Physics 111: Elementary Physics
Laboratory C
Speed of Sound
1.
Introduction
The transmission of energy by propagation of waves is one of the most common phenomena
which we experience. Our lives would be radically different if we could not communicate with each other
readily through the use of sound waves. Life would be impossible if it were not for the energy coming to
the Earth from the Sun in the form of electromagnetic waves. While the effects of waves can be intriguing
and sometimes beautiful, as in the case of light wave phenomena, all waves share characteristics which can
be better understood by studying sound waves. In particular, all sound and light waves are oscillatory
phenomena, and so they both have the characteristic of frequency, the number of cycles or oscillations per
second, commonly expressed in the unit hertz (Hz). The waves propagate with characteristic speeds. All
light, regardless of frequency (or color), propagates in the absence of matter with a speed of approximately
3.00 x 108 m/s. When passing through matter, the speed of light depends both on the nature of the material
and on the frequency of the light. Sound also propagates with a speed which depends on the nature of the
material through which the sound is passing, but to a high degree, the speed of the sound is independent of
the frequency f, of the sound. A person’s voice is made up of sounds of different frequencies uttered
simultaneously. If these different components did propagate at different speeds, a person’s voice would
sound different if the person speaking were close to us rather than far away. Far away, the more slowly
moving sound components would reach us later than the faster moving ones, and a word spoken at a
distance would sound slurred and of variable pitch. For sound in dry air at 0° C the speed of sound is about
332 m/s. Since changes in temperature affect gases readily, the medium changes according to the
temperature, and one can show that over the normal range of temperature of air, the speed of sound in dry
air at a Celsius temperature t is given by the expression
v(t) = 332(1+ t/546)m/s .
(1)
The period T, the time required for a wave to complete one whole cycle, is just the inverse of the
frequency; T, equals 1/f. The distance which the wave travels in a single period is called the wavelength, λ,
and so λ equals the product of the period T and the speed v, vT. Replacement of T by 1/f and
rearrangement gives the fundamental relation
v = λf .
(2)
Another important property which all waves have in common is that of resonance. When the waves
are in a cavity of a particular size, the wave will reflect from the walls of the cavity and the reflected wave
can interact with the original wave to augment or to diminish it. This is called, respectively, constructive or
destructive interference and is responsible for standing waves similar to those observed on strings. This
wave phenomenon is important in giving musical instruments their characteristic tones or pitches. It is
possible to use a source of sound of known frequency and the standing wave pattern created by the
resonance phenomenon to determine the wavelength of the sound. Using Eq. (2) an experimental value
vexp, for the speed of sound can be obtained. This measured value of v can then be compared with the
theoretical value vtheo, given by Eq. (1).
2.
Procedure
When a source of sound, such as a vibrating tuning fork, is held at the end of a tube, the sound
wave will propagate to the other end of the tube and be reflected. This reflection is different if the tube is
closed at the far end than if it is open. In either case, the reflected wave usually will interfere destructively
with the original wave, and the sound in the tube is diminished. However at certain characteristic lengths,
related to the wavelength of the sound, the interaction produces a notably augmented volume of sound in
the tube. If the tube is closed at the end away from the sound source, the characteristic lengths at which
resonance occurs are λ/4, 3 λ/4, 5 λ/4, 7 λ/4, etc., assuming that the physical length of the tube is great
enough to accommodate all these resonances. (Actually, the lengths of the resonant cavities are just
slightly less than these ideal lengths, but the differences are approximately independent of the particular
length and so will cancel out in the procedure to be used here.)
There will be available a plastic tube on which there is a stripe calibrated in cm. The tube is fitted
with a plunger, and the length of the cavity then can be adjusted by moving the plunger to different
positions. The experiment proceeds by mounting the tube on the work table and fitting the plunger into the
tube almost to the end. A tuning fork is struck and held at the end away from the plunger. As the fork
continues to vibrate, the plunger is moved back into the tube to locate the first position when a strong
resonance is observed. Since the sound increases in strength and then diminishes as resonance is past, you
will need to judge where the resonance is strongest. Note the approximate uncertainty in the position of the
resonance. Record the distance of the plunger from the open end of the tube as λ/4. Go on to the next
resonances, at 3λ/4, at 5λ/4, and (if the length of the tube permits for the frequency of the fork) 7λ/4.
Subtracting successive values will then give values for λ/2 which can be averaged. From this, an average
value of λ is obtained which can be multiplied by the known frequency of the tuning fork to find the speed
of the sound wave. Repeat this procedure with a tuning fork of a different frequency. This will let you
compare the speed of sound as observed for different frequencies.
Physics 111: Elementary Physics
Pre-Lab Exercise
Speed of Sound
Name: ______________________
1.
Section: ________
A typical data set might read as follows. Complete the calculations.
Tuning fork frequency = 512 Hz
λ/4 =
17.5±0.5 cm
3λ/4 =
51.5±0.5 cm
5λ/4 =
85.2±0.5 cm
7λ/4 =
119.0±0.5 cm
3λ/4 - λ/4 = λ/2 __________________
5λ/4 - 3λ/4 = λ/2 __________________
7λ/4 - 5λ/4 = λ/2 __________________
(λ/2)avg ____________
vexp = λf = ___________
Air temperature = 22.5º C
vtheo(t) ___________
(from Eq.(1))
∆v(%) = 100[(vexp - vtheo)/ vtheo] = ______________
2.
Recall the uncertainty in locating the point at which a particular resonance was strongest. If the error
bounds on the initial data (±0.5 cm) are used most accurately, the corresponding bounds on λ/2 are ±
1.0 cm. Use the upper and lower values of this range of values of λ, and find the range of values which
this implies for vexp.
3.
Sketch the standing wave pattern of the sound wave in the tube from the tip of the plunger to the open
end, for the first two resonances. Remember, the closed end of the tube must be a node and anti-nodes
in this pattern represent positions of maximum volume (i.e. the resonances).
4.
If the frequency of the tuning fork were twice that of the one which was used, what changes in the
positions of the resonances would have occurred?
5. State briefly (100 words or less) the objectives of this experiment and the hypothesis which is to be
tested.
Physics 111: Elementary Physics
Lab Report
Speed of Sound
Investigators:
________________________ ,
_______________________
________________________ ,
_______________________
________________________
Date: _____________
Procedure: Describe briefly (200 words or less) the procedures used in this experiment.
Data:
Tuning fork frequency ________
λ/4 _____±_____
3λ/4 - λ/4 = λ/2 __________
3λ/4 _____±_____
5λ/4 - 3λ/4 = λ/2 __________
5λ/4 _____±_____
7λ/4 - 5λ/4 = λ/2 __________
7λ/4 _____±_____
(λ/2)avg _________
vexp = λf = _____±_____
Tuning fork frequency ________
λ/4 _____±_____
3λ/4 - λ/4 = λ/2 __________
3λ/4 _____±_____
5λ/4 - 3λ/4 = λ/2 __________
5λ/4 _____±_____
7λ/4 - 5λ/4 = λ/2 __________
7λ/4 _____±_____
(λ/2)avg _________
vexp = λf = _____±_____
Average vexp from the two forks = ___________±________
Air temperature __________
vtheo(t) = __________ (from Eq. (1))
∆v(%) = 100 [|vexp - vtheo|/ vtheo] = ______________
(Use the average vexp in the above formula)
Conclusions: State your conclusions about the outcome of the experiment, in particular
whether or not the data support the idea that the speed of sound is independent of
frequency.