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Physics 272 Laboratory Experiments
Dr. Greg Severn
Spring 2016
What are the characteristic or allowed resonant frequencies of the standing waves on
a stretched string of given µ, L, F and so on? In making the measurements described
below, try to discover experimentally what these are and how to predict them. In
order to do this well, you will need to make careful measurements, grasp conceptual
ideas, and compose a brief mathematical argument to make your model explicit.
You have a variable frequency function generator to drive waves on the string.
How will you bend it to your purpose? You will need to assess the correspondence
between your model predictions and your experimental measurements (discrepancy)
and, separately, assess the extent to which the measured quantities are reliable in
themselves (uncertainty).
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1.1
Mechanical standing waves on a stretched string
Introduction
I want to draw your attention to mechanical waves on a stretched string. The transverse waves characteristic of such
systems exhibit many fundamentally important phenomena: constructive and destructive interference, phase shifts,
and resonance to name a few. But they also possess a discrete1 set of characteristic frequencies and wavelengths
due to boundary conditions. For a given set of fundamental parameter having to do with the mass of the string, its
length, its tension, and so forth. Not just any frequency of vibration will excite a resonance (also called a ’mode’).
In this laboratory exercise, you will try to discover what these resonance frequencies are for a given tension, mass
density (it’s a linear one...), and the boundary conditions. You will play around looking for the allowed modes, and,
we’ll define ‘playing’ as the process of doing careful experiments, varying things, trying things, observing things, and
finding things out, having ideas and testing them, and keeping good records to aid in doing all of the above.
It has been shown that the speed of such waves can be written
s
F
,
(1)
v=
µ
where F is the tension2 in the string and µ is the mass density. The spatial periodicity of the waves is given by the
wave length λ and the temporal periodicity by the period T . These periodicities have important reciprocals which
give the number of complete periods per meter and the number of complete periods per second. These are the wave
number and angular frequency, respectively, k = 2π/λ, and ω = 2π/T , permit the following expression for traveling
waves,
y(x, t) = sin (kx − ωt),
(2)
ω
= λf = v,
(3)
k
where v is the speed of transverse wave defined above, and the frequency f is ω/2π. If then boundary conditions are
(experimentally) enforced, namely that
y(x = 0, t) = y(x = L, t) = 0,
(4)
1 Very soon we will be able to appreciate the correspondence between this sort of discreteness with certain properties of quantum
mechanical systems, and we will be able to draw analogies between quantum and classical systems like standing waves on a stretched
string in order to obtain a better conceptual grasp of certain quantum concepts. File this away for the near future!
2 Many text books choose T to denote the tension in the string, which is fair enough. I chose to use the more familiar nomenclature for
force, F , so you wouldn’t confuse that physical quantity for the temporal period, T . It is almost impossible however to avoid all collisions
with meanings and symbols. For example, you may find yourself wondering about the relationship between the k defined in this lab and
k you learned about in simple harmonic motion. They are different!
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then standing waves occur, may occur, of the form
y(x, t) = A sin (kx) cos (ωt),
(5)
where only a certain discrete set of wavenumbers are allowed, and, because of (3), only a certain discrete set of
frequencies are allowed. The amplitude is the continuous thing that depends on the energy of the excitation, how
far you twist a knob. But the allowed wavenumbers and frequencies are characteristic of the system. All of this
is familiar already, of course you will be expected to know the meaning of the words ‘node’ and ‘antinode’ before
arriving at the lab! Equations (1)-(5) encode in mathematical expressions an abstract model of the real system we
aim to understand.
1.2
Experimental background, and the question to be answered
We will use a ‘diving board’ style vibrator to excite transverse oscillations near one boundary of the stretched string.
A partial set up is shown in figure 1. A piston style driver is shown in the figure, but that makes no essential
difference in understanding how the experiment works.
Figure 1: The left most figure shows a driver exciting a resonance, or standing wave, on the stretched string, and the
right most figure is a cartoon of the experimental set up. It’s mostly complete. What is left out? Have a complete
set up in your lab notebook!
1.2.1
The experimental question
What are the characteristic or allowed resonant frequencies of this system (given µ, L, F and so on)? In making
the measurements described below, try to discover experimentally what these are and how to predict them. In
order to do this well, you will need to make careful measurements, grasp conceptual ideas, and compose a brief
mathematical argument. And you will need to assess the correspondence between your model predictions and your
experimental measurements (discrepancy) and, separately, assess the extent to which the measured quantities are
reliable in themselves (uncertainty).
Measure, then, the length and the mass density of the string, and set its tension. Set the output voltage on the
frequency generator to 10Vpp − ish, and slowly ramp up the frequency until the first resonance is found.
1. Measure, observe, record, the frequencies of the fundamental and the next 4 harmonics, for at least 3 different
tensions. Be careful to observe the spatial shape of each mode with a sketch or cartoon that exhibits the
essential features. Record your results in a reasonable data table in your lab notebook. Leave columns for
expected frequency, uncertainty, discrepancy, and so on.
2. What is the frequency width of the resonance? How sharp are they? Try to quantify this somehow and explain
(this will have an experimental answer). I am not looking for something rigorously precise, but rather something
useful. Explain how you made your determination.
3. How might you display the relationship between experiment and theory in the most direct way? Might a
table be just as good? What is the basic import of the graph? What can you say about the sequence of
eigenfrequencies? Discrete? Continuous? Linearly spaced? Quadratically? Exponentially? What is the most
important thing to observe? Make a graph that compares the model prediction (the abstract representation of
the real thing....) with your measurements. What is the most direct way to do this? A question for you: will
the ’theory curve’ have uncertainty associated with it. How does one exhibit this graphically?
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4. What measurements must be made to calculate the velocity of the waves on the string?
√ Can this be measured
directly? Can you make a separate plot of the results, either as v 2 vs. T , or v vs. T . Again, how best to
compare theory to the model prediction? On page 2 of your report, you must indicate the message encoded in
the (properly labeled) plot.
1.3
Questions and Calculations
1. In your own words, explain why are only certain (i.e. a discrete set, not a continuous set) resonant frequencies
possible on the stretched string?
2. How would you have to alter the system to make an arbitrarily chosen frequency into a resonant frequency?
3. Exhibit a calculation for the 5th normal mode’s resonant frequency for this system.
4. Is the percentage discrepancy about as big as the percentage uncertainty? How would you estimate the
percentage uncertainty?
5. For the second resonance, there is a node precisely midway between the two boundaries. That means there
are two antinodes which are exactly λ/4 from the boundaries. Here is my problem: a wave heading for the
boundary travels only λ/4 and then reflects, and then travels another distance of λ/4 where it interferes with
itself. So the path length difference is half of a wavelength! But shouldn’t that lead to destructive interference?!
Please resolve this paradox.
6. Okay, just humor me. Perform the following addition and interpret:
y(x, t) = A sin (kx − ωt) + A sin (kx − ωt + 2(L − x)k + π),
simplifying as much as possible, where x is any point on the string of length L, and where k = 2L/n, and
where n is some integer. Yes, this k is the general expression for all the resonant wavenumbers the boundary
conditions for our system.
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