# The Principle of Superposition

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The Principle of Superposition
Introduction and The Principle of Superposition

What is a wave? A wave is a transfer of energy through a medium.
Examples: Waves on a string, Water waves, Sound waves, Electromagnetic waves (light) There are two major types of waves. Mechanical Wave - Made up of oscillators that are interconnected in some way. Each oscillator undergoes SHM, and is able to influence the adjacent oscillator. It is basically transferred SHM with some “lag” Requires a medium through which it can travel (eg. Particles colliding with each other to transfer energy)

The second type of wave is an electromagnetic wave (light).
This type of wave is self-propagating, and does not require a medium to travel. Electromagnetic waves can travel through a vacuum indefinitely without losing any energy whatsoever! They are the most fundamental unit of energy!

Longitudinal and Transverse Waves
Longitudinal Wave – the individual oscillators move parallel and antiparallel to the wave Examples: Sound waves, compression waves Transverse Wave – the individual oscillators move perpendicular to the wave A wave is the transfer of energy through a medium – it is not an object! It is an abstract phenomenon.

Waves can be categorized in two ways.
All waves are either mechanical or electromagnetic. All waves are either longitudinal or transverse. Sound waves are… longitudinal mechanical waves Waves on a string are… transverse mechanical waves Light is… a transverse electromagnetic wave

Waves are weird! When a mechanical wave travels through a medium, the individual oscillators actually move back and forth, while the wave itself travels through at constant speed!

A transverse pulse is traveling through a rope, as shown below.
Wacky Wave Whiteboard! A transverse pulse is traveling through a rope, as shown below. v A B C D E F G H I At the instant shown, indicate which direction each of the labeled points of the rope is moving. If the point is instantaneously at rest, state so.

At rest: A, C, G, I Moving upward: B, H Moving downward: D, E, F G F H

Every wave has some key properties!
Wavelength – the length of one full wave, from crest to crest or from trough to trough. Period – the amount of time that it takes one of the carriers of the wave to make one full cycle. Frequency – the number of cycles per second made by the carriers of the wave Amplitude – the maximum displacement from equilibrium that the wave carriers have Speed of the wave (waves travel at a constant speed, unless they change their medium).

λ A A Frequency = how many waves pass a certain point in one second

v = λf v = Δx/Δt v = λ/T T = 1/f v λ
Since waves travel at a constant speed, we can use some basic kinematics… λ v = Δx/Δt v = λ/T T = 1/f v = λf v

f = frequency of the wave
v = λf v = speed of the wave λ = wavelength f = frequency of the wave This is the most important relationship that you will need to know when analyzing wave phenomena. Know it like you know your birthday.

The Principle of Superposition
What happens when two waves occupy the same location at the same time? They add together, according to The Principle of Superposition When two waves are superimposed on each other, you must add together the displacements at each point on the wave. Superimpose the waves on each other, and add the displacement from equilibrium at each point.

Two square waves are traveling along a string, as shown above
Two square waves are traveling along a string, as shown above. Draw the shape of the string 3 seconds later.

These parts negate each other

Superposition, Beats and Wave Speed

The Principle of Superposition
When two waves are superimposed on each other, you must add together the displacements at each point on the wave. Superimpose the waves on each other, and add the displacement from equilibrium at each point. It helps to draw the contribution from each wave, and then drawing the net displacement of the medium.

Wave Whiteboard Warmup
Two wave pulses are incident on a string, as shown below. Draw the shape of the string four seconds later.

Contributions from each pulse:

A Little Bit Tougher Now
Each end of the rope shown below is oscillated with a frequency of 0.75 Hz, producing the waves shown. Draw the shape of the string one second later (assuming that the rope continues to be oscillated after the instant shown.)

Using v = λf, we find that the speed of the waves is 3 m/s.
Contributions from each wave:

Result:

Beats! When two waves of different frequencies are superimposed upon each other, the result is an amazing phenomenon known as beats. The result will alternate between constructive and destructive interference, as shown.

Beat frequency The frequency of the combined wave depends on how different the two waves are initially!

Conceptual Tune-Up! When string musicians tune their instruments, often they will and play a string, and simultaneously use a tuner to play the note that they want the string to make. (eg. They would play the ‘A’ string simultaneously with a known ‘A’ note) How can they use beats to determine how close their string is to the correct note?

What affects the speed of a wave?
The medium!!! Properties of the medium are the only thing that determines wave speed. For a given medium, v = λf Wavelength and frequency are inversely proportional Speed is constant

For a mechanical wave traveling through a rope or spring,
the speed of waves in the medium depends on only two things: - Tension in the rope or spring - Linear density of the medium (mass per unit length)

Conceptually, how does tension affect wave speed?
The more tension in a medium, the more strongly the oscillators interact with one another. In a high tension medium, waves travel very quickly. The tighter a string is pulled, the more quickly the adjacent oscillators will transfer energy to one another, resulting in a higher wave speed.

Conceptually, how does linear density affect wave speed?
The more dense the medium, the slower the wave will propagate. It takes more time for each oscillator to become displaced in a medium with high linear density (mass per unit length). Since more dense ropes have more inertia per unit length, waves will travel slower in them.

How does linear density affect wave speed?
Linear density (μ) is the mass per unit length of the medium.

Mathematical Model for Wave Speed in a Rope or Spring