Chapter 2
(Particle Properties of Waves)
Review: Wave Properties
of Waves
Chapter 2
Particle Properties of Waves
Overview
Heinrich Hertz was a German physicist who
showed that radio waves (not known as such
back then, of course) and light are both
examples of the electromagnetic waves
described by Maxwell’s equations.
During his investigations of the photoelectric effect ca. 18871888, Hertz made some observations about light which could
not be explained by the wave properties of electromagnetic
radiation. You’ll hear more about the photoelectric effect in
the next lecture.
The Compton effect, explained by Compton in
1923 while he was at Washington University in
St. Louis, involves the change in wavelength of
x-rays when they are scattered.*
The Compton effect cannot be explained by the wave
properties of electromagnetic radiation. We will skip this
section of the text.
We have some unsolved “issues” with electromagnetic
radation. But first, let’s review what you already know.
*Nobel Prize, 1927.
2.1 Electromagnetic Waves
This section is a review of material you should already know.
Consider a wave described by
y(x,t)  A sin (kx  ωt) .
The phase of this wave is
θ(x,t)  kx  ωt .
dθ
dx
Also
k
ω .
dt
dt
If  is constant with time (i.e., d/dt=0), then we are moving
with the wave, and
dx 

.
dt k
The phase velocity, vp, is given by
ω
vp 
.
k
Imagine yourself riding on any point on this
wave. The point you are riding moves to the
right. The velocity it moves at is vp.
Note that k=2/, so if  is a nonlinear function of k, then
the phase velocity depends on .
y(x,t)  A sin (kx  ωt)
ω
vp 
k
The spatial coordinate of any point of constant phase travels
in the +x direction when /k is positive, and in the -x
direction when /k is negative.
In other words, waves travel to the right when /k is positive,
and to the left when /k is negative. Thus, the signs of 
and k tell the direction of motion of the phase of a wave.
The wave is moving from left to
right so /k must be positive.
In Physics 201, you may have derived a classical expression
for the energy per unit time carried by a wave. The result
is
ρ 2 2
Power 
ω A vp.
2
The power depends on 2 (or f2) if the amplitude A is
constant, or on A2 if  is constant. As we will see in the next
section, the experimental results of the photoelectric effect
are in disagreement with this equation
Superposition -- a characteristic of all waves.
When waves of the same nature travel past some point at the
same time, the amplitude at that point is the sum of the
amplitudes of all the waves
The amplitude of the electric field at a point is found by adding
the instantaneous amplitudes, including the phase, of all
electric waves at that point.
If A = a + jb is an amplitude, then A* = a - jb and
AA *   a  jb 
2
2
a-jb

a

b
.
 
Remember that power (or intensity) is proportional to
amplitude squared.
Superposition is a result of the linear relations between electric
field and polarization and between magnetic field and
magnetization.
The magnitude of a wave is found by multiplying the amplitude
by its complex conjugate.
The intensity of the superposed waves is proportional to the
square of the amplitude of the resulting sum of waves.
Interference -- a result of the superposition of waves.
Constructive Interference: If the waves are in phase, they
reinforce to produce a wave of greater amplitude.
Destructive Interference: If the waves are out of phase,
they reinforce to produce a wave of reduced amplitude.
Web pages on interference:
http://micro.magnet.fsu.edu/primer/java/interference/waveinter
actions/index.html
http://www.colorado.edu/physics/2000/schroedinger/index.html
Young’s Double Slit Experiment (1801)
This experiment demonstrates the
wave nature of light.
Consider a single light source, and
two slits. Each slit acts as a
secondary source of light (this is a
result of diffraction; the light
waves bend around corners at the
slits).
Light waves from secondary slits interfere to produce
alternating maxima and minima in the intensity.
Reference and “toy:” http://micro.magnet.fsu.edu/primer/java/
interference/doubleslit/index.html.
How does this work?
Light waves from the two slits arriving at the detection screen in
phase will interfere constructively and light waves arriving out of
phase will interfere destructively.
In phase—
constructive.
Out of phase—
destructive.

 is the path length difference.
Interference is constructive when =, 2, 3...
Interference is destructive when =/2, 3/2, 5/2…
For path lengths in between, the interference is only partial.
In fact, the familiar n = a sin applies here, where 2 is the
angle between the two rays at the screen.
More “toys:”
http://www.colorado.edu/physics/2000/schroedinger/
two-slit2.html
http://webphysics.ph.msstate.edu/jc/library/24-3b/
Useful reference page:
http://www.colorado.edu/physics/2000/applets_AL.html
So What?
Interference and diffraction are exclusively wave-like
properties. So is refraction (also a property of light and
waves). We conclude that light is clearly a wave.
Maxwell studied electromagnetic waves. Maxwell showed
what kind of wave light is.
Maxwell, 1864, suggested that accelerated electric charges
generate electromagnetic waves.
Faraday then showed that a changing magnetic field can
induce a current.
Maxwell proposed that a changing electric field has an
associated magnetic field. This was confirmed only after his
unfortunate death.
Maxwell showed that electromagnetic waves propagated
with the speed
1
c=
= 2.998×108 m/s .
ε0μ0
This is exactly the speed of light, and Maxwell concluded light
consists of electromagnetic waves. Note that light waves are
only a small part of the electromagnetic spectrum.
Electromagnetic waves propagate at the speed of light in a
vacuum, but slower in materials, which makes the index of
refraction greater than unity (n = c/v).
Hertz provided experimental confirmation of electromagnetic
waves in 1887-8. The experimental confirmation was found in
the photoelectric effect.
When asked by a student what these experiments meant, Hertz supposedly said
“It’s of no use whatsoever. It just proves Maxwell was right.” When the student
then asked “so, what next?” Hertz supposedly replied “nothing, I guess.”*
*Hertz was brilliant but excessively modest. Every day you benefit from dozens of
applications based on his work. I’m not sure I believe this “story” but you can
search for sources on the web if you are curious. Unfortunately, Hertz died at age
37 of blood poisoning.