Introduction To Waves
A wave is several pulses generated at regular intervals which move energy from point to point without
transporting matter. Wave motion is a mechanism for the transfer of energy from one point to another.
Mechanical Waves require a medium to transfer this energy while Electromagnetic Waves do not.
All waves behave the same. Water waves, sound waves, and light waves all follow the same rules for behavior.
The wave shown below is a transverse wave (it looks like a sine curve). In transverse waves, the particles in
the medium move perpendicular to the wave’s motion. (Remember: Transverse Perpendicular!!)
The diagram above can be used to define several wave characteristics. Wavelength () is the distance in
meters from any point on the wave to the same point on the next wave. (B on the diagram) As a definition it is
the distance between consecutive points on a wave that are in phase.
The amplitude of the wave is the distance from the base of the wave to the point of maximum displacement,
either above or below (C on the diagram or half of D).
The speed of all waves is given by v = f . The wavelength (m) multiplied by frequency (cycles per second) =
velocity (meters/second). (Cycles aren’t really a unit in physics.)
The frequency (f) of a wave is equal to the number of waves per second (waves/sec) and is measured in units
called Hertz. The inverse of frequency is called the period (T) of the wave and is defined as the time for one
wave (seconds/wave) and is given by the equation T = 1 / f.
In the diagram at the right, the period of the wave is 0.5
seconds and the frequency (1/T) is equal to 2 cycles per
second (Hz). This means that two full waves pass a given
point in one second.
Points B and G are in phase with each other (same point
on the next wave) which means they are exactly one
wavelength () apart.
Points C and E are exactly out of phase (or 180 out of phase).
They are two points which are ½  apart.
Behavior of Waves at a Boundary
When a wave reaches a boundary with another medium, part of the wave is reflected from the boundary, part of
the wave is absorbed into the medium (we often ignore this part), and the rest of the wave is transmitted into the
new medium.
*Entering a new medium is the only way a wave’s speed may change, since wave speed depends only on the
medium.
*Frequency does NOT change when you enter a new medium, since the frequency depends only on the source
of the wave.
What do these two statements (in combination with the wave equation) imply about what happens to
wavelength when you enter a new medium?
Reflection Off a Boundary
When a wave is reflected a boundary that is “harder” (or in light’s case more “optically dense), the wave will be
inverted (180 out of phase):
Transverse and Longitudinal Waves
In transverse waves the particles in the medium vibrate perpendicular to the wave’s velocity. (Ex’s
include electromagnetic (light) waves, waves on a rope)
In longitudinal waves the particles in the medium vibrate parallel to the wave’s velocity. (Ex’s include sound
waves, compression waves on a spring)
Sound Waves
Sound is one of the first waves studied by scientists and everyday people who had a love of music. A vibratory
disturbance in air, water, or even steel, can produce a sound wave.

Sound cannot travel through a vacuum, it can only travel through a medium (it is a mechanical wave) In
outer space, or even on the moon, you can scream at the top of your lungs and no one will hear you ! (You
cannot hear explosions either - the movies lie!)

Sound waves are longitudinal
Because sound is a compressional wave, it is produced by the alternating compression and spreading out of the
molecules of the medium through which it travels. It makes sense then that sound would travel faster in
mediums in which the molecules are closer together (solids). It also travels faster in a medium whose molecules
are moving faster (higher temperature). We can calculate the speed of sound in air at a given temperature by
using the equation:
Vair = 331 m/s + 0.6 m/s (°C)
The speed in water is approximately: 1,500m/s
The speed in steel is approximately: 5,000 m/s
Other characteristics of sound waves are their frequency, amplitude, and wavelength.

