Waves are vibrations

Steps to a vibration:

  • Disturbance caused by a force
  • Restoring force brings it back to its original position
  • Inertia keeps it moving past its original position
  • Friction slows it and stops it- dampening.

Waves transfer energy only (not the medium)
Pulses- single disturbance
Periodic waves- wave train
Propagation- the direction of wave travel.


As the wave above moves, what will happen to point A?
Point B?
Point C?

A will go down and then up.
B will go up and then down.
C will go down.

It is not the particle that moves over the wave but it is the wave that travels under the particle.

Types of wave motion:

  • Direction of vibration is parallel to the direction of propagation.
  • Vibration goes the LONG way LONGitudinal- get it?
  • Alternating compression and rarefaction or high pressure and low pressure.
  • A wavelength is the distance from one compression to another.

Ex: Sound, earthquake "P" wave

Speed Links

Characteristics of Waves
The Wave Equation

Standing Waves
Doppler Effect
Electromagnetic Spectrum
Index of Refraction


Compatability Pack for converting docx files to doc files

Note: Sound waves are longitudinal as shown above but are often shown as transverse. Here’s why:

  • Transverse waves are very difficult to draw
  • It is difficult to see where one longitudinal wave ends and the next begins
  • Amplitude of a longitudinal wave is not obvious (it is the contrast between how compressed the compressional areas are with how rarefied the rarefied parts are)

For these reasons sound is often showed as the oscilloscope representation which shows it as a sinesoidal (transvberse) wave.



  • Direction of vibration is perpendicular to the direction that the wave moves.
  • Vibration is perpendicular to propagation.
  • Trans- means “across”. The vibration is “across” the wave.

Ex: Water, light, earthquake "S" wave

Characteristics of waves

Wavelength (λ): the distance for one complete vibration. Once you go past one wavelength the pattern starts to repeat.
Amplitude (A): the height of the wave above (or below) the rest position. It is related to the energy of the wave. For example, a louder sound will have a greater amplitude. For a transverse wave it will be the height of the wave from its rest position.
Crest: high point of a wave
Trough: low point of a wave
Propagation: the traveling movement of the wave

Phase: The position and direction of a point of a wave.
Since many waves are sine waves, they can be described using the degrees of a circle.
Using A (below) as a starting point:
F is 360˚ away from A: it is “in phase”
C is 180˚ away from A: it is “out of phase”
B is 90˚ out of phase from A


1. Which pairs of points are in phase with each other?
2. Which pairs of points are 180° out of phase of each other?
3. Which points are 90 ° out of phase of E?
4. Which points are 180 ° out of phase of C?

1: BF, DH, CG ; 2: BD, DF, FH, CE, EG ; 3: D & F ; 4:B & D

Frequency (f):
“how many waves per second”
cycles/second or vibrations per second (Hz)
Hz = “waves per second” (1/sec)
For sound frequency is the pitch

Period (T):
how long it takes for one wave

f and Hz are inverses of each other.
What does that mean!?

Frequency and period are said to be “inverses of each other”.

You’ll often get it delivered this way: “Frequency and Period are inverses of each other”. And that would be that. No explanation. No clarification. Not even an indication of why it is important that they are inverses of each other.

This means that as one goes up, the other goes down. Another way of thinking of it would be that as the frequency goes down it takes longer for each wave (less waves per second means more time for each wave).

If you look at the units for frequency (Hz means “cycles per second”) it also helps to clear it up a bit:
If you flip cycles per second (f) it becomes seconds for each cycle (the period).

Probably the best way to keep these relationships clear is to always pronounce Hz as “cycles per second” and period as “how long it takes for one wave.”

1. Which wave has the largest λ?
2. Which wave has the highest f?
3. Which wave has the highest amplitude?
4. Which waves have the same λ?

1:B ; 2:D ; 3:A ; 4:A&C

Since waves are usually shown as sine waves, the phase of a wave is often described in terms of degrees of a circle.

  • If you start the cycle at A, one complete cycle will be 360°. Half a cycle will be 180° and a quarter will be 90°.
  • If two points are 360° away from each other (A & F; B & G; E & H)) they are in the same part of the cycle and are said to be “in phase”.
  • Any other point will be “out of phase”.
  • A and C are 180° out of phase (opposite phases)
  • A and B are 90° out of phase.


