WAVES
Everyone has a basic idea of what a “wave” is. We see examples of waves throughout our lives.
We’ve all been to the ocean and have observed ocean waves. If you drop a rock into a pond,
you can observe the waves ripple out from where the rock hit the water. In Earth Science you
studied the types of earthquake waves, P -Waves and S-Waves. Most people know that sound
and light are other examples of waves. In this unit, you will study the properties of waves. In
doing so, you will be able to understand specific examples of waves, and will be able to explain
why they behave as they do. The most common examples of waves that we will study in detail
are sound and light.
Wave Definition
A wave can be described as a disturbance that “propagates” (propagation means to “move”)
from one location to another location and transmits energy, but not matter. Let’s break this
definition down to see what it means. First, all waves begin with some sort of disturbance. Look
at the picture below. What is the disturbance that is causing this wave to be made?
Answer:___________________________________________________________________
Let’s think about other types of waves. What causes earthquake waves? A disturbance in the
earth’s crust caused by the release of built up energy due to plate tectonics. Think about your
voice. When you speak, your vocal cords vibrate and disturb the air molecules in your throat.
That disturbance then propagates to another person’s ears. What makes a light wave? The
disturbance of a charged particle is what causes it. Remember from the Electrostatics Unit that
moving charged particles produce Electric and Magnetic Fields. Another way to say this is that
moving (disturbed) charged particles produce Electromagnetic Energy. Any type of wave you
can imagine always starts from a disturbance. Next, the definition states that energy is
transmitted, but not matter. Look again at the picture of the rippling wave above. If there was a
leaf near where the rock hit the water, what would the leaf do? Would it move away from the
source of the disturbance? No, it would simply bob up and down as the wave passed through it.
In other words, the water itself does not propagate, but only the energy of the wave. Let’s now
think about sound. When you talk, do the actual air molecules travel to another person’s ear?
No, only the energy propagates. Below is a picture from a very famous advertising campaign
from years ago for Maxell Tapes. The person is sitting in front of a speaker which is getting its
sound from a Maxell Tape. The company wanted to convey the idea that the person is getting
“blown away” by the sound produced by the Maxell Tapes.
Click on this link http://www.youtube.com/watch?v=B2WcBi9mu6A to see the actual Maxell
Tape commercial that this poster originates from. View it and then describe below the incorrect
physics shown in the commercial.
Answer:
The Regents Exam likes to test this concept by showing a picture of a wave and then asking the
student what would a point on the wave be doing in the next instant. Here is an example:
The answer is choice (A). The first part of the wave to pass through is the trough, which is a
downward movement. Then, the crest will cause an upward movement. Note that there is no
left/right movement in a transverse wave. The key is to notice if a crest or a trough is about to
move through that point on the rope. To see this in action, click on the following link, which
simulates the motion of waves in a rope https://phet.colorado.edu/sims/wave-on-astring/wave-on-a-string_en.html . Set the damping to zero, and the tension to half way
between low and high. Send pulses down the rope, and notice how the particles vibrate as the
energy passes through them.
Finally, there are two vocabulary words that the Regents Curriculum wants you to know. If a
wave is produced by a singular disturbance (like the rock falling into the water), it is known as a
wave “pulse”. If the disturbance is repeated at a regular interval, it is known as a “periodic”
wave.
Wave Vocabulary
We will now review some words associated with Waves. Examine the following diagram to help
understand the four vocabulary words.
Crest= the high point in the wave
Trough= the low point in the wave
Amplitude (A)= the distance from equilibrium to the crest or to the trough (not from crest to
trough!)
Wavelength ( is the Greek letter Lambda)= the distance through one cycle of a wave.
Now, examine the following diagram to help understand two time related wave properties:
Period (T) = the time it takes a wave to complete one cycle (Units= Cycles/Sec – but, Cycles is
just a descriptive term, so the actual unit is 1/sec (or sec-1) which is defined as a Hertz (Hz)).
Frequency (f) = the number of cycles a wave completes each second (Units= Sec/Cycle = Sec).
From these last two quantities, we can see that Period [sec/cycle] and Frequency [cycles/sec]
are inverses of each other. This leads us to the first formula on the Waves section of the
Reference Table: T= 1/f. Therefore, if you know one of these, the other can be easily found.
Practice Problem: The Period of a wave is 5 seconds. What is its frequency?
Answer here:
How do these Characteristics Apply to Light (EM Radiation) and Sound?
Sound: Frequency: Humans perceive different frequencies of sound waves as different
pitches. Very fast vibrations of air molecules are described by humans as having high pitches.
Amplitude: Humans perceive different amplitudes of sound waves as the loudness of
the sound. In loud sounds, the air molecules vibrate a large distance from equilibrium.
Light:
Frequency: For visible light, humans perceive different frequencies as different colors.
