Jason Fong
702847140
date of experiment: 11-28-00
partner: Rosanne Hong and Victor Hwei
Physics 4AL Lab 5
Experiment 7
Abstract and Introduction
The purpose of this experiment was to investigate traveling and standing waves on a
string. Various modes of standing waves were created and examined. The predicted and
measured frequencies of the sample of modes that were examined were found to agree fairly
well.
The velocity of a traveling wave in a string is determined by the tension in the string and
the density of the string. The tension of the string affects the velocity because it provides the
force to return the string to its original position after it is displaced. The tension on each particle
of the string comes from the neighboring particles of the string. When a particle of the string is
displaced up, the neighboring particles pull down on the displaced particle while they are being
pulled up by the displaced particle. The greater the tension, the greater the pulling force on each
particle will be, and the displacement of the particles of the string will occur more quickly.
Thus, a greater tension on the string will result in a greater wave velocity on the string.
The density of the string affects the velocity of the traveling wave because it effects the
mass of each particle of the string. The greater the density of the string, the greater the mass of
particles of the string that are of equivalent volume. The greater mass means that each particle
will have more inertia and will accelerate slower. The slower acceleration of each particle
causes the traveling wave to propagate through the string slower because the force of the wave is
more slowly transferred to each neighboring particle of the string. Thus, a greater density of the
string will result in a lower velocity of a traveling wave.
Standing waves are formed by the interference of the waves traveling in opposite
directions along the string. When a standing wave is present on the string, there are points of
maximum amplitude called antinodes and points of no movement called nodes. The
displacement at each point of the combined wave is found by summing together the
displacements caused by each individual wave. The antinodes are formed at points where the
phases of the two interfering waves cause the maximum constructive interference, and the nodes
are formed at the points of complete destructive interference so that the displacements of the
waves cancel each other out.
Different frequencies of the wave driver will create different modes of standing waves in
the string. A mode of the string is a certain standing wave pattern created on the string. Each of
the modes are identified by a n number. n=1 signifies the fundamental standing wave of the
string. Each of the modes are characterized by a certain pattern of nodes and antinodes. The
modes are different from nodes in that a mode is a certain standing wave pattern and a node is a
point of zero displacement on the string.
Formulas
The following are the formulas used in this lab.
The tension T in the string of mass m is given by:
T  mg
The uncertainty in that is given by:
T  gm
The density μ of the mass m string based on the tension T on the string and the total length of the
stretched string Ltotal is given by:

m
Ltotal
The uncertainty in that is given by:
 
  

1
 m

  
m   
L    m    2 L 
 m
  L

L
 L

2
2
2
2
The formula for the predicted velocity v of the wave based on the tension T on the string and the
density μ of the string is:
v predicted 
T

The uncertainty in this is:
2
v predicted
2
2
 1
  T


 v
  v
 
T   
   
T   
 
3
 T
  


 2 T
  4
2
The measured velocity of the traveling wave can be obtain by using the total length traveled and
the time taken to travel that distance.
vmeasured 
L
t
The uncertainty in this is:
 v
  v 
1   L 
  L    t    L    2 t 
 L   t 
t
 t

2
vmeasured
2
2
2
The formula relating wavelength λ of mode n to the length of the string L:
n 
2L
n
The formula relating the frequency f of the wave to the velocity v of a traveling wave and the
wavelength λ:
fn 
v
n
The uncertainty in this is:
v
 f

f n   v  
n
 v 
2
Using the formula relating the string length to wavelength gives:
fn 
nv
2L
For a predicted frequency, the uncertainty would be:
f pred 
v pred
n

nv pred
2L
2
n

2L
 1
  T


 


