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Computational
Oscillations and Waves
Seamus Lagan, Dept. of Physics and Astronomy,
Whittier College, Whittier, CA 90608
[email protected]
This sophomore-level course is not a programming course nor is
it a course in numerical methods but elements of both are
present. It is a physics course covering oscillatory motion and
waves. However the mathematics used in these topics is carried
out almost exclusively on the computer using symbolic (Maple)
and numerical (IDL) software. Students use the software to:
evaluate integrals, solve differential equations, solve
simultaneous equations, plot functions and/or data, create series
expansions of functions, find roots of equations and work with
complex numbers.
It is hoped that the students will be motivated and
prepared to use these tools in later courses. Therefore it
is a pre-requisite for almost all other upper-division
physics courses.
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The text for this course is “Computation and Problem Solving
in Undergraduate Physics” by David M. Cook.
(available from David Cook at Lawrence University,
[email protected])
The traditional topics for such a course are covered, beginning
with the block-spring system and adding standard damping and
forcing. However, finding the solutions to the equations is
carried out using the computational tools. The goal is to get the
students familiar with the software and its usefulness as a
tool to understanding the physics, not as an end in itself.
Emphasis is placed on getting the students to use the on-line
help available in both software packages so that they can
become somewhat self-sufficient.
Week 1: they do not solve the SHO problem but simply
investigate the solution that is found from inspection. They
learn how to create simple plots in both programs. This involves
learning how to manipulate arrays in IDL. They also learn how
to import data into IDL (motion detector data for a
block/spring). They investigate the phase relationships between
position, velocity and acceleration by using Maple to
differentiate the position function and studying the plots. They
can do the same with real data and numerical differentiation in
IDL.
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Weeks 2-3: the emphasis is on actually solving ODEs.
This is straightforward enough in Maple using the dsolve
command. The initial value problem introduces them to
solving simultaneous equations in Maple. However the
students must also learn how to solve a simple ODE
numerically. They are introduced to the simple Euler
method and use it to solve the SHO equation. This is a
good vehicle for illustrating the limitations of the method
and the care that must be taken to achieve sufficient
accuracy when solving equations numerically. At this
point they are learning programming basics such as
defining variable types, creating functions and procedures,
and using loops. They write their own procedure for
implementing the Euler method and use it to reproduce the
solution to the SHO problem that they have already
obtained in Maple.
Weeks 3 - 7: Students are introduced to built-in
procedures for solving ODEs in IDL. They use these to
study other oscillatory systems without closed-form
solutions. Examining these solutions requires learning to
manipulate arrays in IDL, e.g. accessing part of an array to
eliminate initial transients, finding max and min values,
calculating the average value of a variable over one period.
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Week 5: Maple is very useful when dealing with
molecular vibrations. Using various expressions for the
interatomic potential, students can examine the form of the
potential well, find the equilibrium position, use series
expansions to help find small amplitude solutions. They
can differentiate the potential energy function to get the
force expression and then use IDL to solve the equation of
motion numerically.
Weeks 8 – 11: Maple is used extensively to deal with
waves, in particular to gain insight into the difference
between the functions representing traveling waves and
standing waves, as well as combinations of the two. Using
Maple’s animation capabilities, students can visually
represent the time-dependent solutions. They can also
create animations corresponding to reflection and
transmission at boundaries between strings.
(These animation capabilities are used again in our
Quantum Mechanics course to examine the timedependence of wave functions for one-dimensional
systems e.g. particle-in-a-box and the quantum oscillator.)
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Week
1
Topic
Block/spring system. Analytic solution of the equation of
motion by inspection. The initial value problem.
Intro to the software. Plotting data. Solving simultaneous
equations using Maple. Using arrays in IDL.
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Solving the diff. equ. symbolically and numerically.
Euler method, accuracy, programming basics in IDL.
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Other systems that oscillate harmonically (simple & physical
pendulum, LC circuit, floating cylinder).
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Non-linear systems (exact pendulum, floating cone).
Numerical solutions using IDL.
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Molecular vibrations. Potential wells.
