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Physics 211 Vibrations, Waves and Optics
Project 3 Presentation The Slinky Machine
The Slinky Machine Experiment : Analysis and Understanding
Ilia Papa
Abstract
The following article is an analysis and explanation of the dispersion relation for
a mechanical wave on a slinky. The remarkable aspect of the article is the way in which
the waves frequency is correlated to its wavenumber and hence, to its wavelength. In
order to do that we set up the experiment in which a slinky was attached on a rod by
several strings and we started driving it at different rates. After we conducted the
experiment and we got our results, we plotted the graph of angular frequency vs. the
wavenumber which also helped us estimate the cutoff frequency. Also, from the
theoretical analysis we managed to determine the equations for the group and phase
velocity and the cutoff frequency. We compared the cutoff frequency with the
experimental one that we determined from the graph and we were finally able to draw
some conclusions for all the possible errors.
I. Introduction
The term dispersion refers to a mechanical wave which results when different
frequencies propagate at different velocities. The dispersion relation can be illustrated
in a plot of vs. k, where is the angular frequency and k is the wavenumber.
Therefore, from the graph we can see how the frequency depends on the wavenumber
(=(k)). We distinguish the velocity of the wave into two different ones which
represent the high-frequency carriers propagation (phase velocity) and the low-
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frequency envelopes propagation (group velocity). In order to write down the exact
formula for phase and group velocity we first need to define (k), something which is
shown below in the theoretical analysis. Moreover, it is feasible to obtain both the
phase and the group velocity from the graph, besides calculating them with the use of
formulas (the way is explained in more detail below). Furthermore, before proceeding
to the experimental procedure, an understanding of the wave cutoff frequency is
necessary. If the strings supporting the slinky were too short then it would be
impossible to observe a wave propagation for a very low frequency 1. Therefore, the
critical frequency which determines whether there will be propagation or not is called
cutoff frequency.
Finally, it is reasonable to assume that the motion of the slinky satisfies an equation
which will be similar to the wave equation
0
2 22
2 2
y yv
t x
=
(1).
However, since there is presence of gravity and the strings, the wave equation that
satisfies the above motion would be
0
2 22 2
02 2
y yv y
t x
=
(2)
in which
2
0
s
g
L = (3)
where 0 is the angular frequency, g is the gravity of Earth and Ls is the length of the
string,
0
2 T
v = (4)
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where v0 is the velocity of the wave, T is the period and is the mass density,
provided that sy L=1
II. Theoretica l Analysis
The next step that would complete our understanding of the dispersion relation is to
theoretically analyze the aspects of this experiment based on equation 2. The
difference between equation 2 and the wave equation is that in this case we take into
consideration the presence of gravity and the strings. As it is explained in the Wave
cutoff on a suspended slinkyarticle, the string connecting the slinky with the rod has a
mass of dx and experiences a restoring force of Td(dy/dx), where T is the tension
and y is the displacement. Additionally, there is a restoring force due to gravity
which is illustrated as (dx)gy/L1. The results of the addition of these two forces are
equations 3 and 4, while the wave equation takes the form of equation 2. Nevertheless,
the real challenge of the theoretical aspect of the experiment lies on the fact that there
is need to find the equations which will determine the phase and group velocities. For
this reason, we need to guess a solution which we will derive and plug in the equation
2. After that, by solving in terms of , we will find the dispersion relation of (k).
At first we have to guess a solution. Since we are dealing with waves, the most
appropriate equation which could illustrate a wave propagating to the right would be a
sinusoidal function of the form2:
( , ) sin( ) y x t A kx t = + (5)
Before we plug it in equation 2 we need to calculate the second partial derivative with
respect to time and space. Hence, we get:
With respect to time t:
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Taking into consideration the above information and before we proceeded to our
experiment the main purpose was to show that the dispersion relation would appear as
a hyperbolic function from which we would be able to calculate the phase and group
velocity. Furthermore, when we observed the motion of the wave we wanted to make
sure that the periods for each trial seemed consistent with the rate at which we moved
the slinky. Thus, trying to be as careful as possible while taking our measurements, our
expectations were to show that the frequency of the wave was positively correlated to
the wavenumber through a hyperbolic function which would allow us also determine
phase and group velocity.