The frequency of a sound wave tells us its pitch

The amplitude of a sound wave determines the loudness

When sound hits a hard surface and bounces off, the reflection is called an echo
Electromagnetic Waves
These waves make up the ‘electromagnetic spectrum’ which consists of lots of different waves like visible light,
x-rays, radio waves, etc. Electromagnetic waves are created by accelerating charges, such as electrons
moving back and forth in a broadcast antenna to produce radio waves. Electromagnetic waves consist of an
oscillating electric field and an oscillating magnetic field at right angles to each other, and both fields are at
right angles to the velocity of the wave:
Since the fields “produce each other”, electromagnetic waves do not require a medium to travel through, so
they can propagate through the vacuum of space.
*All electromagnetic waves travel at the same speed in a vacuum: c = 3.0x108 m/s
(Nothing in the universe can travel faster than c.)
ELECTROMAGNETIC SPECTRUM
Gamma
X-rays
Ultraviolet
Visible
Microwaves
Short wavelength
High Frequency
High Energy
Infrared
Radio
Long wavelength
Low frequency
Low Energy
If we enlarge the visible light portion:
Violet
Indigo
Blue
Green
Short wavelength
High Frequency
High Energy
Yellow
Orange
Red
Long wavelength
Low frequency
Low Energy
“ROY G. BIV” is a mnemonic often used to remember the colors of the spectrum.
Electromagnetic waves at the high frequency end of the electromagnetic spectrum have enough energy to break
chemical bonds and therefore can damage living cells. This is why we limit our exposure to ultraviolet light
from the sun.
Some applications of electromagnetic waves:
Visible light ( = 400-700 nanometers) – sight!
Radio waves – communication
Microwaves – cooking (makes water molecules rotate through resonance)
Infrared waves – thermal vision (objects at ordinary temperatures emit electromagnetic waves in this
range)
Wave Phenomena
Superposition and Interference
The ‘principle of superposition’ states that when two waves overlap (interfere), the result is a wave
whose amplitude is equal to the sum of the amplitudes of both waves. This gives rise to what we call
constructive interference or destructive interference:
In constructive interference, the amplitudes of the waves reinforce each other:
Notice that the waves don’t bounce off each other, they pass through each other.
In destructive interference, the amplitudes of the waves cancel each other:
The picture above shows complete destructive interference. If A’s amplitude was a little bigger than B’s
amplitude, how would the picture in the middle be different?
Effects of Interference
Beats
When two sound waves with slightly different frequencies are played you can hear oscillations in
amplitude (loudness) of the resultant wave due to a changing phase difference:
Standing Waves
When two waves with the same amplitude and frequency are moving in opposite directions, a wave with
stationary crests and troughs, called a standing wave is formed.
Resonance
All objects have a ‘natural frequency’ they like to vibrate at which depends on the shape and material
the object is made from. (You can find the natural frequency of a wine glass by ‘twanging’ it). When
waves matching an object’s natural frequency are incident upon the object, the waves that get absorbed
end up interfering constructively with newly incident waves and standing waves are produced inside the
object. This is called resonance. Objects vibrating in resonance have a noticeably larger amplitude than
the original waves.
(Analogy: Pushing someone in a swing)
Examples of resonance: singer and wine glass; radio tuners; tuning forks; soldiers’ route step; wind
instruments
Wind instrument make use of the standing waves in an open or closed pipe:
Representation of Waves
In one dimension, waves are easy to represent by just drawing the wave with a single line:
In two dimensions the crests of the waves from lines (wave fronts). Think of a pebble dropping into a pond:
(In three dimensions the wave fronts would be spherical.)
When we get to optics, we will use arrows (called ‘rays’) to represent wave fronts:
Doppler Effect
You may have noticed that when a car passes by you on the street with its horn blowing you can hear a change
in the pitch of the car horn as it passes you. It has a higher pitch as it is approaching you and a lower pitch as it
moves away from you. The Doppler Effect is a phenomenon that occurs whenever a source of waves is in
motion relative to an observer:
*If the wave source is approaching the observer, the frequency of the waves in increased (and therefore
the wavelength is shorter)
For sound waves this means the pitch is higher, for light waves it means the color is more blue (blue-shifted)
*If the wave source is moving away from the observer, the frequency of the waves in decreased (and
therefore the wavelength is longer)
