97
Vibrating Spring
98
Vibrating Spring
99
Vibrating Spring
100
Vibrating Spring
101
Vibrating Spring
102
Vibrating Spring
103
Vibrating Spring
104
Vibrating Spring

That was the last frame!
105
Spring Model Error

We find that over approximately three
periods, the MSSE is 0.000174036 m2
or 0.0130551 m.
106
Conclusions and Further
Questions


Using “inexpensive”, modern equipment
(rope, spring, video camera, and computer
software), we’ve been able to show that the
wave equation works!
As is often the case in modeling, we had to
revise our initial model or experimental
setup to get a model that matches reality.
107
Conclusions and Further
Questions (cont.)




How would the model work without
“wobbly” poles?
Would a thinner string reduce
damping?
What is really going on with the
spring?
Would adding an external force to the
models reduce error?
108
Conclusions and Further
Questions (cont.)

One Final Question!


Did d’Alembert, Euler, Bernoulli, or
Lagrange ever verify these models via
experiment?
If so, how?
109
References





William E. Boyce and Richard C. Diprima,
Elementary Differential Equations and Boundary
Value Problems (8th ed).
David Halliday and Robert Resnick, Fundamentals
of Physics (2cd ed).
David L. Powers, Boundary Value Problems (3rd
ed).
Raymond A. Serway, Physics for Scientists and
Engineers with Modern Physics (3rd ed).
St. Andrews History of Math Website: http://wwwgroups.dcs.st-and.ac.uk/~history/
110