Brown University
Physics Department
PHYS 0160
LAB B - 255
Waves and Resonances
References:
Blatt, Frank J., Principles of Physics 3rd edition, 1989 chapters 15,16.
Pasco Instruction Manual and Experiment Guide for Resonance Tube (WA-9612)
Equipment:
a) Pasco Digital Function Generator/Amplifier
b) Pasco Resonance Tube
c) Pasco Variable Frequency Mechanical Wave Driver
d) Tektronix Digital Oscilloscope (TDS 2012B)
Introduction:
Any kind of periodic wave - mechanical, acoustical, electromagnetic, even de Broglie’s matter
waves, can be brought into resonance by suitably limiting the region in which the waves exist.
Mechanical waves on a string achieve resonance when the proper length of string is used;
acoustic waves in a gas resonator in a tube of proper length. For all harmonic waves of single
frequency f, the wave speed v and the wavelength λ are related by
v = fλ
The achievement of resonance makes it possible to determine λ (or λ/2, or λ/4) by geometrical
measurements. Then knowledge of either v or f allows the other property to be measured.
In this experiment we examine the resonance of two types of waves - the transverse waves
of a vibrating string and the longitudinal sound wave. Studying these two physically very
different phenomena together emphasizes how similarly they are described from the point of
view of wave motion.
Theory:
Part I. Transverse Waves (The Stretched Vibrating String)
Consider a uniform string of linear density 𝜌 (mass per unit length), held in a horizontal
position. One end of the string is fastened to an electric vibrator, which can move that end
vertically in simple harmonic motion of small amplitude, at a frequency f of 120 cycles per
second. The other end of the string passes over a pulley, and from this end hangs a weight pan,
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on which known weights can maintain a tension force T in the string. The vibrations we will
observe take place over the length L of string between the pulley and the vibrator. When the vibrator is turned on, its first deflection sends a transverse wave, in the vertical
plane, traveling along the string at a speed v given by
v = (T/ρ)1/2
(1)
Let us say that this first displacement of the string is upward, although it does not matter
in the final outcome. Regardless of what else the vibrator does, this wave (which has the very
small amplitude of the vibrator) travels to the pulley. Here the string is held essentially
motionless, as if it were clamped. The wave is reflected with a reversal of phase; that is, having
come to the pulley as an upward displacement of the string, it returns to the vibrator as a
downward displacement.
Upon reaching the vibrator, this reflected wave is again reflected (another change of
phase) and starts back, once more as an upward displacement, toward the pulley. Since the
vibrator is moving meanwhile, in general a jumble of such waves is created, in no particular
phase relation with each other; consequently the string vibrates only slightly and irregularly.
Suppose, however, that the first wave reflected from the pulley reaches the vibrator, to be
reflected as an upward displacement, exactly when the vibrator is again sending out a wave of
upward displacement. Now a wave of doubled amplitude goes to the pulley end, is reflected to
the vibrator, and is there joined by another upward deflection. After many cycles of such
continually augmented deflection, yet in a time short when measured in seconds, the string
acquires fairly large amplitude. We say that it is in resonance.
The build-up of amplitude eventually ceases and a steady state is reached, for two main
reasons. First, there are dissipative losses of energy within the constantly distorted string, and at
the ends where the string slaps against the constraining pulley or vibrator. These losses become
greater as the amplitude grows. Second, when the amplitude becomes large the string is
appreciably stretched from its quiescent length; the wave speed changes somewhat from that
given by Eq. 1, and the many reflected waves no longer reach the vibrator at quite the right time
to be completely reinforced.
The mathematical condition for resonance is easily seen. The time for a single wave to
travel down the string and back, a time equal to 2L/v, must equal some integral number n of the
vibrator’s period 1/f. The equation is therefore
2L/v = n/f
(n = 1, 2, 3, ...)
