Lab #2: Transverse Waves
Introduction:
The speed, v, of a wave depends upon the properties of the medium through which it propagates. This is often a
misunderstood feature of wave phenomena. Instead, it is commonly thought that the frequency of a wave determines
the speed with which it travels. In particular, high frequency waves are thought to travel faster than low frequency
waves. However, if this were true, then listening to a music concert would not be possible. If higher pitched notes
traveled faster than lower pitched notes, then they would reach the back of the auditorium prior to the arrival of the
lower pitched note created at the same time. The resulting sound heard by the listener would be a haphazard mix of
notes that could hardly be called “music”.
If an oscillating source, such as someone’s hand, is put into contact with the end of a rope it will raise the
end of the rope by exerting an upward force on the rope. The resulting acceleration of the rope is inversely
proportional to the mass of the segment of the rope upon which the force acts. Since the force does not act on the
entire rope at the same time, we can approximate the response by dividing the rope into many small segments of
rope having length dl and mass dm. The mass per unit length of the rope is just dm/dl = m/l. In the diagram below a
pulse encounters a segment of stretched rope.
The rope rises in response to the force but the inertia of the segment of the medium prevents it from rising
instantaneously. Once the mass reaches the maximum displacement, it falls due to the restoring force of the medium,
FT = Tension. Experimentally and theoretically, it can be shown that the speed of propagation of a wave pulse is
determined by the following:
v=
FT
m
l
The source of a periodic oscillation determines the frequency at which pulses are produced in the medium. For
example, a person who shakes his/her hand back and forth determines the frequency at which the pulses enter the
rope. The frequency of the source is the same as the frequency of the resulting wave pulses.
It is a combination of the frequency of the source and the speed of the medium that determines the wavelength ( λ )
of the periodic wave that enter the medium. If the source sends a pulse into a relatively fast medium then the first
part of the pulse will have the opportunity to travel a great distance before the next pulse is sent behind it, causing a
longer wavelength. Whereas, if the medium is slow, the first part of the pulse will not travel as far into the medium
before the next one is sent behind it, causing a shorter wavelength. A simple relation can be obtained that
summarizes these relationships.
v = λf
Lab #12 – Transverse Waves
This equation should not be misinterpreted. It states mathematically that the product of the frequency and the
wavelength determines the speed, v, of the waves. However, as stated above, it is actually the combination of the
frequency of the source and the speed of the medium that determines the wavelength, λ, of the wave.
Since
v = λf
v=
and
FT
m
l
, it must follow that:
FT
m
= λf
l
In this experiment we investigate this relationship.
Another known property of waves is that they reflect off of a boundary. A boundary is defined as the place
where a medium changes or ends. The reflected waves interfere with the waves that are still traveling toward the
boundary. Under certain conditions, the superposition of the incident and reflected waves produce what is known as
standing wave patterns. For an oscillating medium fixed at both ends, the waves continue to bounce back and forth
between two boundaries. The condition that must be met to produce standing waves in a fixed-fixed medium is that
the total length, L, of the medium must be an integer multiple of a half wavelength of the propagating wave:
L = n2λ
, where n is any integer.
λ
L
For a string of fixed mass per unit length (non-stretchable) that has a fixed frequency source of oscillation attached
to it, there should be multiple lengths of that string that permit standing waves for any given tension. That is:
v=λ⋅ f
FT
m
l
=
2L
⋅f
n
 F
1 
L = n⋅ T ⋅

m
 l 2f 
Other lengths of the string will produce superposition patterns that are haphazard in appearance.
Lab #2 – Transverse Waves
Lab #2: Transverse Waves
Goals:
•
•
•
To determine the speed of waves propagating in a string.
To experimentally verify the relationship between wave speed and tension.
To predict new modes of oscillation for vibrating strings.
Equipment List:
String Vibrator
String Pulley
Clamps
Hanging Mass Set
Meter Stick
Activity 1: Standing Waves on a String
1.
Set up the apparatus. The vibrator sends a succession of waves along the
string. If the length of the string is just right, then the reflected waves will produce standing waves, which can
easily be observed. Initially, set the vibrator to be approximately 1 meter from the pulley.
2.
Add 750g to the end of the string not connected to the vibrator
3.
Record the first observable resonance frequency (the frequency when the smallest number of complete
loops with maximal amplitiude are observable)
4.
Find and record the next two resonance frequencies
5.
Find the velocity of the waves in the string by taking the average of your three observations, show
your calculations
f
(Hz)
λ
(m)
FT
(N)
v
(m/sec)
Lab #2 – Transverse Waves
Activity 2: Speed and Tension
1.
Repeat Activity 1 for four more tensions. Be sure to vary the tension significantly.
2nd Tension
f
FT
v
λ
(N)
(Hz)
(m/sec)
(m)
3rd Tension
f
(Hz)
4th Tension
f
(Hz)
5th Tension
f
(Hz)
2.
λ
FT
(N)
v
(m/sec)
λ
FT
(N)
v
(m/sec)
λ
FT
(N)
v
(m/sec)
(m)
(m)
(m)
Record the values for the Tensions used and the (average) Speeds obtained in the chart below.
Tension (N)
Speed (m/sec)
3.
Make a graph, using Excel, of Speed vs. Tension.
4.
Make a graph, using Excel, of Speed Squared vs. Tension.
5.
Using the graphs above as a reference, answer the following questions:
•
Do your graphs verify the relationship given by the following equation? Explain.
v=
•
FT
m
l
Compare the mass per unit length of your string obtained from the graph to the value obained
from the scale.
Lab #2 – Transverse Waves