Red light is the lowest frequency human eyes can perceive. Violet is the highest frequency
human eyes can perceive. Outside the visible spectrum, different frequencies correspond to
different types of EM waves (X-rays versus Radio Waves, for example).
Amplitude: For visible light, humans perceive different amplitudes by the brightness
of the light. Brighter light waves have greater amplitudes than dimmer light waves. For EM
waves outside the visible spectrum, amplitude corresponds to the energy of the wave.
Ways to Classify Waves
Scientists often classify phenomena in order to better understand them as well as to
communicate important similarities of whatever they are studying. Waves are classified into
two important categories. Let’s look at both of them:
Category #1: Mechanical or Electromagnetic (classified by whether they need a medium)
Mechanical: A “mechanical” wave is any wave that needs a material “medium” through
which to propagate. An example of a mechanical wave is sound. It cannot propagate through
empty space. It needs a solid, liquid, or a gas through which to propagate. Humans normally use
a gas (our atmosphere) to transmit sound waves. However, sound actually travels faster in
solids and liquids than through air. Other examples of mechanical waves are earthquake waves
and waves through a slinky. An interesting example of this concept was used in the 1979 movie
“Alien”. It was a horror movie set in space. Click on the following link and watch the trailer
http://www.youtube.com/watch?v=LjLamj-b0I8 . At the end is the famous marketing line from
the movie. Write it below, and explain how it relates to mechanical waves.
Marketing Slogan:
Physics Meaning:
Here is another video clip that demonstrates that sound is a mechanical wave. View it and then
summarize what you see in terms of sound being a mechanical wave.
http://www.youtube.com/watch?v=ce7AMJdq0Gw .
Electromagnetic: An “electromagnetic” wave does not need a material medium through
which to propagate. An electromagnetic wave is any wave on the electromagnetic spectrum.
Below is the EM spectrum as it appears in your Reference Tables:
So, for instance, visible light can travel from the sun to the earth through the vastness of empty
space. Anything on the electromagnetic spectrum can also travel through material mediums
(light can travel through air and water, for example), but they don’t need them to propagate.
Let’s analyze the EM chart above in a little more detail by answering some questions:
Question1: What makes an X-ray different than a microwave?
Answer 1: This can be answered by saying they have different frequencies or different
wavelengths. A typical x-ray might have a wavelength of 10-9 m and a frequency of 1017 Hz. A
typical microwave might have a wavelength of 10-2 m and a frequency of 1010 Hz. In other
words, microwaves are EM waves that are lower frequency and longer wavelength than x-rays.
Question 2: Does visible light make up a large portion of the EM spectrum?
Answer: No! As you can see, it makes up a very small portion. So small, in fact, that the
Reference table had to magnify the different frequencies of visible light in order to analyze
them. Think about this crazy idea: Do humans “see” most of the world around us? Not at all!
Question 3: What makes red light a different color than green light?
Answer: Different colors are different frequencies. For example, the chart shows us that
scientists define violet light as electromagnetic radiation that is between the frequencies of
7.69 x 1014 Hz and 6.59 x 10 14 Hz. Orange light is a lower set of frequencies- those between
5.03 x 1014 Hz and 4.82 x 1014 Hz.
Question 4: What are Ultraviolet (UV) waves?
Answer: Ultraviolet waves are EM radiation that is just higher a frequency than our human eyes
can detect. It is responsible for sun tans/burns. What if humans could “see” UV light? One
weird consequence would be that humans wouldn’t be able to see through glass because UV
light cannot pass through glass!
Question 4: What are Infrared (IR) waves?
Answer: They are EM waves that are of a frequency that is just below what human eyes can
detect. They are basically heat. Night vision goggles take advantage of this in order to “see’ at
night. Visible light is gone, but objects still radiate heat. Some nocturnal animals can detect IR
radiation.
Category #2: Transverse or Longitudinal (classified by the direction the wave vibrates)
Transverse: A “transverse” wave contains oscillations that are perpendicular to the
direction of propagation of the wave itself. A transverse wave looks like a sine wave from math.
Below is an illustration of a transverse wave:
All Electromagnetic Waves are transverse.
Longitudinal: A “longitudinal” wave contains oscillations that are parallel to the
direction of propagation of the wave itself. Longitudinal waves are much trickier to draw, so
examine the diagram below. It shows how the particles of a longitudinal wave would move as
compared to a transverse wave. Sound is an important example of a longitudinal wave.