(2)
Solving Eq. 2 for f and using Eq. 1 yields
f = (n/2L) (T/ρ)1/2
(3)
If n = 1, its lowest value, the first returning wave reaches the vibrator at its first upward swing
after the initial one, and this continues for all later motions of the waves. Therefore each wave
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going toward the pulley is always an upward displacement if the first one was, and each reflected
wave traveling back to the vibrator is always a downward displacement. Vibrator end Vibrator end L n=1 L n=2 n=3 Fig. 1 The whole string swings first up, then down, producing a blur indicated by the shading in Fig. 1.
Use of Eq. 2 shows that v/f, the wavelength λ of the wave, satisfies
or
λ = 2L
(4)
L = λ/2 (5)
Equation 5 evidently agrees with the half-wavelength pattern in Fig.l. If n = 2, Eqs. 4 and 5 are altered to give
λ = L (n=2)
(6)
which is also depicted in Fig.1. The physical reason for this new pattern is this. It now takes two
periods (n = 2) of the vibrator for the wave to go to the pulley and back, and (v being always
constant) one period to reach the pulley. Just when the wave is reflected at the pulley, as a
downward displacement, the next wave of upward displacement starts from the vibrator and goes
toward the pulley. The first reflected wave and the second outgoing wave meet halfway along the
string, with opposite displacements. There is therefore a node, a point of zero displacement, at
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the midpoint of the string. Successive reflected and outgoing waves that are nearest in time
continue to behave in this way, and so the node remains as the amplitude builds up. There is no
contradiction with our previous description of resonance; each reflected wave reaches, and is
reflected at, the vibrator just in time to join the next-but-one outgoing wave.
In each case, the pattern is called a standing wave, or stationary wave, a superposition of
waves traveling in both directions, with certain definite speeds and wavelengths.
You can work out for yourself, in both mathematical and physical terms, what happens when n =
3 (see Fig. 1), or any higher integer.
Procedure: The density of the string used in this experiment will be given you by the
instructor.
(a) Start with a weight (suggested by the instructor) on the string to be vibrated. Remember to
include the weight pan itself as part of the total weight. Turn on the vibrator and adjust L
carefully, to obtain resonance with n = 1. After recording the data, leave L fixed and decrease the
suspended weight (thus changing T), to obtain successively the patterns for n = 2, 3, and higher
values if possible. Again work carefully to obtain maximum amplitude.
(b) Next, using only the smallest suspended weight from Part (a), vary L to obtain patterns for
several values of n.
(c) Finally, use a doubled string, slightly twisted so as to vibrate as a single string with doubled
density 𝜌. Repeat Part (a) for at least two values of n.
Since the generalization of Eqs. 4 and 6 is
λ = 2L/n
(7)
(T/ρ)1/2 = 2fL/n
(8)
and since fλ = v, Eqs. 1 and 7 yield
Compute the two sides of Eq. 8, which are respectively the theoretical and experimental values of v, and tabulate them for each measured case. Tabulate parts (a), (b), and (c) separately, and
within each part according to the variable being altered. Discuss the extent of agreement; if there
is any definite trend of discrepancy, discuss it also (for example, might ρ always have been a
little wrong?). Regard f and n as perfectly definite numbers, not subject to error.
Part II. Longitudinal Waves (Sound Waves In a Tube)
In this part of the experiment, the speed of sound in air is determined by producing resonance of
standing waves in a tube, closed at one end by a movable piston and open at the other to allow
stimulation of the acoustic waves by means of a sound source driven by an electronic oscillator.
In its crude form, the experiment attains only modest accuracy in determining the speed of
sound. The main aim is to demonstrate principles; in particular, the similarity of behavior when
compared with the visible waves of Part I.
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A sound wave in a gas can be measured and described in various terms, such as the
displacement of particles in the gas, the associated particle velocity, or changes in the pressure
and density of the gas. For our purposes, it is convenient to describe the behavior of the particle
velocity.
