Physics 125 – Seeing the Light
University of Massachusetts at Amherst
Department of Physics
Fall 2015 – Some Brief Notes on Waves and Wave Motion
Robert B. Hallock
version 11/1/15
A vibration is the to-and-fro motion of an object. Good examples of
vibrating objects are (1) diving boards after the diver has left, (2) a bell that
has been struck (although in this case you can’t see the vibration), (3) a mass
moving up and down hanging from a spring, etc. Pendulum motion is also an
example of vibration – the to-and-fro motion of an object that hangs from a
string or a rod.
We can define the period (symbol T) of the motion as the time to
complete one complete vibration or cycle.
T = time for one complete vibration (i.e. to-and-fro motion) or cycle
T = (total time) / (total number of complete vibrations)
We can also define the frequency, f
f = (number of complete vibrations) / (total time)
f = 1/T
Units: T has units of seconds, and f has units of 1/sec, or Hertz, which
we can abbreviate at Hz. So, 1 Hz = 1 / sec. Sometimes 1 /sec is written as
cycles /sec, and called “cycles per second”. The word “cycle” has no units.
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So, a musical note of middle C is a note of 256 Hz, or 256 cycles (to-and-fro
motions) per second; 256 Hz = 256 / seconds.
Example: A bell rings at a frequency A, which is 440 Hz. How
much time is required for one cycle of oscillation of the bell?
T = 1/f = 1/ 440 vibrations/sec
= 0.002273 sec / vibration
A wave is a distortion (or change) in something, and the distortion
propagates (i.e. moves). Examples are waves on strings (in which the shape
of the string changes, and moves along), on water (in which the surface of
the water changes shape), in the air (where you can’t visibly see what it is
that is changing) etc. Water wave are the easiest to picture in your mind.
Here the distortion propagates along the surface of the water.
Transverse Wave: a disturbance that causes a distortion at right
angles to the propagation. Examples include water waves, waves on strings
etc. [Note: A transverse wave that reflects from a boundary where the
string, or spring, is tied down, is reflected upside down.]
Moving in this direction.
Amplitude
amplitude: The distance from equilibrium to the maximum deviation.
Wavelength and amplitude are shown on the figure here.
wavelength
amplitude
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wavelength: The distance between identical looking locations on the
wave shape. Generally to properly define a wavelength, there typically is a
large number of repetitions of the basic wave shape. In the figure above we
have only shown a couple of these. We denote wavelength by the Greek
symbol .
For all waves, there is a relationship between the wavelength, the
speed of the wave and the frequency of the wave. This rule is v = f. Here
the velocity is represented by v, the frequency is f, and the wavelength is
represented by the Greek symbol . This rule is independent of the
amplitude of the wave in all cases we will consider.
Example: If the speed of a wave on the water is 4 m/sec, and 5
seconds elapse between breakers on the beach, what is the wavelength of the
waves?
v = f and so
 = v / f = (4 m/sec) / f
Note that to get this, we need to obtain the frequency. We were given the
period of the wave motion - the time between successive “breaks” on the
beach. Thus, f = 1 / T and f = 1 / 5 sec = 0.2 /sec, or 0.2 Hertz. It is
important to keep straight the difference between the concepts of frequency
and period. So, we have,
 = v / f = (4 m/sec) / (0.2 Hz) = 20 meters. (distance between breaks)
Example: A particular orange light has a wavelength of 600 nm in
vacuum. What is the frequency of this light? (1 nm = 1 x 10-9 m)
v = f and so
f = v /  = 3 x 108 m/sec / 6.00 x 10-7 m = 5.00 x 1014 Hz
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Longitudinal Wave: a disturbance which causes a distortion along
the direction of the propagation. A compression in a spring is a good
example.
Sound waves are good examples of longitudinal waves and so are
waves which move along a stretched spring. In the case of the spring we can
see what is moving, but in the case of sound waves, we can’t see what is
going on. In that case, it is the density of the air that is changing. To better
picture this in our minds, we can consider a tuning fork.
In the case of a tuning fork, the tines move, and press back and forth
on the air molecules - These locations of compression move away from the
tuning fork. We draw a sketch of the compressional waves. And, we can
indicate on the sketch the wavelength for the waves.
wavelength
The intensity of any wave is the power it delivers to a unit of area.
I = P/A = power / area
The units are watts / m2
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Examples: Sunlight: Light from the sun deposits 1000 W / m2 (for a
typical noon sun exposure on the equator). That is, 1000 watts of power are
deposited on each square meter of the Earth’s surface. Sound: The ear can
hear sounds over a tremendous range: typically taken to be 1 - 10-12 W / m2
(The smaller number here is called the threshold of hearing, the minimum
intensity the human ear can just perceive as sound. Technically this is the
definition of the threshold of hearing at 1000 Hz. To reach the human
threshold requires a bit more intensity at lower and higher frequencies.)
Optional – the ear: (mentioned FYI in lecture but not required):
The range for the ear is actually quite remarkable. The ear can hear
over the huge range of intensity values shown below. For the most part we
don't even notice. The table below gives the intensity values for typical
sounds. The dB, the deciBel, is another unit of sound intensity.
description
(W/m2)
(dB)
threshold of hearing
whisper at 1 m
average office
loud radio
average factory
loud rock concert
jet plane at close range
bursting of eardrum
1 x 10-12
1 x 10-10
1 x 10-7
1 x 10-4
1 x 10-2
1
1 x 10+2
1 x 10+4
0
20
50
80
100
120
140
160
End of the Optional.
Superposition - the addition of amplitudes of two or more
simultaneous waves. The waves can add strongly, or they can cancel out
depending on how well the crests and troughs line up. Examples:
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+
+
=
=
Standing Waves: a repeating wave form which fluctuates in space and time.
Shown here are three examples of standing waves (you might see this is you
took a very fast photograph with a strobe light that “froze” a vibrating rope
or spring in position in the photograph).
fundamental, 1st harmonic
2nd harmonic (2 bumps)
3rd harmonic (3 bumps)
Note  is the distance between successive identical locations (we will
draw some sketches on the blackboard), the node locations (are places where
there is no vertical motion), and antinode locations (are places where there is
maximum vertical motion). A node is a position in the vibration pattern
which appears to stand still. An antinode is a place where the disturbance is
a maximum. Musical instruments that have strings (like guitars) rely on
frets to set the lengths of the strings and setting the length for a given tension
in the string sets the wavelength and thus the frequency with which the
string vibrates. A violin has no frets and the position of the fingers has to be
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quite precise to produce notes that are “on key”. We may not specifically
use the words “node” and “antinode” in lecture, but they are straightforward
definitions.
[As a comment, (not covered in detail in lecture) we note the following: You
have heard the superposition of two simultaneous notes many times, but may
not have noticed. A touch tone phone of a certain style works by using two
tones for each key. That is why the keys sound a bit odd to you when you
hear the sound when keys are pressed. Here are the frequencies.
1
2
0
9
1
3
3
6
1
4
7
7
1
2
3
697
4
5
6
770
7
8
9
852
0
941
With this system, touching any key produces the two tones shown. So, for
example, touching a 5 key produces the two tones 770 Hz and 1336 Hz;
touching a 3 key produces the two tones 697 and 1477. I have no idea why
these particular frequencies were chosen.]