Series expansions of functions in Maple.
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Standard damping. Light, critical and heavy damping.
Solving the ODE. Complex representation in Maple.
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Driven oscillations.
Solving the ODE. Creating resonance and phase plots.
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Waves and their mathematical representation.
Animated plots in Maple.
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Travelling and standing waves. Waves on a string. Sound
waves.
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The wave equation.
Solving the PDE in Maple.
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Reflection and transmission of waves. Characteristic
impedance.
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Special topics/Projects
Table 1: Semester Schedule
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USE IN OTHER COURSES
PHYS330 Electromagnetic Theory: Students use Maple
extensively to evaluate integrals and the LinearAlgebra and
VectorCalculus packages are employed to take gradient,
divergence and curl of vector fields and also to plot vector
fields.
PHYS350 Quantum Mechanics: Maple is used heavily for
integral evaluations (expectation values and inner products).
Students also use it to approximate the time dependence of
wave functions and then create animations e.g. wave packets for
a particle-in-a-box or quantum oscillator. Boundary value
problems for the finite square well and variations thereon have
been solved.
PHYS360 Astrophysics: Some IDL assignments have been
incorporated, e.g. creating an H-R diagram using data from the
Yale Bright Star Catalog, and producing a simulated image of
the Sun that reproduces the limb darkening effect.
PHYS310 Mechanics: Students have been asked to use IDL to
calculate the motion of three bodies interacting gravitationally.
IDL is used heavily by students doing astronomy research.
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General Comments
 We removed oscillations and waves from both our introductory
(Freshman) courses and upper-division Mechanics (PHYS310)
since the material is covered in some detail in this course. This
has given us extra flexibility in the content of these other courses.
 Some interesting physics associated with oscillations and waves is
not covered. For instance, plasma oscillations, acoustic vibrations,
light scattering, evanescent waves etc. We used to have an
elective course on oscillations and waves that covered these
topics. We no longer offer this course.
 Some students find it quite difficult to learn physics and
computation in the same course. Indeed, I continue to struggle
with the problem of how to divide classroom time between
description of the physics and instruction on use of the
computational tools. It is a delicate balance.
 There is also a problem of using two different computing
environments at the same time. Students tend to get the two
confused as regards syntax. The learning curve for IDL is steeper
than for Maple.
 This course counts towards our Scientific Computing Minor.
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 We still need to do more to incorporate these computational
tools into the upper-division courses. The difficulty is that
not all our faculty members are familiar with all of the tools.
I will be making more efforts to inform my colleagues what
the students can do with Maple and IDL after having taken
this course and how I have incorporated their use into my
own upper-division courses.
STUDENT COMMENTS
When asked how they feel about using computational tools
some comments were:
“Computational tools take some effort to familiarize
with but once you do they are addicting.”
“I think it is a must and IDL/Maple could be used more
commonly in more courses.”
“Computational tools demonstrate different physical
phenomena very effectively and it’s very easy to see
what is happening (for example, waves and potential
wells).”
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Interactive Data Language (IDL)
This is an interactive software package that can be run under Unix,
Windows and Mac operating systems. It is very powerful for
manipulating arrays and for data visualization. It is available through
ITT Visual Information Solutions (www.ittvis.com/idl/)
Below is shown the Graphical User Interface for IDL. At the top is the
text editing window. Below that is a variable window showing defined
variable types and values. Below that is the log window recording the
commands that were input from the command line at the very bottom.
Fig. 2 : GUI interface for IDL in Windows.
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IDL Homework Solution: Driven oscillator with a third
power damping force (Fdamp = b*v3).
First write the function file that defines the two differential
equations. Note the third power dependence of the damping
term.
function driven2,t,n
common param,k,m,b,F,w
return,[n[1],(-k/m)*n[0]-(b/m)*n[1]^3+(F/m)*cos(w*t)]
end
Now set up the common area of memory and input the values of
the parameters.
IDL> .COMPILE "E:\computational\driven2.pro"
Compiled module: DRIVEN2.
