Download Sound, Resonance and the Ear

640-151: Laboratory Exercise 1
Sound, Resonance and the Ear
Preparation
Read the background information that leads into Experiment 1, ie. pages 1 to 4, before the
laboratory session.
Introduction
In today’s experiment you will explore the nature of sound and the idea of resonance. When
a flute player blows across the mouthpiece of a flute “white noise” is produced. That is, a
jumble of sinusoidal pressure waves with frequencies across the frequency spectrum and
having a variety of amplitudes is produced. Of course, the sound the flute sends to the
hearer is quite different. Only a few frequencies of sound are sustained in the flute while the
rest die out very quickly. The flute seems to choose some of the frequencies in the white
noise. These are called the resonant frequencies and their values depend on a number of
parameters of the flute — most significantly on the length of the vibrating air column and
the fact that this column is effectively open at both ends.
This process of resonance is crucial to the production of sound, by your body as well as by
musical instruments. It is also the key to the first stage of sound detection by your ears. By
the end of today’s experiment you should have increased your understanding of both of
these aspects of sound’s behaviour.
Aims: To study the nature of sound, the importance of resonance, and applications to
production and detection of sound in the human body.
Reference
Serway, R A Principles of Physics (2nd edition) Chapters 13 and 14
Part A: The Nature of Sound-Waves
Waves transport energy and momentum through space without transporting matter. In
essence, the transport of the wave motion through a medium requires coupling between the
elements of the medium.
When a spring is momentarily compressed at one end, the elastic coupling between the coils
leads to transmission of a pulse along the spring.
Figure 1. A longitudinal wave pulse on a spring.
In the case of a sound wave, it is the elasticity of the air that leads to transmission of the
pulse. In effect, a pressure pulse moves through the air.
Sound, Resonance & The Ear
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Demonstration
A long tube with rubber diaphragms will be provided. You can demonstrate how the
elasticity of air leads to transmission of a pressure pulse by hitting the diaphragm at one
end of the tube, and observing the pressure pulse at the other end using a sensitive detector
like a very light lever.
The response at the far end of the tube is not simultaneous with the impulse applied at the
front. The speed with which the pressure pulse moves through the air depends on the
elasticity and density of the air (both dependent on the pressure and temperature). In air at
room temperature the speed of the sound pulse is c = 344 m s-1.
Later in this exercise you will measure the speed of sound.
If, instead of a single tap on the diaphragm, it is moved back and forth in a regular way,
with a frequency f s -1, a periodic set of pressure maxima and minima move down the tube
with a speed c. The spacing between any successive pressure maxima is called the
wavelength, λ, and the relation between the frequency, the wavelength, λ, and the wave
speed is simply
λf=c
(1)
Figure 2 below shows a representation of the situation at an instant of time.
(a)
Displacement from equilibrium of air molecules in a
sinusoidal sound wave versus position at some instant.
Points x 1 and x 3 are points of zero displacement.
(b)
Some representative molecules equally spaced at their
equilibrium positions before the sound wave arrives.
The arrows indicate the direction of displacement that
will be caused by the sound wave when it arrives.
(c)
Molecules near points x 1, x 2, and x 3 after the sound
wave arrives. Just to the left of x 1, the displacement is
negative, indicating the gas molecules are displaced to
the left, away from the point x 1, at this time. Just to the
right of x 1, the displacement is positive, indicating that
the molecules are displaced to the right, which is again
away from the point x 1. At x 2 the displacement is
positive and has its maximum value. Molecules on both
sides of x 2 are displaced to the right.
(d)
Density of air at this time. At point x 1, the density is a
minimum because the gas molecules on both sides are
displaced away from that point. At point x 3, the density
is a maximum because the molecules on both sides of
that point are displaced toward point x 3. Both are points
of zero displacement. At the point x 2 the density does
not change because the gas molecules on both sides of
that point have equal displacements in the same
direction. At x 2 the density is equal to the equilibrium
density and there is a maximum in displacement.
