Waves
• Traveling waves: Traveling, periodic, sinusoidal (Shaped
like a mathematical sine wave) water waves, sound waves, electromagnetic waves, and others, share certain features shown in
the figure. The distance from crest to crest is the wavelength
λ. They pass an oberver with speed v. The time between the
passage of one crest and the next crest is the period P of the
wave, so λ = vP .
Traveling wave
wavelength λ
Amplitude of the wave A
λ
crest
A
trough
v
observer
Period of the wave P
speed of the wave v
frequency f,
f = 1/P
λ= v P
v =λ f
The frequency f of a periodic wave is the reciprocal of the
period, f = 1/P . So for example if P = 1/10 second, then
f = 10 cycles/second. However, we use a different name for
the unit to measure frequency. 1Hertz, abbreviated 1 Hz, is
the same as 1 cycle/second. So, in the previous example, we
say f = 10 Hz.
• Transverse Waves:
One can imagine that the wave in the figure above is a transverse wave traveling on a rope. The wave travels to the right
1
but the motion of the rope is up and down, perpendicularly
(transverse) to the direction of motion. Water ripples, shown
in the figure below, and electromagnetic waves are also transverse.
TRANSVERSE WAVES
water wave: surface motion
electromagnetic wave: Electric field
amplitude
wavefront
Energy is carried along rays
ray perpendicular to
wavefront
radius = r
Water ripples travel on the surface of the water. Wave fronts
are an important concept for waves. In this case the wave fronts
are circles that follow the crests of the wave.
• Longitudinal Waves:
Sound waves and waves on a spring, shown in the figure below,
are longitudinal waves because the motion of the material particles is back and forth from left to right as the wave travels
to the right. In sound through air it is the air molecules that
move back and forth from left to right as the wave travels to
the right. Sound waves also travel through solids and liquids.
If the figure represents a sound wave, the compressed regions
have higher pressure and density than the stretched regions.
2
In the spring regions of compression and stretching move
back and forth from left to right as the wave travels to the
right.
LONGITUDINAL WAVE
v
a long spring vibrating as a
logitudinal wave passes
• Standing waves or normal modes of oscillation:
A rope or a string tied is tied down at both ends, like in a guitar
or violin. We can make waves travel back and forth on it with
a speed v in a complicated pattern, with different parts of the
string moving in different ways. However, when the frequency
is right, the string can vibrate in a standing wave, called a
normal mode, in which every point of the string oscillates at
the same special frequency. There are many normal modes,
each with its own frequency. The first few normal modes of a
string are shown below.
L
λ/2 = L or
λ1=2L
f1
2λ/2 = L or
λ2=L
2f1
3λ/2 = L or λ3=2L/3
3f1
With a wave speed v, f1 = v because λ= 2L
2L
There are many more normal modes each with
a frequency nf1 where n=1, 2, 3, 4,...
Normal modes or standing
waves on a string of length L
tied at both ends
3
The frequencies of oscillation are multiples of the first harmonic f1 , 2f1 , 3f1 , 4f1 , ..... Other vibrating systems may have
a different family of harmonics, such as only the odd ones,
f1 , 3f1 , 5f1 , ..... This idea that only certain special characteristic frequencies are possible, is fundamental in music. But it is
also fundamental in Quantum Mechanics.
• Electromagnetic waves:
Electromagnetic waves travel in all directions, so their wave
fronts are spheres. Their amplitude represents the strength
of the electric field. Their intensity, which is proportional to
the square of the amplitude, falls as 1/r2 because the energy
is spread over a sphere of radius r. In a vacuum all electromagnetic waves, radio, microwave, light, X-rays, etc., all travel
with the speed of light c = 3 × 105 km/s, which is actually the
fastest speed possible in the universe. An electromagnetic wave
consists of oscillating electric fields transverse to the direction
of propagation, shown in red in the figure below, together with
magnetic fields, shown in blue, that are also transverse and
perpendicular to the electric fields. We say that such a wave is
vertically polarized.
An electric dipole antenna is shown in the figure below. The
antenna radiates strongly in a direction perpendicular to the
antenna and very weakly in the direction along the antenna
wires. A vertical electric dipole antenna produces a vertically
4
polarized wave as shown above. The horizontal dipole antenna
produces a horizontally polarized wave.
Electric dipole antenna
direction of strong wave perpendicular to antenna
oscillating voltage
wire
zero radiation in this
direction prallel to antenna
zero radiation in
this
direction parallel to antenna
λ/2
• Polarization of Light:
Visible light is emitted by atoms. In ordinary light sources the
electric field changes direction randomly, because the atoms
radiate independently of each other, and we say the light is
unpolarized. This is why our eyes have not evolved to detect
the polarization of light. However reflected light, and blue sky
light are partially polarized. Polarized sunglasses shown below
have a polarizing filter that only allows vertically polarized light
to pass through.
vertically polarized light passes through and enters eyes
pass axis
absorption axis
polarizing filter
horizontally polarized light is absorbed and does not pass
vertical incoming E
field
vertical E field
passed
horizontal
incoming E field
horizontal E field
absorbed and does
not pass
polarizing filter
pass axis
5
They are useful because they tend to eliminate the glare produced by light reflected from horizontal surfaces. If a vertical
filter is put in front of a horizontal filter, then no light passes
through at all. The polarization of light is used in a clever way
to produce the bright and dark pixels in LCD TV monitors.
