Physical Science 1422 Standing Waves and Velocity of Sound in Air

Physical Science 1422
Department of Physics and Geology
Standing Waves and Velocity of Sound in Air
Equipment Needed
String Vibrator
Pulley with clamp
Balance
Sine Wave Generator
Speaker
Qty
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1
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Equipment Needed
Mass and Hanger Set (ME-8967)
String
Universal Table Clamp
Patch wires
Resonance tube set
Qty
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Background
Standing Waves
We are all familiar with waves in one form or another. We shall define a wave as a disturbance or
vibration propagating through space. In general, waves can be classified as being either transverse or
longitudinal. In a longitudinal wave, the medium vibrates in a direction parallel to the direction the wave
is traveling. Sound is an example of a longitudinal wave. In a transverse wave, the medium vibrates in a
direction perpendicular to the direction the wave is traveling. Figure 1a shows a wave with its
corresponding wavelength, amplitude and crest. The wavelength (λ) is the distance of one complete
“wave”. The amplitude of a wave is the maximum displacement of any part of the wave from its
equilibrium position. The time it takes for a wave to travel a distance of one wavelength is called the
period (T). The frequency (f) is the number of oscillations or cycles that occur during a given period of
time. Frequency is usually measured in cycles per second, or Hertz. The period (T) is usually measured
in seconds. The frequency (f) and the period (T) are related by the following expression:
Period (T) = 1/ Frequency (f)
One of the properties of a wave is that the velocity of a wave is related with its frequency and
wavelength by the following equation.
Velocity of a wave (v) = frequency (f) × wavelength (λ)
A wave traveling along a plucked string is transverse in nature. Note that as the wave travels to the
right, the medium (the string) is actually only moving up and down.
Figure 1. (a) Description of wavelength, amplitude and crest. (b) Description of nodes and antinodes.
One of the properties of waves is interference. When different waves in a medium meet, they
superpose one another. If two crests meet, then the net effect is increased amplitude (this is known as
constructive interference). If a crest meets a trough, the net effect is reduced amplitude (this is known as
destructive interference). A special case of interference, known as a standing wave, occurs when two
identical waves traveling in opposite directions meet, resulting in the creation of nodes (null points,
point for which there is no motion, no displacement from its equilibrium position) and antinodes (points
with greatest amplitude). In this lab we will generate a standing wave in a string. The position of these
nodes depends on the frequency of vibration, tension in the string, and density of string.
Tension is a type of force acting on string. Since the amount of tension is equal to the gravitational
force in our case, Tension = mass × gravity = mass × 9.8 m/s2
.
Tension = (mass of hanger + amount of mass added) × 9.8
During this experiment, the wavelength will be calculated as:
wavelength = 2 × (distance between two consecutive nodes)
Velocity of Sound in Air
Sound travels at 331.5 m/sec in air at 0°C. As the temperature of the air increases, the speed of
sound also increases. The velocity of sound at any temperature T, can be calculated using the following
linear equation.
vT = (0.6 * T) + v0°C
In the above equation v is the velocity of sound at any temperature T and v
T
0°C
is the velocity of
sound at 0°C. For example, the velocity of sound at 10°C is v = (331.5 + 0.6×10) = 337.5 m/sec.
T
As mentioned above, the velocity of a wave is related with its frequency and wavelength by the
following equation.
velocity of a wave = frequency × wavelength
In order to find the velocity of sound using this equation, we need to know both the frequency and
the wavelength of sound.
In this lab, we provide sound waves by a speaker. The frequency of the speaker is given and set by
the sine wave generator. When a speaker vibrates over the air column in the resonance tube, sound
propagates through the air into the air column, bounces off the boundary and comes back out of the air
column.
