Standing Waves on a String
Casey Robinson
Robert Bianchini
29 January 2009
Version 1
Abstract
This experiment is designed to investigate the properties of standing waves on a
string. There are many properties to investigate; propagation velocity and vibration
modes are two that come to mind. The equation for propagation velocity/frequency was
verified, and the relationship between vibration modes was confirmed.
Introduction
When a string stretched between two fixed endpoints is perturbed a wave
travels across the string. When this propagating wave reaches an endpoint it is
reflected and returns inverted. If another wave is coming towards the reflected wave,
interference will occur. Interference can be constructive or destructive. Constructive
creates a larger wave and destructive creates a smaller wave.
Any wave propagating across a string can be described as a combination of the
string’s fundamental frequency and its harmonic frequencies. Interference of identical
waves at one of these frequencies will result in a standing wave on the string.
Standing waves have many unique characteristics. Standing waves are static
across the length of the string and only oscillate perpendicular to the string. Nodes,
points where the string does not move, are formed at even intervals. Anti-nodes, the
points at which the string oscillations have the largest amplitude, occur exactly in
between each of the nodes.
Theoretical
Waves on a string follow the Wave Equation (1), which can be found in any introductory
physics text. In the wave equation c is the speed and u(x,t) is the traveling wave.
𝜕2𝑢
𝜕2𝑢
2
=
𝑐
𝜕𝑡 2
𝜕𝑥 2
(1)
For waves on a string
𝑐=
𝑇𝑒𝑛𝑠𝑖𝑜𝑛
=
𝐿𝑖𝑛𝑒𝑎𝑟 𝐷𝑒𝑛𝑠𝑖𝑡𝑦
𝑇
𝜇
(2)
One solution to equation 1 results in standing waves on a string with fixed endpoints at
for certain frequencies fn. These frequencies are the topic of this experiment.
𝑢𝑛 𝑥, 𝑡 = 𝐴 ∗ 𝑠𝑖𝑛 𝑓𝑛 𝑡 sin
𝑛𝜋𝑥
𝐿
(3)
As with all waves the propagation velocity of waves on a string depends on frequency
and wavelentgth.
(4)
𝑣 = 𝑓λ
By rearranging and combining equations 2 and 4 we get that
𝑇
𝑇
𝑣
𝜇 𝑛∗ 𝜇
𝑓= =
=
λ
λ
2𝐿
(5)
n defines the harmonic, n = 1 is the fundamental, n = 2 is the first harmonic and so on.
The vibration modes are defined by n, where n = 1 is the fundamental mode, n = 2 is the
first harmonic, and so on.
𝑓𝑛 = 𝑛𝑓1
(6)
Experimental Method
clamp
string
driver
detector
mass
Figure 1: Experimental Setup
A Pasco Sonometer (WA-9611/9613) was setup to facilitate our investigation
into the properties of standing waves on a string. The sonometer consists of four parts;
the clamp which holds the string in place, the bridge supports which designate the
endpoints of the string, the tensioning lever which provides a location to attach mass
for tension, and the base which holds it all together.
The sonometer was placed on a workbench in the lab. Four guitar strings of
different linear density were used in this experiment to investigate the effect of
linear density on propagation velocity. See table 1 for further details. Varying masses
were attached to the tensioning lever to apply various amounts of tension to the string.
These are discussed in table 5.
An audio driver was connected to a frequency generator through an audio
amplifier. The driver was placed underneath the string 5cm to the right of the left
bridge support. The electric field produced from the driver forced the guitar string to
oscillate. This oscillation was captured by the detector, placed 5cm to the left of the
right bridge support. Both the frequency generator and the detector were displayed on
an oscilloscope.
This lab tries to answer many questions about standing waves and each one
requires a different variation in the setup of the experiment.
Does the location of the nodes and antinodes depend on the tension and linear
density? What is the mathematical relationship between propagation velocity, tension,
and linear density? To answer these questions the sonometer was set up to have a
string length of 50cm with the driver and detector 5cm from opposite endpoints.
Masses ranging from 200g to 1kg were placed on the fifth notch of the tensioning
lever, resulting in tensions from 9.81 N to 49.05 N. Adjusting the frequency of the
driver allowed for observation of standing waves on the string at each tension. After
standing waves at the fundamental frequency were found for each tension, the string
was swapped for one with smaller linear density. The experiment was repeated with
three different strings. The results of this portion of the experiment can be found in
table 5.
What is the mathematical relationship between the fundamental frequency and
the harmonic frequencies? In order to answer this question string one was stretched
across the sonometer and one kilogram was placed on the first notch of the tensioning
lever. The bridge supports were placed at three different distances; 50cm for setup
#1, 30cm for setup #2, and 40cm for setup #3. For each distance the driver and
detector were placed 5cm from opposite endpoints. This was also to prove that the
change in length did not affect the relationship between the frequencies. The results
of this portion of the experiment can be found in tables 2, 3, and 4.