T


3

 2 T
  4


 
2
Procedure
This experiment was set up by taking a long string, clamping one end down, running it
horizontally over a table, and placing the other end over a pulley and vertically hanging a weight
off the end of the string. The density of the string was found by weighing the string and
measuring the length of the string after the weight was attached, and then using those values to
calculate the density. A device to drive the wave motion was attached to the end near the clamp,
and a laser and a light detector were placed at the end near the pulley. The wave driver could
drive the string at an inputted frequency. The light detector would measure the displacement of
the string by measuring the intensity of the laser light reflected. The laser was positioned so that
more of the beam would be intercepted when the string rose higher. So, the intensity of the light
reflected would indicate what position the string was in.
In order to measure the velocity of a traveling wave, the wave driver sent a single pulse
down the string and then the time for several return pulses of the wave was taken from the scope
readings of the laser light intensity. The number of pulses detected and the length of the string
were used to calculate the velocity.
The velocity of the wave and the length of the string was then used to calculate predicted
values for the frequencies of various modes of standing waves. Then those modes were
produced and the actual frequencies needed to produce them were noted. The frequencies were
found by adjusting the frequency of the wave driver until the amplitude on the scope was at a
relative maximum and the expected number of nodes were visible along the string.
Data Analysis and Error Analysis
A predicted velocity of the wave was first calculated by using the tension on the string
and the density of the string. The tension was supplied by the weight hanging on the end of the
string over the pulley, so the tension force is the force of gravity pulling on the weight.
T  Mg
where T is the tension force in the string, M is the mass of the weight used, and g is the
acceleration due to gravity. The density of the string was more difficult to calculate since
depending on the weight used, the string would stretch different amounts so that the length and
density of the string would be different. The length of the string up to the pulley is the same for
all weights used, so only the length from the pulley to the weight would change. The length
from the driver end to the pulley was found to be 230 ± 1 cm, and since the radius of the pulley
was 2.5 cm, the length of the string over the pulley was a quarter of the circumference of the
circle, or 0.039 ± 0.001 m. The density could then be found by using the stretched length of the
string and the measured mass of the string. The mass of the string was found to be 13.6 ± 0.1 g,
or 0.0136 ± 0.0001 kg.
The tension and density of the string could then be used to find the
velocity of a traveling wave along the string. The following table summarizes the resulting
densities and predicted velocities for each of the different weights.
Mass
Tension
Total Length
Density
Predicted
Velocity
.3000 ± 0.0001 kg
2.955 ± 0.001 N 2.93 ± 0.01 m
.00464 ± 0.00004 kg/m
25.2 ± 0.1 m/s
.2570 ± 0.0001 kg
2.534 ± 0.001 N 2.62 ± 0.01 m
.00519 ± 0.00004 kg/m
22.1 ± 0.1 m/s
.2500 ± 0.0001 kg
2.463 ± 0.001 N 2.53 ± 0.01 m
.00538 ± 0.00004 kg/m
21.4 ± 0.1 m/s
This table shows that for increasing masses, the velocity of a traveling wave increases. This
makes sense since a greater mass creates a greater tension in the string. As previously discussed,
the greater tension will lead to a greater traveling wave velocity.
For the first part of the lab, traveling waves were used to measure the velocity of waves
on the string. The measured velocity could then be compared to the velocity predicted. A single
pulse was sent from the wave driver, and that pulse was allowed to reflect a few times. The
wave travels two lengths of the string between each time it reaches the laser detector. The scope
readings of the reflected laser lights showed downward spikes each time the pulse reached the
other end. The scope shows a spike because the string will have much increased displacement
when the pulse arrives at the laser spot. Using the scope readings, the time was taken for a
certain number of spikes in the graph. The distance between each spike represents twice the
length of the string since the pulse must travel back to the driver and then to the laser again
between each spike. The velocity could then be calculated by using the time and the distance for
the number lengths needed to create that number of spikes. The following table summarizes the
results.
Mass
Time
Spikes
Lengths
Total Length
Measured
Traveled
Traveled
Velocity
.3000 ± 0.0001 kg
1.741 ± 0.001 s
8
16
36.80 ± 0.01 m
21.14 ± 0.01 m/s
.2570 ± 0.0001 kg
1.718 ± 0.001 s
7
14
33.20 ± 0.01 m
19.32 ± 0.01 m/s
.2500 ± 0.0001 kg
1.764 ± 0.001 s
7
14
33.20 ± 0.01 m
18.82 ± 0.01 m/s
The predicted velocities were faster than the measured velocities. This makes sense since the
wave will lose energy as it moves up and down the string. The energy loss will be due to air
resistance, heat, and the imperfect transfer of energy between each particle of the string and the
imperfect reflection of the wave at the endpoints of the string. The energy loss will result in a
reduced measured speed of the wave as seen in the tables. The following table summarizes the
difference between the predicted and measured velocities. The percentage difference is given by
