Transverse Standing Waves
Introduction
A mechanical wave is the propagation through a medium of a distu
ance of the medium. Consider the string under tension shown below. In this case, the string itself is the medium that ca
ies the wave.
Sketch
A continuous distu
ance of the stretched string by the oscillator creates a train of transverse waves that travels along the string. The waves are reflected at the pulley. The leftward traveling reflected waves interfere with the rightward traveling incident waves. The result under certain conditions is a standing wave pattern along the length of the string.
The system resonates when the frequency of the oscillator matches a natural frequency of the string under tension. Natural frequencies, in this case, are those wave frequencies for which an integer number of halfwavelengths (segments) fit along the string (which is "fixed" at both ends). Resonance co
esponds to the appearance of standing wave patterns.
To establish resonance, it is necessary and sufficient to adjust the length of the traveling waves until
(1)
where N is the number of segments in the standing wave pattern, each segment co
esponding to a halfwavelength and is the wavelength.
From equation (1), wavelength can be a function of N (N=1, 2, 3, 4, 5, …) with a fixed L, we rewrite it as
XXXXXXXXXX2)
Therefore, with different segments –N of standing waves, co
esponding to a set of discrete wavelengths. In our lab, the frequency f is supplied by an oscillator with a fixed value -- f= 125 Hz.
Now, wavelength and wave frequency, f, are related by
(3)
where is a set of discrete velocities of propagation of the waves traveling along the string. We will treat these as intrinsic wave values.
On the other side, the tension force applied to the string also create a mechanic wave, with the wave speed satisfies an empirical equation
(4)
where is the linear mass density of the string. The tension in the string is created by hanging a mass on the end of the string therefore FT = mg.
Based on the equation (4), we can change the tension force continuously as we want, if we match one the of discrete intrinsic speed values (equation (3)), then we have the transverse standing wave! Standing waves have the biggest magnitude. That is how we do this lab.
Simulation site
Standing Wave Simulation
(Please note: when count the number of segments—N, count both +max and -max. For example, for the figure above, N=8.)
Procedure
1) Click the link to go to the site
2) Carefully change the tension force, observing the wave magnitude getting bigger, stop when the magnitude is the maximum (keep increasing when it become smaller, then go backward to where the maximum is). Note, when closing in to the maximum, using the right (or left) navigation a
ow ˃ key to adjust the tension—each click is only 0.01 N.
3) When you decide you have the standing wave (the wave has the maximized magnitude), click pause (optional) and record your wave segment number and tension force value in table2.
4) Repeat until you fill up table2.
Data
Table1—the fixed values for our trials
Length -L (m)
Oscillation frequency f (Hz)
String linear density (kg/m)
4
125
3.2*
XXXXXXXXXXTable2 – Standing Wave Recording
Tension force F (N)
N
(Number of segments)
16.33
14
19-
13
22+
12
26+
11
32
10
40-
9
50
8
65+
7
88.89
6
Analysis
For each standing wave, compute using Equation (2), then v using Equation (3). Computing the next V using equation (4). Also compute percent difference.
XXXXXXXXXXλ=2L/N XXXXXXXXXXv=λf XXXXXXXXXXv comparison
XXXXXXXXXXTable3
# of wave segment
N
Wavelength--λ (m)
Wave Speed v from oscillator (m/s)
Wave speed v from tension force (m/s)
Percent diff.
14
0.57
71.25
71.43
0.25%
13
12
11
10
9
8
7
6
1.33
166.67
166.67
0%
many discrete intrinsic waves by oscillator
One wave to match by tension force
100%
x
two
the
of
average
value
2
-
value
1
Difference
%
nd
st
=
L
2
N
=
l
m
T
F
v
=