The Wave Equation

The “wave equation” gives us the relationship between speed, frequency and wavelength:

v = fλ

(Which I think is one of the prettier equations!) If the velocity of a wave is to stay constant, then the frequency must go up as the wavelength goes down- and vice versa.

For example: Light in a vacuum must always be the same speed (c)

c = 3.00 x 108m/s

If you increase the frequency of the light, say from red light to blue light, then the wavelength must go down.

Since f is also equal to 1/T then the equation can be written as

v = λ/T

(Not as pretty)

1. What is the frequency of a wave if 4.0 waves pass a fixed point in 10 seconds?
2. What is its period?
3. What will happen to the period of a wave if the frequency is doubled?
4. What is the frequency of a wave if its period is 0.25 second?

1: 4/10 or .4Hz ; 2: 1/.4 or 2.5sec ; 3: halved ; 4: 1/.25 or 4Hz




Material through which a wave passes.

Mechanical waves such as sound and water waves need a medium.
Electromagnetic waves (light) do not need a medium although they CAN travel through media.
Ex.: Light can travel through glass and water.



Resonance: sympathetic vibration
The vibration of a body at its natural frequency caused by a vibrating source at the same frequency.
Another interpretation of this could be to give a series of well-timed pushes to get something going. You will be pushing at the natural frequency of the object.

Examples of resonance:

  • Pushing someone on a swing.
  • That annoying guy next to you at the red light with the speakers so loud you can feel it with your windows. His rattling fenders are resonating from the music.
  • Rocking a stuck car to get it out of the snow.
  • Something vibrating in your car only when you are at a certain speed.
  • Water molecules when microwaves them.
  • Swirling a cup of liquid to get it to spin.
  • The classic opera-star-hits-the-high-note-and-shatters-the-wine-glass effect. That is, of course, if she can somehow find the exact unique natural frequency of that specific glass.


The effect of two or more waves passing simultaneously through the same region of a medium
Superposition is where two waves are in the same place at the same time.
Two waves in superposition will interfere with each other.

Constructive- two waves that are "in phase" at the same place and same time will add energy to each other making the wave stronger. (Adding +2 and +2)

Destructive- two waves that are "out of phase" at the same place and same time will take away energy from each other making the wave weaker. (Adding +2 and -2)

Bose ™ noise cancelling headphones. The headphones have a microphone on the outside to pick up outside noise- specifically the steady drone of machines and aircraft engines for example. The sound waves are then reproduced but in the opposite phase to cancel out the noise.

Similar to the Bose™ noise cancelling headphones but much cooler! Microphones mounted on the outside of a helicopter pick up the engine and rotor noise. Speakers on the outside of the craft produce the exact sound but, again, 180 º out of phase to cancel out the noise making it almost silent.

(This was used on the SEAL Team Six raid into Osama Bin Laden’s stronghold on May 2, 2011.)



A beat is an alternating loud and soft sound. It comes from two waves of slightly different frequencies interfering with each other. As the two waves interact, eventually they will sync up constructively then destructively. The mixed sound gets louder as they constructively interfere and get quiet when they are out of phase.

A cycle of loud/quiet is called a beat.
If you have a tone of 250Hz and the other is 254Hz, you will get 4 beats per second which is the difference between the two frequencies. It will sound like “wow-wow wow-wow” (if you say that in 1 second).

Side note: When a musician tunes an instrument, she does not simply adjust the pitch based on some memory of what the sound sounded like. He will make a tone of a known frequency, say a middle-C for example (256Hz) using a tuning fork or synthesizer. She will then listen for beats if the instrument is out-of-tune. As he gets the tuning closer and closer, the beats will get slower and slower until they are the same.
(I switched the gender of the tuner as not to offend anyone who is overly-sensitive over such trivial things as this. I figured it was the best way without referring to a single human (or is it huperson?)as “it” or “they” or “hir” which my spell checker doesn’t recognize.)


Standing Waves

Any pattern of wave crests or troughs that remains stationary in a medium. They form when two waves with the same f and amplitude travel in opposite directions and interfere both constructively and destructively. The waves don’t actually stay in place. They move up and down (or in circles) but they pivot around the nodes which do not move at all.

A common example of seeing standing waves would be to hold a rope or chain and letting it hang. If you spin with just the right speed, you get two loops shaped like a figure-8 with a spot in the middle that stays the same.