For sound waves this means the pitch is lower, for light waves it means the color is more red (red-shifted)
Applications: Police speed traps, Doppler radar for meteorology, Red-shift of starlight
Reflection
Reflection comes in two “flavors” – Regular and Diffuse:
Both types of reflection obey the Law of Reflection:
Angle of incidence = Angle of Reflection (i = r)
*The biggest mistake students make is that they
call the other two angles (the ones not labeled) i
and r. Don’t make this mistake! Remember
that all angles we deal with are measured FROM
THE NORMAL LINE. If a picture does not
have a normal line drawn, you need to add it!
Refraction
When waves enter a new medium at an angle they will be bent in a direction that depends on whether the wave
speeds up or slows down in the new medium. Waves that slow down bend toward the normal, and if the wave
speeds up it bends away from the normal:
Index of Refraction is a relative measure of how much light is slowed down in an object. The higher the index
of refraction, the slower the light travels:
n = index of refraction of the medium
v = wave speed in the medium
c = 3.0x10^8 m/s
n
c
v
n = 1 for a vacuum (or approximately 1 for air) and is higher in all other mediums
(Why can’t n be less than 1??)
Snell’s Law – Learn it, love it!
n1 sin 1  n2 sin  2
Critical Angle – The angle of incidence for which the angle of refraction is 90.
(Notice this can only occur if n2 < n1) If the
angle of incidence is greater than the critical angle, the ENTIRE wave is reflected off the boundary and no transmission occurs. This
is called total internal reflection and is the phenomenon that makes fiber optics possible.
Dispersion
You may (or may not) have noticed that the indices of refraction given in your reference table are for a specific
frequency, which happens to be yellow light. This is because the index of refraction for any medium has some
small variation which depends on which color of light is passing through it. We usually ignore this variation,
but this fact is the reason why white light exhibits dispersion in a prism. Dispersion refers to the separation of
colors when white light refracts through something (like a prism or raindrop). Since each color experiences a
slightly different index of refraction, each color refracts at a slightly different angle (notice that violet light
bends the most):
Rainbows are formed from dispersion of sunlight through water droplets in the air. This is why you only see
rainbows when it has been raining.
Diffraction
Diffraction is the spreading out of waves when passing through small openings or around sharp objects. The
smaller the opening is in relation to the size of the wavelength, the more diffraction (spreading out) that occurs.
For example, in this room you can hear someone’s footsteps walking down the hallway because the sound
waves bend around the corner of the doorway. However if there was light coming from the hallway it would
NOT bend around the corner. It would trace a straight pattern of the doorway on the floor. This is because the
wavelength of light is much too small to be diffracted through an opening as large as the doorway. (Sound
waves, on the other hand, have wavelengths that are anywhere from about 17 cm to about 17 m.)
If light is diffracting from two (or more) openings at the same time, an interference pattern forms. This pattern
consists of a series of bright and dark fringes. The bright fringes occur where light from the two sources is
interfering constructively, and the dark fringes are points where destructive interference is occurring:
Polarization
The transverse nature of light can be demonstrated with a “plane polarizing filter”. Light has vibrations
occurring in all directions perpendicular to the direction of the wave motion. When a polarizing filter is placed
in front of a beam of light, the filter removes all the light except those waves vibrating in the plane of
polarization of the filter. Longitudinal waves, such as sound waves, exhibit no such filtering. On the other
hand, light does which is evidence of the transverse nature of light and other electromagnetic waves. Polaroid
or polarizing sun glasses block or filter light by plane polarizing incoming light, thereby reducing its intensity.
It is worth mentioning that polarization is the only wave phenomena that does not apply to both transverse
and longitudinal waves.
Blue Sky and Red Sunsets
A clear cloudless day-time sky is blue because molecules in the air scatter (reflect) blue light from the sun more
than they scatter red light. This has to do with the shape and size of air molecules. When we look towards the
sun at sunset, we see red and orange colors because the blue light has been scattered out and away from the line
of sight. In the picture below, light reaching the observer on the left will be mostly blue, while the light
reaching the observer on the right will be mostly red (because most of the blue light has already been scattered.)