Experimental Results
Table 1: Linear Density
String
1
2
3
Mass (g)
0.1736 ± .0001
0.1221 ± .0001
0.1289 ± .0001
Length (cm)
3.07 ± 0.02
6.12 ± 0.02
3.83 ± 0.02
Density (g/cm)
0.0565 ± .007
0.01995 ± .003
0.03655 ± .005
Table 2: Frequency Dependence, Setup #1
Frequency (Hz)
Number of Nodes
Wavelength (cm)
20.385
0
100.00 ± .02
40.793
1
50.00 ± .02
61.370
2
33.33 ± .02
83.579
3
25.00 ± .02
Table 3: Frequency Dependence, Setup #2
Frequency (Hz)
Number of Nodes
Wavelength (cm)
33.975
0
60.00 ± .02
69.093
1
30.00 ± .02
105.104
2
20.00 ± .02
141.514
3
15.00 ± .02
Table 4: Frequency Dependence, Setup #3
Frequency (Hz)
Number of Nodes
Wavelength (cm)
25.684
0
80.00 ± .02
52.944
1
40.00 ± .02
79.231
2
26.67 ± .02
105.708
3
20.00 ± .02
Table 5: Experimental frequency dependence on density and applied mass
String 1
String 2
String 3
200g
20.745 ± 0.001 Hz
34.872 ± 0.001 Hz
26.193 ± 0.001 Hz
400g
28.368 ± 0.001 Hz
48.384 ± 0.001 Hz
36.753 ± 0.001 Hz
600g
35.122 ± 0.001 Hz
59.328 ± 0.001 Hz
46.063 ± 0.01 Hz
800g
40.197 ± 0.01 Hz
68.267 ± 0.01 Hz
51.635 ± 0.01 Hz
1000g
44.403 ± 0.01 Hz
75.557 ± 0.01 Hz
58.495 ± 0.01 Hz
Table 6: Theoretical frequency dependence on density and applied mass
String 1
String 2
String 3
200g
20.826 ± 0.06 Hz
35.061 ± 0.06 Hz
26.995 ± 0.06 Hz
400g
29.452 ± 0.06 Hz
49.584 ± 0.06 Hz
38.176 ± 0.06 Hz
600g
36.071 ± 0.06 Hz
60.727 ± 0.06 Hz
46.756 ± 0.06 Hz
800g
41.651 ± 0.06 Hz
70.122 ± 0.06 Hz
53.989 ± 0.06 Hz
1000g
46.568 ± 0.06 Hz
78.398 ± 0.06 Hz
60.362 ± 0.06 Hz
Analysis of data
Linear Density
Mass
µ = 𝐿𝑒𝑛𝑔𝑡 𝑕 =
ΔMass 2
Δµ =
𝑀𝑎𝑠𝑠
0.1736 g
2
ΔLength
+
= 0.0565 g/cm
3.07 𝑐𝑚
.0001
=
𝐿𝑒𝑛𝑔𝑡 𝑕
2
0.1736
+
0.02 2
3.07
= .007
Tension
𝑚
T = (notch position)*(mass)*(gravity) = 5 * 200g * 9.81
ΔMass 2
ΔT =
𝑀𝑎𝑠𝑠
ΔGravity
+
2
=
𝐺𝑟𝑎𝑣𝑖𝑡𝑦
Δ1 2
200
+
.01
9.81
𝑠2
2
= 9.81 N
= .005
Frequency
f=
Δf =
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑊𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡 𝑕
ΔT 2
2∗𝑇
+
=
Δµ 2
2∗µ
𝑇𝑒𝑛𝑠𝑖𝑜𝑛
𝐿𝑖𝑛𝑒𝑎𝑟 𝐷𝑒𝑛𝑠𝑖𝑡𝑦
2∗𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
+
ΔD 2
𝐷
=
=
.005
2∗9.81
9.81 𝑁
0.0565 𝑔/𝑐𝑚
2∗50 𝑐𝑚
2
+
=20.826 Hz
.007
2∗.0565
2
+
.1
50.0
2
= .06
Uncertainty Evaluation
There are a few fundamental sources of error in this experiment. The limitation
of measurement devices is the most obvious one. Air resistance is not included in the
simple theoretical model described above, which of course exists in the lab. Also
magnetic force holding the bridge supports was not strong enough to create a perfect
fixed endpoint. At times, particularly with high frequency, the string tended to bounce
at the ends. This resulted in difficulty obtaining an accurate measurement of standing
wave frequencies.
To overcome the limitation of the bridge supports, mass was placed on each of
the supports. The gravitational force of the masses helped to fix the problem of
bouncing strings.
Discussion/Conclusion
This experiment shows that the mathematical relationship for standing wave
frequency derived above is accurate. The results of the experiment are not exact due
to some limitations in the setup. Also the relationship between the fundamental
frequency and the successive harmonic frequencies is correct.
If this experiment were to be repeated a few things could be different to
improve the accuracy of the results. Placing the sonometer into a vacuum would
eliminate air resistance, just as in the wave equation. Adding a stronger bonding force
between the bridge supports and the base would eliminate some of the bounce in the
string. This would make finding a frequency for standing waves more accurate.