Figure 2 sketches the arrangement. For all practical purposes, the particle velocity in the gas at
the piston face must be zero. At the open end of the tube, 0, the particle velocity can have as
great a magnitude as the wave can produce. With the piston in position A, a quarter-wavelength
back from the open end, resonant standing waves are set up in this part of the tube, because, in an
acoustic wave, adjacent positions of zero and maximum particle velocity are a quarterwavelength apart. Just as the resonance of standing waves on a stretched string is shown by
maximum amplitude of the waves, acoustic resonance is shown by maximum sound - and this
will be the basis for determining resonance in this experiment. A microphone, placed at the open
end of the tube and directed to receive the reflected waves emerging from the tube, is monitored
by an oscilloscope. Therefore λ/4 can be determined as approximately the distance A-0. The
determination is only approximate because the standing waves “spill out” of the open end of the
tube to some extent, and the distance A-0 is a little less than λ/4.
Next, let the piston be moved back to the next position, B that produces resonance,
detected in the same way. From the sketch of particle velocity in Fig.2, it can be seen that the
distance A-B equals λ/2. There is no end effect between A and B, but the total distance B-0 is
still a little less than 3λ/4. Given a long enough tube, other positions C, D, ... can be found that
produce resonance and maximum sound. Each successive point is a distance λ/2 beyond the
preceding one.
When the piston is set halfway between A and B, it is at a position that for resonance would
require maximum particle velocity at the piston face. This being impossible, the halfway position
is marked by a minimum in the detected sound. Halfway between B and C the same thing occurs,
and so on. You will recognize these points as the node points of the acoustic wave.
To determine the wave peed from this information, given the rather limited accuracy of the data,
a graphical method that averages the information from several maxima and minima is the best
approach. (Recall that, with the frequency known, we can find v from the best value of λ
determined by the experiment.) The axis of ordinates (see Fig. 3) is marked off in equal intervals
1, 2, 3... indicating the number of quarter wavelengths in the open (right-hand) part of the tube.
The axis of abscissas is marked in actual lengths of the open part, which are easily measured on
the rod that moves the piston. In Fig. 3, crosses represent positions of resonance (maximum
sound), while circles show positions of nodes (minimum sound). A straight line is drawn to represent all points as well as possible, with equal scatter above and below the line, on the
average. This line, which because of the “end effect” will not pass through the point of zero
distance, should be extrapolated to two points. On the axis of abscissa it will intersect at point a;
and at a point which we call b, it will reach an exact integral number N of quarter-wavelengths
(N=5 in Fig. 3). The distance a-b, which is 5λ/4 in the example, is in general Nλ/4, and gives us
the best average value of λ to be determined from the data. In your own procedure, b should of
course he related to the maximum value of quarter wavelengths that you can detect.
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Physics Department
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LAB B - 255
Tube Piston B A λ/2 Sound Source Particle λ/4 Position Fig. 2 Procedure:
1. The frequency dial of the oscillator should be calibrated by comparing the sound from the
speaker with that from each of several tuning forks. Adjust the output of the oscillator until it
produces about the same loudness as the fork. Use the “zero-beat” method (cf. the
measurement of the velocity of light) to match the frequency of the oscillator to that of the
fork, and record the actual and dial-setting frequencies.
2. Set the oscillator to a frequency of a few hundred cycles, and scan through by moving the
piston without trying to measure distances. There are two purposes for this scan. The first is
to adjust the position of the microphone and the settings of the oscilloscope for the best
detection of maxima and minima. The second is to insure that at least a half-dozen such
points can be detected over the range of the piston positions. Once the frequency has been set
to achieve this condition, record it, then measure the distances as indicated in Fig. 3.
Also record room temperature, for you will find when you compare your measured value of
the speed of sound in air with that from accurate experiments (see, for example, the
“Chemical Rubber” handbook) that it is a function of temperature.
Determine by the graphical method described, and so calculate v. From the scatter of the
points on the graph, estimate the precision achieved in your measurement of the speed of
sound. From your graph, determine the end-effect length, the distance by which A-O falls
short of λ/4.
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3. Repeat the procedure at another frequency, differing by at least fifty-percent from the
frequency used in part (2).
N Length of open tube a A B C b Fig. 3 140516 7