Although it is often shown with standing waves going from a single to a double to a triple, etc. I think it makes it easier to see the progression if we start with a double.

Pictured above is a string that is being vibrated and attached on both ends. It is being vibrated at just the right frequency so that one complete wave fits between the two ends.

Above, we see that the length of the string (l) is equal to the wavelength ( λ):

l = λ (l = 2/2λ)

But, I refer to it as 2/2λ so that it fits into the pattern that will be established below.

One way to describe standing waves is to count how many waves fit between the two ends. In this picture above, only half of the wave’s length fits between the ends but it is still stable. Standing waves will only work if you can fit half a wave, or multiples of a half wave. For that reason, the length of the string is usually shown in relation to the half waves.

1/2 of the wave is equal to the length of the string.
l = 1/2λ

Now the pattern becomes a little clearer. The length of string can now support three half waves. Rather than call it 1-1/2λ it makes the pattern easier to handle if we just call it 3/2 λ.

The pattern now lets us count it as:

1/2, 2/2, 3/2, 4/2, 5/2 and so on. We can take care of the improper fractions later if need be.

The fundamental frequency is the lowest frequency that an instrument can play.
It will be determined by the longest standing wave that will fit, from node-to-node, within the instrument.

To change the frequency of wave made by a wind instrument, the wavelength of the fundamental tone is shortened by making openings in the tube. This is done by uncovering holes closer to the mouthpiece.
Similarly, in stringed instruments, higher frequencies are made by “shortening” the string by clamping a finger closer to where the string is played.


Doppler Effect

... the apparent change in frequency due to the relative velocity between the observer and the source.
As the source and observer approach the frequency gets higher.
As the source and observer move away the frequency gets lower.
Faster movement means more shifting.

When sound is made by a stationary object, you hear a certain frequency and it would look like this...

Wave front diagram.
Lines = wave fronts
Spaces = troughs

In front of the source (to the right) you will hear a higher pitch.

Behind the source (to the left), you will hear a lower pitch.

If you are at the bottom or top of the source, you will hear the sound slide from high to low.

Bad boy, bad boy, what-chu gonna do?

Police use the same principle when they use radar guns. The gun sends out a pulse of radio waves. It hits your car and returns to the gun. If the radar signal returns with a higher f, you are moving towards the cop. The bigger the shift, the faster you are going. The same is true when you are driving away: the lower the pitch, the faster you are driving away.

Note: the guns work best when you are driving directly at or away from the gun. The police cannot get an accurate speed if you are directly next to them. The closer you get to them, the less your speed will be towards them (it’s that whole vector, right triangle thing). Your clocked speed will become less accurate. Unfortunately, if you are clocked at 55mph at an extreme angle, it means that you are doing much more than 55. The only way you can make this work if if you introduce reasonable doubt due to the inaccuracy. However, to do so would mean that you admit that you were going faster than recorded.


Examples of the Doppler Shift

When the object moves at the speed of sound (below), the waves overlap and you get a sonic boom.
When the plane is traveling at the speed of sound it is not a good speed to be flying. The shock waves and pressure are right where you are. There are extreme vibrations that can damage the craft…and you don’t want to damage a craft that you are in moving at the speed of sound!


Fly faster than sound (below), and you’re gone before the boom is heard.

As you travel faster, the shape of the cone gets narrower. The sonic boom is along the front edge of the cone and follows the plane (assuming it is a plane). Observers on the ground will hear one boom but that boom will travel with the plane.

This does not have to be a plane flying supersonically. It can also be a boat traveling faster than the water can carry waves. When a boat is traveling at the speed of water waves, it is constantly fighting against the pile of water just in front of the boat. Go faster than the speed of the waves and you are in front of the disturbance and the boat cruises along nicely.


An online visualization:

Doppler Lab

Blank worksheets for the Doppler Lab

vW = 2cm/sec
vB = 0cm/sec, 1cm/sec, 2cm/sec, 3cm/sec
On paper, each center point is separated by the vB

Time (sec)
Radius (cm)

Doppler Lab Write-up:

  • Procedure is how you made the diagrams
  • Data is measurements (given to you)
  • Analysis is explaining your drawings
  • Measure the average wavelength in front and behind
    Calculate frequency:
    v = fλ

vW = 2cm/sec = f•(calculated wavelength)
What happens to frequency in front and behind?