The Speed of a Wave
Just like a car traveling down a road, waves need a certain amount of time to travel from one
point to another. The speed of some waves are well known and can be found on the front of
your Reference Tables. Find them and write them below:
Speed of Light in a Vacuum: _________ m/sec
Speed of Sound in Air at STP:
__________m/sec
Many times, the speed of a wave is not know and must be calculated. There are two ways to
calculate the speed of a wave:
Technique #1:
v=d/t
Just like a car traveling down a road, if the distance a wave travels and the time it takes to
travel are known, the speed can be calculated. In real world situations involving sound, this
technique is used with an echo. Look at the following problem:
A student is standing 400m from a tall cliff. He has a starter’s pistol used in track meets, and
fires a blank from the pistol. He hears the echo in 2.3 sec. Calculate the speed of the pistol’s
sound wave.
Solution: First, we can’t just look up the answer in the Reference Tables (331 m/sec) because
that speed is only valid at STP (Standard Temperature and Pressure). So, we use v=d/t. Watch
out….. the distance traveled in an echo is down and back….. therefore: v=d/t=800m/2.3 sec =
348 m/sec.
Technique #2: Use wave characteriistics
In your Reference Tables in the Waves section is the formula v=f. Let’s show ourselves that
this is a valid way to calculate the speed of a wave by dooing unit analysis:
V=f = [Hz][m] = [cycles/sec][m ]= [1/sec][m] = [m/sec]
Let’s do a simple example: What is the speed of a the sound wave produced by a tuning fork
whose frequency is 440Hz and whose wavelength is 0.79m?
Write out answer here: Formula:
Substitution with Units:
Answer with Units:
Now look at this example: Calculate the wavelength of the FM radio station 104.1 Wild 104.
This is a bit more difficult because we have try to figure out how to determine the given
variables. As it turns out, the 104.1 stands for 104.4MHz, which is 104.1x10 6 Hz. The speed of a
radio wave is the same as the speed of any other frequency of EM radiation, namely 3x10 8
m/sec (within a given medium, the speed of a wave cannot be changed- all frequencies and
amplitudes travel at the same speed- we will learn more about this soon). Now, we can use our
new formula.
v=f= (104.1x106Hz)()= 3x108 m/sec so, = (3x108 m/sec) / (104.1 x 106 Hz)= 2.9 m.
Now, try one on your own:
Calculate the wavelength of the AM radio wave 1430 “The Team”. In AM, 1430 stands for 1430
kHz (not MHz).
Formula:
Substitution with Units:
Answer with Units:
Can the Speed of a Wave be Changed Within a Given Medium?
Think of 2 people singing a song. One person sounds like Darth Vader (a very low voice). The
other person sounds like Mickey Mouse (a very high voice). In other words, the two voices have
different frequencies. Does one voice reach you before the other? What about in a band, where
there are very high frequency and very low frequency instruments. Do they reach the audience
at different times? In both cases, they don’t! Can you see that this shows us that frequency and
amplitude do not affect the speed of a sound wave? What about light? Does blue light travel
faster than red light? Do bright lights travel faster than dim lights? No! This is why all waves on
the EM spectrum travel at the same speed- 3x108 m/sec. From just these observations, it
should be apparent that the speed of a wave with a given medium cannot be changed!
Is it Possible to Change the Speed of a Wave By Some Other Way?
Yes- if the medium is changed. When sound travels from light into water, its speed goes up
dramatically. It is almost five time faster in water than in air. When light travels from air into
glass, it slows down. In a slightly more subtle example, hot air is considered a different medium
than colder air. That is why the speed of sound in the Reference Tables is given at 331 m/sec at
STP (Standard temperature is 0C). The speed of sound goes up by 0.6 m/sec for every degree
above zero, so the speed of sound at 10C is 331 m/sec + 6 m/sec = 337 m/sec. The following
link has been previously seen. It is the animation of waves on a string. Go to this link, set the
damping to zero, and the tension to about 1/3 of maximum. Send a pulse down the rope. Then,
increase the tension in the rope (which changes the medium) to the maximim and send a pulse
down the rope. Describe what you see. https://phet.colorado.edu/sims/wave-on-astring/wave-on-a-string_en.html .
Phase of waves
The phase of a waves relates to where the waves are compared to each other within their
cycles. If two waves are always in “sync” with each other, they are said to have a zero degree
phase difference (also known as a phase shift). See the waves on the left below. If two waves
are at exactly opposite points within their cycles, they are said to have a 180 degree phase
difference (shift) because a crest and a trough are ½ of a cycle, or 180 degrees apart in the
wave cycle. See the waves on the right below.
These are the two extremes of phase. Waves can have any phase difference from 0 degrees to
360 degrees. For example, a 90 phase difference would put the waves at ¼ of a cycle difference.
One more ideas about phase. Points within a a wave itself can be described with a relative
phase. For example, a point at the crest of a wave can be thought of as 180 degrees out of
phase with the adjacent trough of the wave.