IV. Experimental Procedure and Results
The first step for generating a precise experiment was to make sure that the slinky
was set up in the right way. We measured the length of the strings connecting the
metal rod with the slinky to be 35cm. In order to calculate the period of each trial we
used three different stopwatches for more accuracy. We drove the slinky at different
rates for 5 complete oscillations during each trial while we were taking a picture of it
and measuring its period. We conducted the same procedure 10 times having a
respective period and picture for each trial, which would allow us to identify the
different frequencies and wavelengths.
After we collected our data the first thing we did was to examine the pictures and
determine a wavelength for each wave. Using the formula
2k
= (12),
we calculated the corresponding wavenumber for each wavelength. Also, we calculated
the respective average frequencies from each period based on the formula
1f
T= (13),
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and then we found the angular frequencies using the formula
2 f = (14).
Continuing, we plotted vs. k and the graph we got was the following:
Figure 1: Plot of angular frequency (k) vs. the wave number k for a slinky which is
connected with a string of length Ls=35 cm.
Comparing the above graph with a second graph where the length of the strings
was longer, we can see that even if the values are different, the trend of the points is
similar, which means that the corresponding function would be of the same hyperbolic
form:
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Figure 2: Plot of angular velocity (k) vs. wave number k for a slinky which is
connected with a string of length Ls=72 cm.
At this point its convenient to explain how phase and group velocity can be
obtained from the graph. Phase velocity can be determined at any point of the graph
from the division of (k)/k, that is the division of the (k) and k values that represent
that specific point on the graph. Group velocity can be determined graphically from the
slope of the function at any given point since it is theoretically determined by the
derivative d/dk.
Finally, in order to complete our understanding around dispersion we need to find
and compare the cutoff frequency for both graphs, experimentally and theoretically.
Experimentally, the cutoff frequency can be estimated from the graph at the point
where k=0. So, for the first one the cutoff frequency is approximately =3.5 rad/sec
and for the second one is =2.8 rad/sec. These values are approximations since the
line used for the best fit does not represent a hyperbolic function but an exponential.
However, it looks similar enough with a hyperbolic function in order to fit between the
points and to give us an decent approximation. Thus, these estimations are reasonable
enough because the first experiment was conducted with a slinky attached to shorter
strings, which prevents it from propagating as much as the slinky with the longer
strings when both of them are driven at a very slow rate. As a result, the frequency in
which we would not be able to observe a successful propagation would be higher for
the first experiment. Theoretically, since we assume that there isnt any damping, we
can say that =0, and in order to calculate the cutoff frequency we can use equation
3. Thus, for the first experiment we have:
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0
9.81
0.35
5.29 / sec
s
g
L
rad
= =
=
=
For the second experiment we have:
0
9.81
0.72
3.69 / sec
s
g
L
rad
= =
=
=
Therefore, comparing these two values with our estimations from the graphs, the
resulting error we get for the first experiment is 34% and the for the second one is
24%.
V. Conclusion
The purpose of the experiment was to illustrate and analyze the dispersion relation
between the angular frequency and the wavenumber of a slinky. As we showed in the
article, the aim of the experiment was succesfully reached. We managed to plot a
graph which, besides any errors during the measurements, accurately shows the
dispertion realation as a hyperbolic function. Furthermore, the way in which the phase
and group velocity can be determined from the graph and theoretically was analyzed
and explained.
However, even if we consider the above experiment and its analysis succesful there
is an issue that needs to be pointed out. The experiment wasnt conducted in the best
possible way since there wasnt any driver for the motion of the slinky which would
precisely give us the period and therefore the frequency of the propagation. For this
reason, the driving of the slinky was conducted manually, something which led to a
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cutoff frequency when the motion was performed at a really slow rate. Thus, when
examining the picture for the low frequencies we couldnt observe a perfectly
propagated wave. The errors for the two experiments were 34% and 24%, which
justifies the above problem we had during the experiment. This issue was the major
problem which had a negative impact on the data analysis and therefore on the
resulting graph.
VI. References
1. Wave cutoff on a suspended slinky, G. Vandegrift, et al., American Journal of
Physics, v. 57, p. 949-951 (1989)
2. The Slinky as a model for transverse waves in a tenuous plasma, J. Blake and L.N.
Smith, American Journal of Physics, v. 47, p. 807-808, (1979)
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