v pred  vmeas
v pred
.
Mass
vpred
vmeas
Absolute
Percentage
Difference
Difference
.3000 ± 0.0001 kg
25.2 ± 0.1 m/s
21.14 ± 0.01 m/s
4.06
16.1 %
.2570 ± 0.0001 kg
22.1 ± 0.1 m/s
19.32 ± 0.01 m/s
2.78
12.6 %
.2500 ± 0.0001 kg
21.4 ± 0.1 m/s
18.82 ± 0.01 m/s
2.58
12.1 %
The case using the 300 g weight was investigated further. The time for the pulse from the
wave driver to first reach the laser was measured to be 99.6 ± 0.1 ms. The time for the 8 spikes,
or 16 lengths of the string, was 1740.8 ± 0.1 ms; so the time for one length is 108.8 ms. The
time from the 8 spikes is 9.24% off from the time for the pulse to first reach the laser. The time
for the 8 spikes is slower because energy is lost in the later reflections of the wave due to the
reasons mentioned above. The loss of energy means that the wave will travel slower, which
results in a longer time measured for the wave to travel a length of the string.
The frequency of the fundamental mode for each weight is the frequency that will
produce the n=1 standing wave in the string. This standing wave will have nodes at the
endpoints and an antinode at the center of the string. The wavelength is twice the length of the
string since only half of the wavelength will fit on the string. Using this information, the
frequency for the fundamental mode for each weight can be calculated from the measured
velocities of the wave on the string using the formula f n 
v
n
. The following table summarizes
the results.
Mass
Wavelength
Predicted Velocity
Predicted Fundamental
Frequency f1
.3000 ± 0.0001 kg
4.60 ± 0.01 m
25.2 ± 0.1 m/s
5.478 ± 0.002 Hz
.2570 ± 0.0001 kg
4.60 ± 0.01 m
22.1 ± 0.1 m/s
4.804 ± 0.002 Hz
.2500 ± 0.0001 kg
4.60 ± 0.01 m
21.4 ± 0.1 m/s
4.652 ± 0.002 Hz
The frequencies for each of the other modes can by calculated with the n number of each
mode by using f n 
nv
. For the case with the 300 g weight, this was done for various modes as
2L
listed in the following table. Based on previously calculated values, this table uses a velocity of
21.14 ± 0.01 m/s for waves in the string and a length of 2.30 ± 0.01 m for one length of the string
that the wave travels along.
Mode
Predicted frequency
2
9.19 ± 0.04 Hz
3
13.79 ± 0.07 Hz
4
18.38 ± 0.09 Hz
5
22.98 ± 0.11 Hz
10
45.96 ± 0.22 Hz
20
91.91 ± 0.43 Hz
30
137.87 ± 0.65 Hz
The frequency for various modes were found by adjusting the frequency of the wave
driver until the desired mode was found. The mode was considered to have been found when the
amplitude was at a maximum relative to other frequencies near it, and when the proper number
of nodes were visible along the string. The following table summarizes the predicted and
measured frequencies.
Mode
Predicted Frequency
Measured Frequency
3
13.96 ± 0.07 Hz
13.786 ± 0.001 Hz
6
27.91 ± 0.13 Hz
27.572 ± 0.001 Hz
11
51.17 ± 0.24 Hz
50.548 ± 0.001 Hz
16
74.43 ± 0.35 Hz
73.525 ± 0.001 Hz
21
97.70 ± 0.46 Hz
96.501 ± 0.001 Hz
The values for the frequencies agree pretty well. The slight difference is probably caused by
factors such as air resistance, heat, imperfect transfer of energy between particles of the string,
and imperfect reflection of the waves at the endpoints.
For the even modes (n=2,4,6,...), there is a node at the center of the string. When a bar
was placed at that node, the standing wave continued to remain in place on the string. The bar
did not interfere with the wave because the string does not move at the node, so restricting
movement at the node would not affect the wave. For odd modes, however, there is an antinode
at the center of the string. When the bar was left at the same center point and odd modes were
attempted to be created, the wave was disrupted once it reached the bar. Since the wave needs to
move at the antinode, the wave is disrupted because the bar restricts the strings movement. This
was observed as the standing wave being disrupted after passing the bar for odd modes.
When the wave driver frequency was set halfway between the fundamental and the next
higher mode, the resulting wave was not a standing wave. The wave was a jumble of interfering
waves that had a lower amplitude than either of the standing wave frequencies. This is because
the frequency halfway between the standing waves does not cause the outgoing and reflected
waves to undergo the proper amount of constructive interference. The wave does not completely
destructively interfere either, so the result is a jumbled wave that does not resemble a standing
wave.
The laser was placed 14.3 cm away from the pulley. As the frequency is increased, there
will be a point when the wavelength becomes comparable to the distance separating the laser and
pulley. When the wavelength is this distance, there will be a node at the laser spot. Since the
node will not move, the laser light detector will not detect any movement, so no useful
information will be collected by the laser light detector. This frequency can be calculated by
considering what frequency would create a wavelength of the distance between the laser and the
pulley, or .143 m.
f 
v


25.2 m s
 176 Hz
.143 m
Thus, the situation described above will occur at a frequency of around 176 Hz.
Conclusion
This experiment investigated the behavior of traveling and standing waves in a string.
The velocity of traveling waves were found for various tensions on the string. The case with a
300 g mass providing the tension was examined in further detail. Various modes of standing
waves were created and the predicted and measured frequencies of a set of modes were found to
agree fairly well.