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Simple harmonic motion

Interference and superposition

Waves in 2 dimensions, and refraction

What is sound? How do we hear it?

Actually see the speed of sound at a Queen concert

Sources of sound: String instruments, harmonics, wind instruments, quality of sound

Doppler Effect

Sonar, echolocation, and ultrasound

Why pianos are never in tune: Math and Physics.

MCAS Physics exam: sample wave problems


What are waves?

a repeated disturbance that spreads out, and transfers energy as it moves forwards.

How long is a wave? No set answer. A wave can go on for miles, or even infinitely.

But we can ask “What is the distance between successive crests?” That is a wavelength.



Wave properties

A (amplitude) = distance from the center line to the top, crest (or bottom, trough.)
Be careful: the center line is not always drawn.

H (wave height) = Twice the amplitude.
Distance from the top of the crest to the bottom of the trough.

λ (wavelength) is the distance between successive crests
The symbol is the Greek letter lambda, λ

Trough = lowest point of the wave

Crest    = highest point of the wave.


waves have period, frequency and velocity

T = period = time for one complete wave to pass by a standing object.

T = time for 2 crests to pass by a standing object.

f= frequency = number of times a peak passes/second.

f = 1 / T

v = wave velocity = frequency x wavelength

                               v = f • λ

Here, T = time for two crests to pass the center line.

Metric units of the wave equation

wavelength – measured in meters (m)

speed – measured in meters / second = m/s

frequency = waves/second = 1 / second = 1 / s

The honorary unit for frequency is the hertz (Hz)

                    1/s = 1 Hz

Heinrich Rudolf Hertz
Heinrich Rudolf Hertz

Watt is love? Baby Don’t Hertz Me, No Morse 😉

Oh… My head Hertz from the Frequency of These puns…


waves carry energy

Energy is just a property of a system that enables change to occur.
It has no independent existence.
If a wave can push something, or heat something, we it has energy.

This wave applies a force to a buoy, moving the buoy a distance.
So the wave does work on the buoy.  w =f·d
and therefore it has energy.

Here an EM (electromagnetic) wave heats food.
EM waves that we can see are called visible light.
EM waves that we can’t see, but can heat food, we call microwaves.
(Microwaves, therefore, are a color of light!)

Energy from the EM waves is absorbed into the food molecules.
This makes the food molecules vibrate faster.
So EM wave energy -> molecular vibrational energy.

Waves have energy

The bigger the amplitude of the wave, the more energy it carries.

Wave energy is proportional to the amplitude squared

E ≈ A2

If we tripled the wave’s amplitude, what would happen to the energy?

E ≈ (3A) 2 = 9A2

If we cut the amplitude in half, what would happen to the energy?

E ≈ (1/2·A) 2 = 1/4·A2 = A2 /4

2 types of waves: mechanical and light

Mechanical waves need a material (medium) to be transmitted.
Can transmit through solids, liquids and gases

Here we see waves of dirt and rock, in an Earthquake.

Tarbuck & Lutgens

Tarbuck & Lutgens

Electromagnetic waves

Light waves are made of EM (electromagnetic) radiation.
They do not need a classical medium to be transmitted.
They can move through vacuum (empty space)

Here we see light propagating through vacuum.

In the real world one can’t actually see an EM wave from the side.
We can only see light if it hits our eyes.

EM wave from Giancoli Physics

as the wave moves through space,
the E (electric) field increases and decreases,
the B (magnetic) field increases and decreases.


Colors and wavelengths

EM waves can be of many different wavelengths.
Longer wavelengths we perceive as orange and red
Shorter wavelengths are towards the blue end of the spectrum

Our eyes can only see a tiny amount of the EM spectrum.
There are longer and shorter waves as well.

Gamma rays Spectrum Properties NASA

Waves can be  reflected, absorbed, or transmitted

Waves can be reflected from a surface
Here, all of the energy bounces back
None is absorbed into the wall

Waves can be absorbed.
Here, some of the light is reflected (bounces back)
Yet some if absorbed: it goes into material, but not through.
This absorbed EM wave gets turned into heat.

And some of the wave is transmitted (goes through to the other side)

More details on Dan Russell’s tutorial on how waves reflect, when they hit a boundary.

Wave interference

What happens when 2 waves collide with each other? We get superposition, interference, standing waves,  nodes and antinodes. See these resources.

Interference-and-superposition (Kaiserscience)

Dan Russell’s excellent tutorial on wave superposition

more examples of waves interacting with each other.

Longitudinal and Transverse waves


From Dan Russell

While teaching undergraduate physics at Kettering University for 16 years, I was often frustrated with the depiction of standing sound waves in pipes as it was presented in most elementary physics textbooks… The solution: an animation to visualize particle motion and pressure for longitudinal sound waves….I created the animation below and its accompanying description in an attempt to better explain the behavior of a standing sound wave in a pipe.  Standing Sound Waves (Longitudinal Standing Waves)

Transverse waves

The Mexican Wave. First seen internationally at the 1986 World Cup in Mexico City. It appeared at American football games for a few years before that.

Earthquake waves

S-waves (Shear) – Transverse waves that travel through the Earth

S waves

Earth Science, Tarbuck & Lutgens, Chapter 8

P-waves (pressure) – longitudinal waves, aka compression waves.

P waves

Earth Science, Tarbuck & Lutgens, Chapter 8

Only longitudinal waves can propagate through a fluid, because any transverse motion would not experience any restoring force since a fluid is readily deformable. This fact was used by geophysicists to infer that a portion of the Earth’s core must be liquid: after an earthquake, longitudinal waves are detected diametrically across the Earth, but not transverse waves

Then the 2 types of waves can combine

Suface waves are like ocean waves

Tarbuck & Lutgens

External resources

Dan Russell’s excellent tutorial on waves

Physics of sound waves – University of Salford

PhET Radio waves and electric fields app

PhET Sound waves app

Uses of Waves: GCSE Physics review, at S-Cool

Self-guided tutorials

Self guided tutorial on colors, spectrum and EM waves

Basic EM wave properties Physlet

Visulaize propagation of an EM wave I

Visualize propagation of an EM wave II

Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

HS-PS4-1. Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling within various media. Recognize that electromagnetic waves can travel through empty space (without a medium) as compared to mechanical waves that require a medium

SAT subject test in Physics: Waves and optics

• General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect
• Reflection and refraction, such as Snell’s law and changes in wavelength and speed
• Ray optics, such as image formation using pinholes, mirrors, and lenses
• Physical optics, such as single-slit diffraction, double-slit interference, polarization, and color

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