The length of the air column is determined by adjusting the length of the tubes. You can change the
length of the column of air by moving the tubes in and out of each other. Whenever a driving frequency
matches the natural frequency we call this resonance. In this lab, the driving frequency is produced by
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the speaker and we are changing the natural frequency of air column by adjusting its length. Whenever
the length of the air column matches the resonance condition (for the air column this happens when the
length becomes odd fourths of the wavelength, ¼λ, ¾λ,…), you will hear the loudest sound. At that
point, the sound wave is also a standing wave. By finding the resonance points, we can calculate the
wavelength of the sound. And because the frequency of sound is known, we can the speed of sound.
Open
End
Closed
End
Open
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Closed
End
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λ
SAFETY REMINDER
•
Follow the directions for using the equipment.
For You To Do
The purpose of this laboratory activity is to study concepts of waves, standing waves and the speed
of sound in air. The main concepts in this lab are wave properties of sound, resonance, and the standing
wave.
Section A: Standing Waves
PART I: Equipment Setup
1. Cut a piece of string long enough to go from one end of the lab table to the other.
2. Measure the exact length of the piece of string. Measure the mass of the string, calculate the linear
density, ρ (mass/length) and record your measurements in the Lab Report section.
3. Tie one end of string to the vibrator and make a loop
at the other side of string. Clamp the vibrator at one
end of the table and clamp a pulley at the other end
of table. Place the hanger in the string loop and
extend it past the pulley.
4. Turn on the Sine Wave Generator and turn the
Amplitude knob all the way down (counterclockwise). Connect the Sine Wave Generator to the
string vibrator using two banana patch cords. The
polarity does not matter.
5. Set the Amplitude knob about midway. Use the Coarse (1.0) and Fine (0.1) Frequency knobs of the
Sine Wave Generator to adjust the vibrations so that the frequency is set to 60 Hz.
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PART III: Data Recording
1.
Begin adding weight to hanger until a well defined standing wave pattern emerges.
2.
Record the total mass required to get a standing wave pattern in the Data Table in the Lab Report
section. Make sure you include the mass of hanger.
3.
Measure the distance between two (2) consecutive nodes.
4.
Calculate the wavelength by multiplying the distance between two nodes by 2.
5.
Find wavelength squared.
6.
Repeat steps 1-5 until you have 5 different measurements.
Analyzing the Data
1.
Calculate the Tension and Wavelength2.
2.
Using the computer and the data studio software plot a graph of Wavelength (Y-axis; vertical) and
Tension (X-axis; horizontal).
3.
Click ‘Fit’ (
) in the Graph toolbar and select ‘Linear’. This will draw the linear fit line,
and a text box will appear on the graph with the y intercept and slope automatically calculated by
the computer. Record the slope on the lab report section.
Print a copy of the graph and attach to the lab report section.
Using the equations in the background, we can solve for the Frequency and we get the equation
bellow. Using that equation, calculate the frequency and record it in the lab report section.
4.
5.
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Section B: Velocity of Sound In Air
PART I: Equipment Setup
1. Turn on the Sine Wave Generator and turn the
amplitude knob all the way down (counter-clockwise).
2. Connect the generator to the speaker using two banana
patch cords. Polarity does not matter.
3. Place the Resonance Tube horizontally, as shown, with
the speaker near the open end. Place the speaker at a 45°
angle to the end of the tube, not pointed directly into it.
4. The inner (white) tube slides inside the blue tube to
adjust the effective length of the closed tube.
PART III: Data Recording
1. Slide the two tubes together so that the tube length is zero. Set the Sine Wave Generator frequency
and the amplitude on a reasonable level.
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2. Extend the white tube to increase the tube length. The loudness of the sound will noticeably
increase as you approach resonance. Move the tube in and out to pinpoint the position that gives the
loudest tone. Record this position in the lab report section.
3. Continue to extend the white tube and find the next position that cause a resonance and record its
position in the lab report section. Each of these positions represents a node in the standing wave
pattern.
4. Set the Sine Wave Generator frequency for the next frequency value and repeat the above procedure.
Analyzing the Data
1. Calculate the distance between the nodes and take the average if you have more than one value.
2. Use this distance to calculate the wavelength of the sound wave.
3. Use the frequency of the sine wave generator to calculate the speed of the wave.
Record your results in the Lab Report section.
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