Chapter 10
Wave Phenomena
Lab Partner:
Name:
10.1
Section:
Purpose
Wave phenomena using sound waves will be explored in this experiment. Standing waves
and beats will be examined. The speed of sound will be determined.
10.2
Introduction
While we will be concerned primarily with sound waves in this experiment, all waves have
certain common features. There are two general types of waves. A transverse wave is one in
which the disturbance is perpendicular to the direction of travel. A longitudinal wave is one
in which the disturbance is parallel to the line of travel of the wave. Sound is a longitudinal
wave. A wave is periodic if the pattern is repeated over and over. This occurs in both space
and time. There are several features which describe such a periodic wave. The period, T, is
the time it takes for one complete cycle or oscillation of the wave. The units are seconds. The
wavelength, λ, is the distance between crests of the wave. Wavelength has units of distance.
The amplitude is the magnitude of the disturbance. The amplitude can have various units
for different types of waves. For a water wave, the amplitude would be the height in meters
Transverse
Longitudinal
Direction of wave
Figure 10.1: Transverse and longitudinal waves. Sound is a longitudinal wave where the
displacement of the medium is in the same direction as the motion of the wave.
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λ
Velocity
Amplitude
Distance
Figure 10.2: The amplitude and wavelength are represented in this figure.
of the wave. For a sound wave, the amplitude is given in units of pressure (Pascal (Pa) =
Newton/m2 )
The period is related to the frequency by:
f=
1
T
(10.1)
The unit of frequency is the Hertz (Hz = s−1 ).
The relationship between the speed of a wave, the wavelength and the frequency is:
v = λf =
λ
T
(10.2)
This applies to both transverse and longitudinal waves. For sound in air at 1 atmosphere of
pressure (1.01 x 105 Pa) and 20o C temperature, the velocity is 343 m/s.
Waves exhibit interference effects. This is based on the principle of superposition which
can be stated as:
• When two or more waves are present simultaneously at the same place, the resultant
wave is the sum of the individual waves.
If the wavelength and amplitude of the two waves are the same, the two waves can add
up in constructive interference to produce a wave with twice the amplitude of the individual
waves (constructive interference). If the waves are ’out of phase’ by 180o, i.e. one wave is
going ’up’ while the other wave is going ’down’, the result is destructive interference and the
two waves cancel out. See Figure 10.3.
A ’standing wave’ is a stationary pattern produced by interference of a wave with itself
when it reflects off of the opposite end of an enclosure. The wave interferes with its own
reflection. Each standing wave pattern is produced at a unique frequency. This frequency
corresponds to an integer number of wavelengths that will fit into the length, D, of the
enclosure. The positions where there is no amplitude (pressure) is called a node. The
positions with maximum amplitude (pressure) are called anti-nodes. Consider a tube of
length D. The frequencies that produce the ’one loop’, ’two loop’, etc., patterns shown in
figure 10.4 are given by:
64
+
=
+
=
Figure 10.3: In the top schematic, the two wave interfere constructively. In the bottom
schematic, the waves interfere destructively.
Figure 10.4: Standing wave pattern.
v
)
n = 1, 2, 3, . . .
(10.3)
2D
where v is the velocity of the wave and D is the length of the tube. For a given D, a set of
frequencies fn , where n = 1, 2, 3, . . . , produces standing waves. These frequencies are said to
be harmonically related. The lowest frequency f1 (n =1), is usually called the fundamental
frequency, the next lowest frequency, f2 = 2f1 (n = 2), is called the first harmonic.
When two waves of different frequency are added together, the two waves also interfere
with one another. The number of times per second that the loudness rises and falls is the
beat frequency. Consider two waves of the form:
fn = n(
z(t)1 = Asin(2πf1 t)
z(t)2 = Asin(2πf2 t)
(10.4)
Using the trigonometry identity sinα + sinβ = 2 cos 21 (α − β) sin 12 (α + β) the sum of the
two waves is:
2π(f1 − f2 )t
2π(f1 + f2 )t
))sin((
))
(10.5)
2
2
Note the difference of frequencies in the cosine function and the sum of the frequencies in the
sine function. Figure 10.5 shows a plot of a function like equation 10.5. The three variations
in intensity form an ’envelope’ which is determined by the cosine function in equation 10.5.
.
z(t) = 2Acos((
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Figure 10.5: This figure shows a plot of a function like equation 10.5. Note the ’envelope’
which is determined by the cosine function in in equation 10.5.
Transmitter
Closed Tube
Reciever
slide
L
D
Figure 10.6: Schematic diagram of the sound tube used to study standing waves and determine the speed of sound.