IDL> common param,k,m,b,F,w
IDL> m=5.0
IDL> b=20.0
IDL> k=100.0
IDL> F=2.0
IDL> w=3.0
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Solve the differential equation, first using the default tolerance.
IDL>ludiffeq_23,'driven2',t,n,t0=0,tf=200,init=[2.0,0.0],
tol=.001,maxsteps=10000000000
IDL> plot,t,n[0,*],xrange=[150,200],title="w="+string(w)
Fig.3 : Driven oscillator with third-power damping.
Solve with a smaller tolerance just to be sure you can trust the
solution. (There was no significant difference.)
Repeat this process for ten different driving frequencies, noting
the amplitude of the oscillation in each plot. To graph the
response curve, create an array of frequencies and an array of
corresponding amplitudes.
IDL> A=[.025,.03,.035,.11,.07,.014,.0075,.05,.09,.025]
IDL> freq=[2.0,2.5,3.0,4.47,5.0,7.0,8.5,3.5,4.0,6.0]
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Now plot these values and also put the standard damping
response curve on the graph.
IDL> w=findgen(801)/100+2
IDL> Astandard=(F/m)/((w^2-k/m)^2+(b*w/m)^2)^0.5
IDL> plot,freq,A,psym=2,title="response curve for third power
damping !C dashed line is standard damping
response",ymargin=[8,8],xtitle="driving frequency
w",ytitle="amplitude"
IDL> oplot,w,Astandard,linestyle=2
Fig.4 : Response curve for oscillator with third-power damping.
There is a much greater resonance effect for the third-power
damping. The resonance frequency for standard damping
(sqrt(o2-b2/2m2) = 3.46) is quite a bit different from o (= 4.47)
whereas it is very close to o for the third-power damping.
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MAPLE
A typical Maple notebook to solve an initial value problem.
> restart;
Input x(t)
> x(t):=C*cos(1.5*t+phi);
x( t ) := C cos( 1.5 t )
get v(t) by differentiating x(t)
> v(t):=diff(x(t),t);
v( t ) := 1.5 C sin( 1.5 t )
Define eq1 as x(t=0)=3.1 using the eval command.
> eq1:=eval(x(t),t=0)=3.1;
eq1 := C cos(  )3.1
Similarly define eq2 as v(t=0)=-0.5
> eq2:=eval(v(t),t=0)=-0.5;
eq2 := 1.5 C sin(  )-0.5
Now solve the two equations simultaneously for C and phi.
> sol:=solve({eq1,eq2},{C,phi});
sol := { C-3.117869643 , -3.034477330 },
{ C3.117869643 , 0.1071153234 }
There are two solutions and sol is given as a sequence of the two
solutions. Each solution is itself a set. I always want the amplitude C
to be positive so I want to pick the second solution.
> sol[2];
{ C3.117869643 , 0.1071153234 }
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Maple Notebook: Reflection/transmission on a string.
> restart; with(plots):
Consider two strings connected and a sinusoidal wave coming in and
getting reflected off the join. The tension is the same in each string but
the mass per unit lengths are different. Therefore the speeds of
propagation of the wave on the two strings (c1 and c2) are different.
For convenience I will take c1 = 2*pi and then we will look at what
happens when we vary c2.
>c1:=2*Pi;c2:=(ratio)*c1;
Define a traveling wave expression and then convert it to a function so
we can generate the incident, transmitted and reflected waves by
changing the arguments of the function.
>f(z,t):=cos(z-c1*t); f:=unapply(f(z,t),[z,t]);
(Z = characteristic impedance, T = tension, c = speed of propagation,
R = reflection coefficient) Z=T/c and for two strings connected the
tension is the same in both. Therefore R=(Z1-Z2)/(Z1+Z2)=(T/c1 T/c2)/(T/c1 + T/c2)=(1/c1 - 1/c2)/(1/c1 +1/c2) . If we multiply above
and below by c1*c2 then we get R = (c2-c1)/(c2+c1) for the reflection
coefficient.
