Lab Report: Standing Waves
Osvaldo Bambi
University of Arizona
Physics 142
Thermodynamics
February 27, 2019
Abstract
On this lab we investigated the concept of standing waves in a string and the
relationship between the frequency of the waves on the strings and the number of
nodes in the standing wave. Moreover, we also examined how the frequency of
oscillation of the string relates to the tension and length of the rope and used these
relationships to calculate the linear mass density of the string (). The procedure
for this experiment involved using a frequency generator, a string, a mass and a
pulley to generate waves in the string. The calculated value for the linear density of
0.0012 kg/m for the three experiments had an error of approximately .8%
compared to the theoretical value of 0.0012126 kg/m.
Introduction
The main goal of this lab experiment was to find the linear mass density of
the string based on the relationship between frequency and tension, nodes, and
length.
At first instance, the study of waves appeared to be a very challenging task,
u until Michael Faraday introduced the idea of standing waves in 1831. Standing
waves or stationary waves are referred to as the resultant pattern from the
interference of two or more waves that travel at the same channel. The positions
along the medium where the waves stand still are called nodes. Many also refer to
standing waves as the combination of two waves with the same amplitude and
frequency travelling in opposite directions.
Theory and Derivation
For the purpose of this lab, the string used is enclosed within a boundary. Having
the string enclosed causes the waves traveling through the medium to reflect back
when they hit one the boundaries. To clearly see or define the nodes, it is crucial
that that the waves fit within the boundary. See the fig.1 for a clear idea of how
waves behave on an enclosed boundary.
Fig1. Illustration of nodes in a string within an enclosed boundary.
General equation of waves:
2𝜋
𝐴 = 𝐴𝑚𝑎𝑥 × 𝑆𝑖𝑛(  (𝑥 + 𝑣𝑡 + 𝜑
(1)
Where:
A- Amplitude(m)
Amax- peak amplitude(m)
- wavelength(m)
- phase
t- time(s)
x- horizontal shift(m)
-
If we set = 0, we are able to derivate the standing wave equation.
-
Wave equation in terms of length(L) and the mode number(n)
=
2𝐿
𝑛
Description of the procedure
For this lab we used : - A string vibrator, a pulley, a sine wave generator, a string, and
different masses.
-We started by hanging the 500 g mass on the trail for the first part of the experiment and
started changing the frequency on the sine wave generator until we could see all the
requested nodes and the waves were stable. We know that a wave is stable when the
amplitude is not changing, and the aluminum bar attached to the string vibrator is very
steady.
- we repeated the process for the second part but started at 100 g and kept changing the
mass until we reached 1000 g, and the goal was to have 3 nodes at every mass.
-lastly, we changed the length of the string at increments of about .3 meters and recorded
the frequency necessary to generate a wave with 3 nodes.
Fig2. Lab set-up
Results and Sample calculation
Finding the string mass density through the slope of the Frequency vs Number of nodes
plot ( fig. 3)
1 𝐹
𝑆𝑙𝑜𝑝𝑒 = √
𝐿 𝜇
Where: F is the tension(N)
𝜇- String mass density(kg/m)
𝜇=
𝐹
(𝑆𝑙𝑜𝑝𝑒 ∗ 2𝐿)2
Using :
F= 4.9 N
L=1.38m
Slope= 23.135
4.9
𝜇=
(23.135 ∗ 2 ∗ 1.38)2
𝜇 = 0.0012𝑘𝑔/𝑚
𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑡𝑟𝑖𝑛𝑔
Or we can calculate it using the formula 𝜇 = 𝑡𝑜𝑡𝑎𝑙 𝑙𝑒𝑛𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑡𝑟𝑖𝑛𝑔
m= 0.0024 kg
Ltotal= 2.02m
0.0024
𝜇=
2.02
𝜇 = 0.00121𝑘𝑔/𝑚
Graphs and tables
300
250
Length: 1.38m
Tension (N) Frequency
(Hz)
0.98994949
41.2
1.4
57.7
1.71464282
70.4
1.97989899
80.8
2.21359436
90.2
2.42487113
98.8
2.61916017
108.8
2.8
114.4
2.96984848
121.7
3.13049517
127.9
y = 23,135x - 0,2218
200
150
100
50
0
0
2
4
6
8
10
12
nodes
Fig 3. Graph of frequency as a function of the number of nodes
Standing wave graph for diffrent
masses
Frequency (Hz)
Mode: 4
Mass
(Kg)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
standing wave graph for Frequency vs
Number of nodes(m=500g)
Frequency (Hz)
Tension(N) Length(m)4.9
1.38
Mode
Frequency
(Hz)
1
22.6
2
46.6
3
71
4
90.5
5
117
6
135.4
7
159.7
8
189.3
9
207.1
10
230.8
11
254.5
140
120
100
80
60
40
20
0
y = 40,727x + 0,6025
0
0,5
1
1,5
2
2,5
3
Square root of Tension(N)
Fig 4- Graph of the frequency as a function of the square
r
root of tension
3,5
Standing wave graph of Frequency vs
Length(m=500g)
250
200
Frequency(Hz)
Tension:
Mode: 4
4.9 N
1/Length
Frequency
(m)
(Hz)
0.72463768
90.6
0.88495575
110.6
1.13636364
138.7
1.4084507
178.6
1.58730159
199.2
y = 127,05x - 2,3559
150
100
50
0
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1/Length(m)
Fig.5- graph of the frequency as a function of the length
Conclusion and discussion
By the end of this lab experiment we were very satisfied with the results obtained. The
resulting values for the string mass density calculated from the plots look the same as the
theoretical value calculated using the mass and the length of the string. For the three plots of
Frequency as a function of the number of nodes, tension, and length the string mass densities are
0.00121kg/m, 0.00126kg/m, and 0.00121kg/m respectively. The theoretical value of the string
mass density is 0.00121kg/m, which gives us an error percentage between 0.1% and 1% for the
three values. Although the results are very accurate, thinking back to the procedures, one error
that might have influenced the results was probably the measurement of the length of the string.
1,6
1,8