• AM and FM waves:
Audio frequencies for the normal human ear lie in the range
from 20 Hz to 20 KHz. Above 20 kHz it is ultrasound. However,
the frequencies of the electromagnetic waves used for radio and
TV transmission are much higher than audio frequencies. This
is done because one wants an antenna of reasonable size. The
length of a good dipole antenna to generate waves of wavelength
λ should be roughly λ/2. But λ = c/f , where the speed of
light is very large, c = 3 × 105 km/s. Then if we tried to emit
electromagnetic waves with f = 20, 000 Hz, the wavelength
would be about 15 km, so it would require a huge antenna about
7.5 km long. To avoid this AM radio uses high frequency carrier
waves with wavelengths in the range of 100 meters, while FM
and TV use higher frequency carrier waves with wavelengths
in the range of 1 meter. In the figure below the carrier wave
has the frequency of the radio station. In order to transmit the
music, or other audio signal, the carrier wave is modulated. In
AM modulation the amplitude of the carrier wave is multiplied
(modulated) by the low frequency audio signal as shown below.
6
In the case of FM modulation the amplitude and frequency
of the signal is used to vary (modulate) the frequency of the
carrier wave. The receiver filters out the carrier, and amplifies
the audio signal, which is sent to a speaker.
• Interference and Coherent and Incoherent Sources:
Two sources of waves are coherent if they have the same frequency or wave shape, and incoherent if they have different
frequencies or wave shapes. Coherent waves can reinforce or
cancel each other. In constructive interference the wave received is stronger because the crests (or peaks) of the two waves
arrive together in a regular (coherent) manner. Then the amplitude of the resultant wave may be twice that of only one
wave, and the intensity will be 4 times rather than 2 times the
intensity of one wave. In destructive interference the wave received is canceled out because the crests (or peaks) of one wave
arrive together with the troughs of the other wave in a regular
(coherent) manner. Then the amplitude and intensity of the
resultant wave will be zero, as shown below.
7
coherent sources have the same frequency or wave shape
coherent sources, constructive interference =
coherent sources, destructive interference =
incoherent sources: no interference
incoherent sources have different frequencies or wave shapes
Two sources of water ripples, or two speakers driven with exactly the same signal will interfere constructively or destructively at a point in space depending on the different distances
from the point to the two sources as seen below.
If one source produces a wave of a different frequency, or of a
completely different shape, the crests and troughs of the two
waves reinforce at one moment but cancel at another moment
in a complicated or random (incoherent) manner. Then the
interference averages out, and the resultant intensity is merely
8
the sum of the two intensities. This is why two flash lights
shining on a spot do not cancel out, but rather always give you
more the sum of the intensities of each flashlight. Ordinary
sound and light sources are incoherent with each other. Therefore the intensities of the sources add together. They are not
canceled, nor doubled.
• Single Slit Diffraction:
When waves of any kind pass through an opening they open
out, or diffract, producing a pattern that depends on the size
and shape of the opening. The figure below shows the diffraction pattern when water waves pass through an opening.
One might have thought that the waves would be restricted to
a beam the size of the opening, but instead they open out with
maximae and minimae in different directions. The appearance
of sharp shadows tells us that light travels in straight lines, and
it is not obvious that light waves also diffract. What happens is
that the amount of diffraction depends on the ratio λ/a of the
wavelength divided by the width of the opening. If λ/a 1
the diffraction is negligible. This is the usual case with light because its wavelength is small compared to most openings, such
as windows. However, with a narrow slit the diffraction is very
similar to the pattern of water ripples. A laser beam, which
is coherent across its face, its aimed at a single slit produces a
9
diffraction pattern projected on a screen as shown below.
A laser beam aimed at an opaque steel ball produces a diffraction pattern projected on a screen as shown below. Surprisingly
there is no obvious shadow, and there is the Poisson bright spot
at the center of what one would have thought would be the
shadow. This happens because the light is coherent all around
the ball.
• Double Slit Diffraction and Interference: When water
waves, or light waves, go through two slits, they diffract and
interfere as if from two coherent sources as shown below for
water ripples.
When the laser beam falls on a double slit the slits act as coherent sources, and produce the pattern below. We get a complicated pattern. What you see is the double slit interference
modulated by the single slit diffraction pattern.
10
• Quantum Mechanics or Wave Mechanics: Quantum
Mechanics shows that waves van behave like particles and particles can behave like waves. The modern view of quantum
electromagnetic fields is that a field of frequency f has wave
properties, but it can only change its energy when it interacts
with matter (charged particles) in multiples of a basic quantum of energy we call a photon, or “particle” of light. The
energy of a photon in light of frequency f is E = hf , where
h = 6.626 × 10−34 Joule·second. Interestingly, this idea is due
to Einstein, who never really believed in Quantum Mechanics. Phenomena like interference are explained using the wave
properties of light, but it is a peculiarity of quantum mechanics
that lets this happen even if there is only one photon in the
field.
Similarly particles like electrons are also entities in a quantum matter field, and they have wave properties. An electron
moving with linear momentum p has properties like a wave of
wavelength λ = h/p. Experiments analogous to the double slit
can be done not only with light, but also with electrons, and
other particles.
11
The magnitude (the strength) of the wave, usually called ψ,
at a point is a measure of finding the particle at that point if
one does an experiment to find where it is. Therefore quantum physics is probabilistic. Einstein objected to this and said
something like ”God does not play dice”. He proposed the
Einstein-Podolsky-Rosen paradox as an objection. However,
it turns out that the objection fails in experiments that show
the truth of something called ”quantum entanglement” as predicted by the ”Bell inequalities”. An interesting article on this
titled ”Is the Moon there when nobody looks? Reality and the
quantum theory” by the well known physicist David Mermin,
appeared in the April 1985 issue of the magazine ”Physics Today”. Some physicists say that Bell’ theorem is ”the most profound discovery of science.” Others say that ”anybody who’s
not bothered by Bell’s theorem has to have rocks in his head.
12