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>R:=(c2-c1)/(c2+c1);T:=1+R;
Now create the incoming, transmitted and reflected waves.
>yi(z,t):=f(z,t); yr(z,t):=R*f(0,t+z/c1); yt(z,t):=T*f(0,t-z/c2);
>ratio:=4;
We get the full motion of the string by superimposing all three waves
on the string using the “piecewise” command.
>animate(piecewise(z<0,yi(z,t)+yr(z,t),z>0,yt(z,t)),
z=10..10,t=0..3,numpoints=200,frames=20,thickness=3,
view=[-10..10,-5..5]);
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The actual output in Maple is an animated plot. Below I show some
frames from two animations, one with the second string being very
light (high velocity of propagation) so that the junction of the two
strings is like a free end. The second example is one in which the left
side of the junction does not have a pure standing wave on it. Students
can use this animation to investigate the concept of standing wave
ratio.
Fig. 1 : Frames from Maple animations of waves on a string.
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A MID-TERM EXAM
Maple is used for parts (a) - (d) and it involves simple algebraic
manipulation and solving single algebraic equations. Parts (f) and (g)
require the use of IDL to numerically solve the equation of motion.
Students do not have to write their own procedure to solve the ODE
but can use a “canned” procedure (from Lawrence University) that
they have used before in class.
A cone of density 1 is placed in a fluid of density 2 and floats at rest
as shown.
(a) Show (use Maple) that
L = (1/2)1/3 H
The volume of a cone is given by
(1/3)  R2 H = (1/3)  H3 Tan2()
Use the following values for the rest of the
problem: 1 = 700 kg/m3 2 = 1000 kg/m3
 = /6 rad
H = 0.3 m
If the cone is pushed down into the fluid and
released then it will oscillate.
(b) What is the net upward force on the
cone when the cone is displaced a
distance x downward into the fluid?
Answer:
Fnet is approx 728x +2734x2 + 3421x3
For small values of x the cone will perform Simple Harmonic Motion
because the force is “spring-like”.
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(c) How large can x be so that the force does not deviate from a
“spring-like” force by more than 5%
Answer: about 1.3 cm.
(d) For small amplitude oscillations what is the value of the period?
(Hint: the mass of the cone can be found from the values of the
quantities given above.)
Answer: 0.598 s
(e) Write down the equation of motion for the cone using the force
expression in part (b). Rewrite as two first order equations.
(f) Using IDL, create the appropriate function file for this set of
equations and then use ludiffeq_23 to solve the equations. Get x(t)
for t = 0 to 6.0 s and use initial conditions x(0) = 0.08 m and v(0) =
0.0 m/s. Plot the position versus time in a graph with a title and axis
labels. (Make sure that the tolerance is set small enough that you are
getting accurate results. Show how you do this!)
(g) Overplot the approximate Simple Harmonic solution
corresponding to x(0)=0.08 and v(0)=0.0. Explain physically why
the cone’s motion is asymmetric about the equilibrium position.
Fig. 5: Position plot for a cone oscillating in water.
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A STUDENT PROJECT
Thermal expansion of solids due to the non-linearity of the
interatomic force.
When a solid is heated up it expands. See p452 of Giancoli for a short
explanation of how this is related to the oscillations of atoms in a
potential well.
Consider NaCl (common table salt). If we model the potential energy
U(r) between the two atoms by
(Blatt “Modern Physics” p207, A = 10-13 eV. nm20) then we can
determine how much the solid expands as we increase the energy of
the oscillation.
The energy of oscillation (above the bottom of the potential well) is
related to the temperature by equipartition of energy, E ≈ k T where k
is Boltzmann’s constant and T is the temperature in Kelvin. Find the
average interatomic spacing for an isolated Na-Cl oscillating pair at
different temperatures and use this spacing to evaluate the volume of a
unit cell in crystalline NaCl.
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Fig. 6 : Maple plot of exact and approx. potential wells for NaCl.
Fig. 7: Temperature dependence of NaCl volume found by the
students. It corresponds to a coefficient of thermal expansion of
about 3 x 10-5 K-1