+ if wave is traveling to the left, - if wave is traveling to the right.
Demo: Wave Machine
What is a wave? (sound, light, water, string, etc.)
Definition: A wave is the motion of a disturbance.
Types of Waves:
1) Transverse: displacement perpendicular to direction of motion of wave.
2) Longitudinal: displacement parallel to direction of motion of wave.
To describe a wave we want a way of describing the displacement of a point in the medium as a
function of time and position.
+ if wave is traveling to the left, - if wave is traveling to the right.
How do we add waves together?
Two pulses traveling in opposite directions on a string at any point in space and time
Just add displacements (works for linear waves)
typically small displacements
Consider two continuous waves
Constructive Interference - Waves are in Phase
Destructive Interference - 180 degrees out of phase
crest lines up with trough and waves cancel each other out.
Wave Speed
Qualitative: Heavier the material, the slower the wave moves
Higher the tension, the faster the wave moves.
Proof:
Approximate the shape of the wave by a circular arc.
This provides the centripetal force necessary to keep the object
moving in a circle.
Also 
Substituting the expression for q into the centripetal force equation gives us
we can solve this to find the speed of the wave on the string
but m and s are related; the longer the segment of string, the more massive.
What happens when the traveling wave comes to the "end of its rope"? The wave will be reflected with
the behavior of the reflected wave determined by the boundary conditions at the end of the rope.
Fixed End: string pulls up, wall pulls down.
reflected pulse: same size, same speed, moving in opposite direction
wave is inverted (up-side-down), 180 degrees out of phase
Moveable End: string pulls down, ring pulls up.
reflected pulse; same size, same speed , moving in opposite direction
wave is erect (right-side-up), no phase shift.
Boundary between two different types of ropes.
part of the pulse will be transmitted; part of the pulse will be
reflected.
The transmitted pulse will always be upright. The reflected pulse will be inverted if the second
string has a greater linear mass density than the first. The reflected pulse will be upright if the second
string has a smaller linear mass density than the first.
Describing Continuous Waves
Freeze the wave in time
Amplitude (A): distance from crest to equilibrium position or 1/2 crest to trough distance
Wavelength (l): distance between two identical points on the wave.
Frequency (f): number of crests/second.
wave speed (v): how fast the disturbance is moving 
1) Freeze the wave in time
2) Isolate in Space
Combine: 
If we look at a particular point on the string through which a harmonic wave is propagating and
find the transverse velocity and acceleration.
Waves carry energy and momentum.
Consider a sinusoidal wave, as this disturbance travels through the string, individual segments of
the string will undergo simple harmonic motion. The energy associated with an object undergoing
simple harmonic motion is
Thus, each little, tiny segment of the string of length dL and mass dm will have the following
energy
if the string has some linear mass density m, then the mass of that short segment can be written as dm = m dx
The rate at which energy can be transferred is the power