Interference of Waves
When two waves traveling in the opposite directions meet, they pass through each other and
eventually continue on their way. However, as they pass through each other, they will
temporarily influence each other and make a new wave. This new wave is simply the sum (or
difference) of the individual displacements of each wave. The name for how they influence
each other is called interference, and the name for the technique used to find the wave made
by the individual waves is called the Principle of Superposition. Let’s examine these ideas more
closely.

Interference: There are two types of interference.
Type 1: Constructive Interference: When the displacements of the individual waves are
on the same side of equilibrium, they build each other up and make a larger wave that is the
sum of thedisplacements of the individual waves. This is shown below.
The following link shows two wave pulses meeting and undergoing constructive interference.
Click on it and watch. http://www.youtube.com/watch?v=YviTr5tH8jw&feature=related
Type 2: Destructive Interference: When the displacements of the individual waves are on
opposite sides of equiibrium, they work against each other and make a smallerwave that is the
difference of the displacements of the original waves. This is shown below
The type of interference shown is called Total Destructive Interference because the waves are
identical and totally destroy each other. If one wave was bigger than the other, they would
undergo Partial Destructive Interference. For example of one wave had an amplitude of +5m
and the other had an amplitude of -2m, the resulting wave would have an amplitude of
5m-2m=3m. How would this apply to sound and light? Can this phenomenon happer to them.
Believe it or not, yes! You have probably heard of Noise Canceling Headphones. They work by
using a microphone to “listen” and analyze the incoming sound, and then producing an
identical wave which is the opposite of the incoming wave (180 degrees out of phase). This
produces destructive interference, and the incoming sound is “canceled out”. Check out this
website (http://www.youtube.com/watch?v=VTx4JgYsW5s ) Light can be canceled out as well,
although there are not many common examples of this. You will see it done, however, by a
demonstration within this unit. Click on the following link to see destructive interference in two
slinky pulses. See if you can get the video to pause exactly when the pulse are canceleing each
other out. If you do, you should see a very straight slinky. You can’t tell that pulses are passing
through each other. http://www.youtube.com/watch?v=IU8xeJlJ0mk&NR=1
Standing Waves
An interesting interference effect occurs when two identical waves pass through each other.
When this happens, an unchanging interference pattern emerges known as a Standing Wave.
The word “standing” refers to the unchanging nature of the interference pattern. What are
some practical ways to produse standing waves? One way is to send a periodic down a string
that is tied off at both ends. The wave reflects off the opposite end of the string and travels
back to its starting point. While traveling back, it will meet (and interfere) with pulses traveling
up the string. Since they are from the same source, they are identical, and can thus make a
standing wave. Interestingly, only certain wavelengths (frequencies) will produce standing
waves. Watch the video from this link ( http://www.youtube.com/watch?v=j_z88-vvz48 ) , and
answer the following questions:
Q1: a) Draw a picture of the first standing wave or the “fundamental mode”.
b) What fraction of a wave is visible in the fundamental mode?
Q2: a) Draw a picture of the second harmonic.
b) What is a “node” in a standing wave?
c) What is an “anti-node” in a standing wave?
d) How much of a wave is represented in the second harmonic?
Q3: Draw the “third harmonic” and label all nodes and anti-nodes. The video has a slight error.
For a string in this situation, each endpoint is also a node. Be sure to label them.
Why doesn’t the string move at the nodes? Because as the waves pass through each other, they
always produce destructive interference at these points. Why does the string move like crazy at
the anti-nodes? Because the interfering waves undergo constructive interference at these
points.
Q4) What fraction of a wavelength exists from one node to the next?
Answer:
Harmonics are a foundational principle for music. When you observed the vibrating string in the
previous video, only one mode was present at a time. However, when a real stringed
instrument is played, multiple harmonics are present in the sound of the instrument. In fact,
the unique mix of harmonics of any instrument ultimately gives it its distinct sound. This is also
true for wind instruments. This is known as the “timbre” of the sound. This is why, if a flute and
a clarinet both play middle C, they will have different tone qualities. Even non-musicians, can
tell with their eyes closed that a flute and a clarinet produce different tones.
Harmonics can also be produced in light waves. LASERS produce standing harmonic waves of
light. LASERS are identified by the wavelength (harmonic) of light they produce. Because of this,
the light emerging from a LASER is both Monochromatic (one frequency) and Coherent (all the
light waves are in phase). This is in contrast to a flashlight, which produces Polychromatic,
multiple wavelength (otherwise known as white) light that is Incoherent (no constant phase
relationship). The drawing below illustrates these two properties of LASER light.
LASER
Resonance
All objects have a “natural” frequency of vibration (we will discuss what “natural” means in the
next section). If energy is added at the objects natural frequency of vibration, that energy will
be reinforced, and will build up within that object. If the energy happens to be sound, that
sound’s amplitude will build up. Click on the following links to observe some common
examples of resonance.