(e)
Pressure change, which is proportional to the density
change, versus position. The pressure change and
displacement are a quarter cycle out of phase.
An expression for this pressure change, p, is
Figure 2
p = A sin [ 2πx/λ – 2πft – φ ]
see page 367 of Serway
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Sound, Resonance & The Ear
Part B: Standing waves in a tube and flask
A tube
When a pressure pulse reaches the end of the tube, whether the tube is open to the
atmosphere or sealed, it will be reflected. Whether the pulse is reflected as a pulse of excess
pressure or reduced pressure depends on whether the end of the tube at which the reflection
occurs is open to the atmosphere or sealed. Regardless of this, the pressure variation at any
point in the tube will now be the sum of all the component pressures at that point.
Wave pulses moving in opposite
directions on a string. The shape of
the string when the pulses meet is
found by adding the displacements
of each separate pulse.
(a)
Superposition of pulses
having displacements in the
same direction.
(b)
Superposition of pulses
having opposite
displacements. Here the
algebraic addition of the
displacements amounts to a
subtraction of the
magnitudes.
Figure 3
If instead of a single pulse, a periodic wave is travelling within the tube, the resulting
pressure wave resulting from the superposition of the reflections is generally messy.
However, under certain special conditions the pressure variation within the tube no longer
consists of a travelling wave, but a “standing wave”. The frequencies at which this occurs
are called “resonances”.
Sound, Resonance & The Ear
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Figure 4
Figure 4 above shows the maximum pressure variation for air in the tube. At every point
along the tube, the pressure variation is a sinusoidal function that oscillates at the resonant
frequency. Note that when the end of the tube is open the pressure variation at that end is
always zero. That is the pressure is equal to atmospheric pressure. When the end of the
pipe is closed, there is always maximum amplitude at that end of the tube.
It should be clear that for a tube of length L, the standing waves that are allowed have
wavelengths consistent with:
For a tube open at both ends
L= n
λn
where n = 1, 2, 3 …
2
For a tube closed at one end
L= n
λn
where n = 1, 3, 5….
4
Since λf = c (equation 1) the conditions above lead to resonant frequencies given by
fn = n
c
2L
fn = n
c
4L
(2)
When n =1 the frequency is termed the “fundamental frequency”, so that we can see that
the allowed resonance frequencies are related to the fundamental frequency f1 by:
fn = n f1 where n = 1, 2, 3 …
fn = n f1 where n = 1, 3, 5 …
(3)
You will check these relationships during this experiment.
Note that it is possible to determine the speed of sound, c, by measuring the resonant
frequencies and applying the equations above.
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Sound, Resonance & The Ear
Experiment 1
a.
Resonance frequencies of tubes
(and measuring the speed of sound)
Equipment:
Two cardboard tubes, one open at both ends, one closed at one end
Signal generator
Small loudspeaker
Microphone and Cathode Ray Oscilloscope (CRO)
b.
Method:
Connect the loudspeaker to the signal generator and adjust the sound level so that it is
audible.
Using the open-ended tube, place the loudspeaker near one end and adjust the frequency
of the sound. You will hear the sound intensity increase at certain frequencies … these are
the resonant frequencies.
Adjust the frequency to the lowest resonant frequency (the fundamental). You should be
able to feel this one! Another check on whether the increase in loudness is really a resonance
of the tube is to rapidly move the tube in and out of the space under the speaker. If the
increase in loudness is a speaker resonance (a possibility) then the tube will make no
difference to the loudness of the sound. If it really is a tube resonance you will hear the
increase in loudness each time the tube is lined up with the speaker.
You can also estimate the fundamental frequency from the equation 2 above knowing L and
assuming the value of c to be 344 m s-1. Record the fundamental frequency, then find and
record as many of the higher resonant frequencies as you are able.
