Illinois
State
University
Department
of
Physics
Physics
112
Physics
for
Scientists
and
Engineers
III
Experimental
Physics
Laboratory
1
Vibrating Strings (BC4)
Introduction
Standing waves on a string of length L are described in Chapter 18 of Physics for Scientists and
Engineers by Serway and Jewett. The basic concept is that a wave will interfere with its own
reflection. When this happens, the wave and its reflection are generally out of phase and will
destructively interfere, leading to partial cancellation. However, when the phase relationship is
just right, constructive interference is maximized and standing waves are observed. Standing
waves are characterized by a set of nodes and antinodes, where the amplitudes of the antinodes
are maximized at resonance. Fig. 1 shows the standing wave patterns for the fundamental
frequency and the second and third harmonics. It is clear that each end of the string is fixed and
there are n–1 nodes and n antinodes, where n is the number of the harmonic (the fundamental is
the first harmonic). The points of greatest constructive interference are called antinodes, and
have the greatest amplitudes. The points of complete destructive interference are called nodes,
and remain stationary. The endpoints are ignored.
Figure 1. The fundamental (first), second, and third harmonics for a string with fixed ends.
Standing waves can be observed in the laboratory using a set-up similar to the one shown in Fig.
2. The experimental set-up consists of a string that is attached to a post at one end of an optical
bench, and it is draped over a pulley at the other end of the bench. A vibrator is placed in
between the post and pulley. When the vibrator is active, standing waves can be created between
the vibrator and pulley by moving the vibrator back and forth with respect to the pulley until the
source wave and reflected wave are in phase with each other.
Vibrator
Table
Mg
Figure 2. Experimental Set-up.
The distance between successive nodes of a standing wave is half the wavelength λ. Standing
waves only occur when
λ
(1)
L=n ,
2
where n can be any positive integer. As we found in class, the wavelength, frequency, f, and
velocity, v, of the wave are related as λ f = v, which can be substituted into Eq. (1) to yield
nv
(2)
f=
.
2L
In the lecture part of the course, we showed that the velocity of the wave on the string is given by
v=
T
,
µ
(3)
where T is the tension and m is the mass density of the string. Equations (2) and (3) can now be
combined to yield
⎛ n ⎞ 1/2
f =⎜
T .
⎝ 2L µ1/2 ⎟⎠
(4)
Equation (4) shows us that the oscillation frequency is linearly dependent on the harmonic
number and inversely proportional to the square root of the mass density. A rearrangement of
this equation might be more useful depending on the measurements that we wish to complete.
We can precisely determine the mass density by carrying out a linear least squares fit of several
pairs of n, f values for the same values of µ, T, and L. The tension can be computed from the
slope of the line. In fact this is generally a very precise way to determine the mass density. Most
commonly available mass scales to not provide sufficient precision to be useful for very low
mass strings. We should also consider that the mass density might change as the tension is
increased. How might Eq. (4) be used to determine the mass density as a function of tension?
Consider the fact that the mass density decreases as tension is increased, consequently, the mass
density in Eq. (4) is not a constant when data is collected for f when T is varied.
Objective
This exercise provides an opportunity to verify many of the equations describing wave motion on
strings. We will experimentally verify the equations describing standing waves and determine
the mass density of a string. Additionally, we will verify the amplitude and frequency
dependence of the power for a wave on a string. Perhaps more importantly, this experiment
gives you a chance to gain practical familiarity with the behavior of a vibrating string.
Procedure
Exploration: Attach the components to the optical bench as shown in Fig. 2. Starting with a
vibrator frequency of 100 Hz and a mass of 150 g, determine the optimal vibrator position for
observing two antinodes. Feel free to vary any value to gain an improved understanding of
standing waves. Does a node always occur at the vibrator?
Measurements: Determine the mass density of one string using as many standing wave
harmonics as possible. Make sure you identify standing wave frequencies to the nearest Hz.
Can you observe a standing wave at the fundamental frequency? Repeat your measurement of µ
for a different tension. Try a different string and determine its mass density. Then repeat that
measurement with the same tension but a different string length L. You now have 4
measurements of mass density and you should be able to take your results and draw some
interesting conclusions.
Tinker with the set-up to make sure that you completely understand it. Before leaving the room,
make sure that your data is self-consistent. If it is not, then identify the problem and correct it.
If you place a finger on the vibrating string, what happens as you move your finger back and
forth along the string? Do you observe any higher harmonics at the same time you observe some
of the lower harmonics?