Does the Wave Equation Really Work

1
Does the Wave Equation Really Work?

• Michael A. Karls
• Ball State University
• November 5, 2005

2
Modeling a Vibrating String

• Donald C. Armstead
• Michael A. Karls

3
The Harmonic Oscillator Problem

• A 20 g mass is attached to the bottom of a
  vertical spring hanging from the ceiling.
• The springs 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.

4
Newtons Second Law

• The (vector) sum of all forces acting on a body
  is equal to the bodys mass times its
  acceleration, i.e. F ma.
• The form Fma was first given by Leonhard Euler
  in 1752, sixty-five years after Newton published
  his Principia.

5
Hookes 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).

6
The Harmonic Oscillator Model

• Using Newtons Second Law and Hookes 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 springs 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

7
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 TIs
  website (http//education.ti.com/) , position
  data can be collected, plotted, and compared to
  solution (3).

8
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.

9
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.

x0
xa
10
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 strings linear density ? and
  tension T.
• Equations (6) - (8) specify boundary and initial
  conditions.

11
The Wave Equation

• Solving the wave equation was one of the major
  mathematical problems of the 18th century.
• First derived and studied by Jean dAlembert in
  1746, it was also looked at by Euler (1748),
  Daniel Bernoulli (1753), and Joseph-Louis
  Lagrange (1759).

12
The Vibrating String Model (cont.)

• Using separation of variables, we find
• where

13
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-locations 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.

14
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
15
Initial String Displacement and Velocity

• For initial displacement we choose the piecewise
  linear function
• For initial velocity, we take g(x)0.

16
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.

17
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 ------
18
Model vs. Experimental Results (cont.)
Model ------ Actual - - - -
Figure 3
19
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.

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
41
Vibrating String

• That was the last frame!

42
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.

43
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.

44
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.

45
Solution to the Revised Model

• Again, using separation of variables, we find
  that the solution to (12)-(15) is
• where

46
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

47
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 centers 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!

48
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 ------
49
Revised Model vs. Experimental Results (cont.)
Model ------ Actual - - - -
Figure 5
50
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.

51
Damped Vibrating String
52
Damped Vibrating String
53
Damped Vibrating String
54
Damped Vibrating String
55
Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
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Damped Vibrating String
68
Damped Vibrating String
69
Damped Vibrating String
70
Damped Vibrating String
71
Damped Vibrating String
72
Damped Vibrating String

• That was the last frame!

73
Damped Model Error

• We find that over approximately four periods, the
  MSSE is 0.000296651 m2 or 0.0172236 m.

74
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.

75
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 springs period in time is about 0.263
  seconds, so using the relationship c 2 a/P, we
  find c 7.6 m/sec.

76
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 ------
77
Spring Model vs. Experimental Results (cont.)
Model ------ Actual ------
Figure 7
78
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.

79
Vibrating Spring
80
Vibrating Spring
81
Vibrating Spring
82
Vibrating Spring
83
Vibrating Spring
84
Vibrating Spring
85
Vibrating Spring
86
Vibrating Spring
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Vibrating Spring
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Vibrating Spring
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Vibrating Spring
90
Vibrating Spring
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Vibrating Spring
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Vibrating Spring
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Vibrating Spring
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Vibrating Spring
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Vibrating Spring
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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
105
Vibrating Spring

• That was the last frame!

106
Spring Model Error

• We find that over approximately three periods,
  the MSSE is 0.000174036 m2 or 0.0130551 m.

107
Conclusions and Further Questions

• Using inexpensive, modern equipment (rope,
  spring, video camera, and computer software),
  weve 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.

108
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?

109
Conclusions and Further Questions (cont.)

• One Final Question!
• Did dAlembert, Euler, Bernoulli, or Lagrange
  ever verify these models via experiment?
• If so, how?

110
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//www-groups.dcs.st-and.ac.uk/history/