If the resonances are difficult to detect you will have access to a microphone and Cathode
Ray Oscilloscope (CRO). The CRO is effectively a voltmeter that displays voltage (the
output of the microphone) as a function of time across the screen. To detect a resonance
you will be seeking the frequencies that provide maximum amplitude of the signal detected
by the microphone. The basic controls of the CRO can be found in the appendix to this
exercise.
Repeat the experiment with the tube that is closed at one end.
c.
1.
Calculations and discussion
Show whether or not the ratios of the observed resonant frequencies to the
fundamental frequency are consistent with equations 3.
2.
From all the resonant frequencies you recorded, use equations 2 to calculate the speed
of sound.
3.
Separately plot the values of c you obtained with the two tubes as a function of the
frequency. Bearing in mind the errors involved, are all the estimates of c consistent, or
is there a systematic trend in the value? If so, suggest a reason for this.
4.
Take the average of the values of c. How does this compare with the known value of
344 m s -1?
Sound, Resonance & The Ear
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Other 3-D Objects
Every solid body has a set of resonant frequencies, for example, a violin, a room, a concert
hall, and importantly the human body. However, none are as simply related to the
fundamental frequency as for a tube of length L (equation 3). Helmholtz did the early
research on this in the 1870’s. He used a series of spherical resonators to determine how the
resonance frequencies depended on the dimensions.
Experiment 2.
a.
Resonant frequencies of a 3-d cavity
Equipment:
Glass Florence flask (ie. an approximately spherical flask)
Signal generator
Small loudspeaker
Microphone and CRO
b.
Method
The method here is the same as for experiment 1, but this time find the series of resonances
with the loudspeaker near the neck of the flask. First find and record the fundamental
frequency, and then as many of the higher resonances as you can. Assign n = 1 to the
fundamental frequency, and increasing values of n to the successive resonances.
c.
1.
Manipulation and discussion
Plot the frequencies observed (on the y-axis) as a function of the number (n) of the
resonance.
2.
Determine if there is a systematic trend in the data.
3.
Do the data conform to the predictions of equations 3?
Part C: Perception of Sound
The human auditory system is sensitive to sound of frequencies from 15 Hz to about
20 kHz. The range of hearing intensities varies by an incredible 12 orders of magnitude:
from the threshold of hearing with a sound intensity as small as 10-12 W m -2 (at a frequency
of 3 kHz), to the threshold of pain at 1 W m-2. Because of this extreme range, it is usual to
express the intensity, I, of sound at a point in space (eg. at the ear) in terms of a relative
logarithmic unit called the decibel (dB). The reference intensity Io is usually taken as the
intensity for the threshold of hearing at 2000 Hz. Thus the intensity level, IL, of sound is
defined as:
IL (dB) = 10 log
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I
I0
(4)
Sound, Resonance & The Ear
In the next experiment you will measure the relative sensitivity of your hearing (hearing
acuity), so it is useful for you to understand the physical processes by which the
longitudinal sound-waves in the elastic medium of air are converted efficiently to waves in
the fluid of the Cochlea. These processes are outlined in figure 5 below.
The anatomical structure of the ear can be conveniently divided into three parts: the outer ear, the middle ear and
the inner ear. The outer ear consists of the external portion called the pinna, the auditory canal that is
approximately 3 cm long, and the membrane at the inner end of the auditory canal called the eardrum. The middle
ear begins just inside the eardrum and consists of a chain of three bones called the ossicles: the hammer, the
anvil and the stirrup. Opening into the middle ear from the throat is the Eustachian tube that permits equal
pressures to be maintained on each side of the eardrum.
The stirrup links the anvil (on the middle ear side) to the round window, which is the beginning of the inner ear. The
inner ear is a liquid-filled, coiled up cavity called the cochlea. If the cochlea were uncoiled, it would have a length
of about 3.5 cm. Dividing the cochlea along its length is the basilar membrane. Hairlike cells line the basilar
membrane and these hair cells are “activated” in the perception process.
Figure 5
Sound, Resonance & The Ear
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Experiment 3:
a.
