Illinois State University
Department of Physics
Physics 112
Physics for Scientists and Engineers III
Experimental Physics Laboratory 4
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.
Fig. 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
. Standing waves only occur when
λ
L=n ,
2
(1)
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  f = v, which can be substituted into Eq. (1)
to yield
f=
nv
.
2L
(2)
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  is the mass density of the string. Eq. (2 ) and (3) can now
be combined to yield
f=


n
T 1/2 .
1/2
2Lμ
(4)
Eq. (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 , T, and L. 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 frequency is observed as a function of tension.
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
Attach the components to the optical bench as shown in Fig. 2. The strings are
guitar strings. Be sure to pick two strings of different mass density. Starting with a
vibrator frequency of 100 Hz and a mass of 150 g, or any other combination of values
that yield clear standing waves, determine the optimal vibrator position for observing
two antinodes. Feel free to vary any value to gain an improved understanding of
standing waves. Determine the mass density of the string and observe 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? You can
check your determination of the mass density of the string by doubling the string or
choosing a different tension and repeating your measurements. Do the new
measurements correspond to the new mass density? Now try a set of measurements
with a different tension. Does a node always occur at the vibrator? You should end up
with four different sets of data.
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?