Standing Waves on a String
R.L.Griffith,M.R.Levi,D.Cartano
ABSTRACT
A test concerning the principles of standing waves on a string was performed. A Pasco
Mechanical Vibrator was used to produce the appropriate wave motion with a known frequency.
Two different types of strings with different linear mass densities were used for this experiment.
Using different tensions and frequencies, we were able to record the distances between each
node and apply the principle equations of standing wave motion. using the equations derived
in the introduction section we were able to verify that the theory is capable of making very
precise approximations. The average error derived was less than 5 percent and comparing it
to the standard deviation we are able to conclude that the experiment was a success.
Subject headings: Waves,sound,standing waves
1.
Introduction
by
v = λn f n
The result of the interference between two
waves can be modeled using the principles of
wave motion. Standing waves are produced
whenever two waves of identical frequency interfere with one another while traveling opposite directions along the same medium. Standing wave
patterns are characterized by certain fixed points
along the medium which undergo no displacement. These points of no displacement are called
nodes (nodes can be remembered as points of
no displacement), The nodes are always located
at the same location along the medium, giving
the entire pattern an appearance of standing still
(thus the name ”standing waves”). The highest
points of displacement occur at the anti-nodes
and those are also located at the same location
along the medium. The modes of vibration are
numbered according to the number of segments
the standing wave has. This number is given the
letter n, and is called the ’harmonic number’.
The wavelength of the standing wave is given by
the formula
2L
λn =
(1)
n
where L represents the length of the strings between the fixed ends. The velocity of a wave on
a string is given by
s
T
v=
(2)
µ
(3)
where fn is the frequency of the wave. If the
string is attached to a mass pulley than the tension is defined as
T = Mg
(4)
where M is the mass of the weight and g is the
gravitational force. using equations 1,2,3, and 4
we get
s
fn =
n
2L
Mg
µ
(5)
we will use equation 5 for our data analysis.
2.
2.1.
Method
Equipment Used
Equipment
Pasco power amp II
Pasco interface II
Pasco Mechanical vibrator
Support rod
Two types of string
Mass set and mass hanger
Meter Sticks
2.2.
Model
CI-6552A
CI-6560
SF-9324
n/a
n/a
n/a
n/a
Procedure
The first step is to set up the equipment as
outlined in LACC physics 103 manual, standing
wave experiment. There were 6 different runs
performed with each string. Taking measurements of the lengths of each segment off the loop
where T is the tension in the string and µ is
the linear mass density of the string. The wavelength, the velocity, and frequency are related
1
number of segments
#
3
2
1
4
5
for each run, we are able to use equation 5 to see
if theory matches experiment. The last test was
performed with both the tension and the length
kept constant. The frequency was changed and
the number of segments was recorded.
3.
4.
Results and Discussion
Plots
An analysis for the relationship between the
mass of the weight and the wavelength can be
modeled using a logarithmic scale. We will be
plotting log λn versus log m if the we rearrange
equation five we get
There are six calculations performed for each
string, using equation 5 we are able to calculate the experimental values for fn and compare them to the theoretical values, which are
acquired from the setting of the mechanical vibrator that was used. The third trial, we were
able to compare the number of nodes per frequency to see if they adhered to the principles
of harmonic motion for waves on a string. The
results we acquired were accurate to a high degree.
s
f n λn =
Mg
µ
(6)
taking the log of both sides we get
log λn =
3.1.
fn
Hz
60
40
20
77
95
1
√
√
log M + log g − log fn µ
2
(7)
where the 1/2 in front of the M is the slope
of the theoretical graph. Finding the slope for
our experimental data and comparing it to the
theoretical slope, we are able to calculate a percentage error.
The error analysis was performed by comparing the standard deviation to the average error
using the experimental and theoretical values.
the average error was calculated using
Data and Calculations
Mass of string 1
2.56 g.
Length of string 1 492 cm
Mass of string 2
.92 g.
Length of string 2 481.6 cm
Part 2 Data
Mass of weight
500 g
Length of string
202 cm
Measured quantities
µ = 5.21 × 10−4 kg/m
Trial m
L
n fn
g
cm
Hz
1
350 350 5 60
2
400 300 4 60
3
450 316 4 60
4
500 250 3 60
5
550 265 3 60
6
600 280 3 60
µ = 1.92 × 10−4 kg/m
1
350 225 2 60
2
400 241 2 60
3
450 257 2 60
4
500 270 2 60
5
550 140 1 60
6
600 147 1 60
Data for part two of this experiment,
where the tension and length were kept
constant
|ftrue − fmean |
× 100 = %
ftrue
(8)
the standard deviation is defined as
s
R2 − R12 /R0
(9)
R0 − 1
P
P 2
where R0 = n, R1 =
xi , and R2 =
xi ,
where n is the number of calculations and xi is
the calculation.
Errorave = .0368
σx = .07156
and since Errorave ≪ σx we can conclude
that the experiment was a success.
String Experimental Theory Error
White .518
.5
3.6 %
Black
.493
.5
1.4 %
Black
18.96
16.59
14.3 %
σx ≡
2
5.
Conclusion
This lab was conducted to get a better understanding on how waves behave on a string. The
results that we acquired were very accurate and
confirmed that the principles that were derived
in the introduction section are very accurate at
making prediction on the behavior of standing
waves on a string. We did acquire some errors
in our experiment. Possible sources of error include, the precision of our measuring equipment
and taking the measurements. We could have
also possibly introduced some error by the precision of the Mechanical vibrator’s output frequency. Our calculations were very accurate to
the theoretical values.
Fig. 1.— graph for trial one, white string.
6.
Acknowledgements
The author would like to thank Roni and
David for their help with this experiment.
REFERENCES
Los Angeles City College Lab Manual Physics
103.
Fig. 2.— graph for trial two, black string.
Fig. 3.— graph for part 2 data.
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