Measurement of Hearing Acuity
Equipment:
Function software
Ear phones
b.
1.
Method
Connect the earphones to the sound output of the laboratory computer, and open the
program Function that can be found in the Lab Tools folder. After clicking OK a
couple of times you will see the control window shown below.
The left-hand controls of Function control the frequency of the sound you will hear
through the earphones, while you can adjust its amplitude, or loudness, using the
right-hand controls.
Note that each set of controls shown in the snapshot above comprises two sets of
arrows. These arrows can be used to increase or decrease the quantity they control,
with the left-hand arrows providing coarse control and the right-hand pair of arrows
in each set providing fine control. You can also use the sliders to quickly change either
frequency or amplitude.
It is possible to choose the units of the display by clicking on the buttons labelled Hz
and kHz for frequency, mV and V for voltage.
If you are interested you can listen to the sound of non-sinusoidal functions, eg.
rectangular or sawtooth, but that is not needed today.
2.
Set the frequency of the sinusoidal function to 2000 Hz at an intensity that is clearly
audible through the earphones. Reduce the intensity carefully until the sound cannot
be heard. Record this limiting voltage.
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Sound, Resonance & The Ear
3.
4.
Repeat this procedure for a range of frequencies from about 50 Hz to 20 kHz, making
sure that for each frequency you record the voltage at the hearing limit as accurately
as possible.
Plot the voltage corresponding to the limit of hearing (vertical scale) versus frequency
on the log-log paper on page 11 which you can detach and paste into your logbook.
Note that your limit of hearing voltage data, as well as the frequencies, will range over
several orders of magnitude. You will therefore find it necessary to use a logarithmic
scale on both axes.
The result should look similar to figure 6 below this box. However since this figure
shows a decibel scale for the Intensity Level, which is already a logarithmic quantity,
the dB scale in figure 6 appears to be linear.
c.
1.
Observations and discussion
Comment on any significant deviation between your results and the standard
response in figure 6 below. In particular, is the frequency range you observe less than
average?
2.
To what extent might the results be affected by the equipment you used? Explain.
3.
Note that the most sensitive frequency occurs at about 3000 Hz. Use the data for the
auditory canal of the outer ear given in the caption of figure 5 to calculate a series of
resonant frequencies for the canal. Now comment on why the ear seems to be most
sensitive at 3000 Hz.
Figure 6. Curves of Equal Loudness
The curve relevant to experiment 3 is the curve marked “Threshold of Hearing”. The scales shown for sound
intensity include the Intensity Level, IL, in dB (LH scale), the intensity in W m-2 (RH scale) and the pressure in
N m -2 (far right)
Sound, Resonance & The Ear
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Appendix: Controlling a CRO
The Cathode Ray Oscilloscope (CRO) that you will be using in the lab today is shown
below.
The controls that you will need to be aware of are:
1. Intensity Control & ON-OFF switch Fully clockwise this control switches the instrument
OFF. When rotated clockwise the instrument is
switched ON and further rotation controls the
brightness of the trace on the screen from zero to
maximum.
2. Focus
Controls the sharpness of the trace on the screen.
3. Vertical input
The input from the amplifier should be connected
here.
4. VOLTS/CM (Attenuator)
This switch adjusts the sensitivity of the vertical
amplifier from 10 mV per cm to 50 V per cm in a 1, 2,
5, 10 series of steps.
5. Vertical Position
Moves the trace vertically on the screen.
6. SEC/CM (Time Base) Switch
When the Time Base Vernier Control is turned
clockwise to the CAL position, the time for the trace to
travel 1 cm horizontally is given by the number
dialled: 10 ms, 1 ms, 100 µs, 10 µs, 1 µs or 0.2 µs.
7. Horizontal position
Moves the trace horizontally on the screen.
8. Trigger level control
Set this on fully counterclockwise so that it triggers
automatically on detecting a signal. Rotation of the
control will vary the position on the waveform that
the trace will start.
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Sound, Resonance & The Ear