10.3
Procedure
10.3.1
Standing Waves
A schematic of the closed sound tube is shown in Figure 10.6. A photograph is shown in
figure 10.7. When a continuous sound wave is generated by the transmitter (speaker) at
one end of the tube, it reflects from the other (closed) end of the tube. The reflected wave
interferes with later waves being generated by the transmitter. A standing wave pattern is
set up in the tube. The movable receiver (microphone) is then used to detect the nodes and
anti-nodes of the standing wave pattern.
• Part 1: Fixed distance with the frequency varied. Open the file ’wave1.cap’ in Capstone. An oscilloscope window will appear. ’Click’ on the signal generator icon ( )
on the tool panel and a signal generator will appear. Pull the receiver to the end of
the tube opposite the transmitter.
• Vary the frequency of the signal generator from 2 kHz to 400 Hz. Record the four
frequencies (f1 ,f2 ,f3 ,f4 ) from the signal generator where the amplitude is a maximum.
The high frequency resonances (harmonics) are easier to see on the display so start at
2 kHz and step the frequency down. You will find f4 first, then f3 , etc.
• Verify the frequencies are related by equation 10.3, i.e., f1 =f4 /4, f1 =f3 /3, etc. Find the
average f1 (fundamental harmonic).
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Figure 10.7: The closed tube apparatus is shown in this photograph. The closed sound tube
speaker is connected to the signal generator output and the voltage input is connected to
the sliding microphone.
Harmonic
f4
f3
f2
f1
Frequency
Average f1
• For the fundamental harmonic, f1 , the wavelength is 2 · D where D is the length of
the tube. Using the average fundamental harmonic, f1 , find the speed of sound from
equation 10.2. Compare the result (percentage error) with the standard value 343 m/s.
Speed of sound (v)
Percent error
• Part 2: Fixed frequency with the distance varied. Find the f4 harmonic around 1.8
kHz. Record the frequency from the Capstone signal generator display.
Frequency (f4 )
• Place the receiver near the transmitter (≈ 2 cm). Slowly move the receiver across the
tube and record the position of maxima (anti-nodes) in the table. The difference in
position between maxima is ∆L = λ2 for f4 . Determine the average λ. Determine the
speed of sound from equation 10.2 and compare to the standard value for the speed of
sound (percent error).
Average wavelength (λ)
Speed of sound (v)
Percent error
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Maximum position (L) difference (∆L)
–
10.3.2
λ
–
Beats
Figure 10.8: The experimental setup for measuring beat frequencies is shown in this photograph.
Figure 10.8 shows the setup to measure the beat frequencies of two tuning forks with
slightly different frequencies. Masking tape on one fork is used to slightly retard the frequency.
• Open the file ’wave2.cap’ with Capstone. Adjust the sound detector to ≈1 cm from the
forks. One fork should have masking tape to retard its frequency from the standard
256 Hz.
• Remove the tuning fork with the masking tape. Strike the tuning fork without the
masking tape and click ’record’.
• Expand the scale if necessary. Fit the curve with a ’sine fit’ and determine the frequency. The fit parameter for the sine fit is the angular frequency, ω = 2πf. Calculate
the frequency from this angular frequency and record it below.
Frequency of fork without tape
• Repeat for the other tuning fork with the masking tape and record the frequency below.
Frequency of fork with tape
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• With both tuning forks in position, strike both tuning forks equally hard with the
rubber mallet and click ’record’. Beats should appear on the display i.e. a variation in
the intensity like figure 10.5. Using the ’smart tool’ function in Capstone, measure and
record the time for four adjacent maxima of the ’beats’. Calculate the time difference
(period) between adjacent maxima and the frequency for these periods. Record the
values in the table and calculate the average frequency.
Maximum time
difference (T)
–
frequency
–
Average frequency
• The frequency of the beats is the difference in the frequency between tuning forks
with and without masking tape. Calculate the difference in frequency between the two
tuning forks. Calculate the percentage difference between this difference in frequency
and the average frequency of the beats calculated in the previous step.
Difference in frequency between forks
Percent difference
• Expand the scale so ≈ 50 cycles are visible near one of the maxima. Using a ’sine fit’,
determine the frequency. Compare (percentage difference) the sine fit frequency to one
half of the sum of the two individual tuning fork frequencies.
Frequency from sine fit
Percent difference
10.3.3
Questions
1. Why does adding masking tape to one of the tuning forks lower the frequency?
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2. Why is the frequency of the beats the difference in the two individual fork frequencies
and not one half of this difference?
10.4
Conclusion
Write a detailed conclusion about what you have learned. Include all relevant numbers you
have measured with errors. Sources of error should also be included.
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