What the drawings mean.
Any highlights for each situation?

Sonic Booms:

Twin Booms from the Shuttle (go to 1:30 for the double BOOM!)


Sonic Booms Compilation


This page is still currently coming together. Below this line most of the information is here but it still needs to be made all pretty-like.



“c” is the speed of light
3.0 x 108 m/sec
Light can only move at one speed (in a vacuum)

The Electromagnetic Spectrum

The Visible Spectrum: White light splits into the colors of the rainbow when passed through a prism.


EM Spectrum


Reflection Lab (To test if the angle of reflection is really equal to the angle of incidence)

Draw a line at half paper
Draw a random arrow at least 4 cm from the center line
Label each corner of the arrow 1-4
Put mirror into a mirror stand and tape it down to the paper along the center line
Place one pin into point one
Close one eye, look along a ruler from any angle at the reflection of the pin (not the real pin!)
Draw a line along the ruler pointing to the image of the pin- the line will not touch the pin but will be stopped by the mirror from reaching the pin’s image.
Label that line #1A
Change your viewing angle and repeat to get a second line for point 1: #1B
Repeat for all other points.
Remove mirror
Extend lines
Follow the paired lines and label where they meet
Draw your image

Part 2 of the lab:
Pick only one line- for example line 1B.
Draw a line from the pinpoint of 1B to the spot where line 1B intersected the mirror.
Measure the angle between the Normal and each of the two lines.
Pick only one line- for example line 1B.
Draw a line from the pinpoint of 1B to the spot where line 1B intersected the mirror.
Measure the angle between the Normal and each of the two lines.

Measure the distances between point 3 and the mirror and the reflected point 3 and the mirror

Questions to address with this lab:

In the step where you had to measure the angle of incidence and angle of reflection (angles from the Normal), which line was the incident ray and which was the reflected:
the one going from the needle into the mirror
the one going from the mirror into your eye
How did the angle of incidence compare to the angle of reflection?
How did the distance between the real point and the mirror compare to the distance between the image and the mirror?
How did the orientation of the image compare to the original:
Up and down?
Left and right?


Natural light is usually “oriented” in all directions: the way that the EM fields vibrate.
Polarized glasses will screen out all light except the rays that are oriented the same as the glass.
If you take two polarized glasses and cross them. It will block out all of the light.

Polarization will only work with transverse waves.

Waves bend around barriers and through small openings. The openings in the barriers must be about the same size as the wavelength of the wave


The bending of light when it enters a new medium.

When a wave enters a new medium, it will travel at a new speed. In order to keep the same frequency, the wavelength must change.
If it goes from a fast medium to a slow one (top to bottom), the wave will slow, wavelength will get smaller.
If it goes from slow to fast, the waves will spread out as the go faster.
Like soldiers marching through mud and pavement.
If the wave hits at an angle, one side of the wave will travel at the new speed which will cause the wave to turn.
The frequency will remain the same which means that the wavelength must change.
If the wave goes from a fast medium to a slow one, the wave will bend towards the normal.
If it goes from slow to fast, it will bend away.


Frequency stays the same when it changes media.
“Red in air is red under water.”
v = fλ if f stay the same λ must increase when v increases.

Absolute Index of Refraction (n)
n = c/v
Which is a ratio of how fast light in a vacuum is compared to the new medium.
(A way of measuring how much the light will slow down)
c = 3.0 x 108m/sec
What is n for air?
n = 3.0 x 108m/sec / 3.0 x 108m/sec = 1

Measured at f = 5.09 x 1014Hz (yellow)

To find v of light in water:

n = c/v
1.33 = (3.0 x 108m/sec)/?
v = 2.25 x 108m/sec

A light ray coming from the top through the air striking the surface at an angle is called the “incident ray”. It strikes the surface at θi or the “angle of incidence”.


Inside the new medium, in this case water, the ray will refract (bend) in a new direction. The new angle is the “angle of refraction” or θr.




White light has all the colors of the rainbow-
Polychromatic light
Monochromatic light has only one color (single frequency)

Different wavelengths bend to different degrees.

Baby Blue Bends Best!


“Light Amplification by Stimulated Emission of Radiation”
Coherent light
light waves are all the same frequency and in phase.
Constructive interference amplifies the intensity of the light.



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