Summary of Waves (Physics 112)

If you multiply the distance between peaks by the number of peaks per second you get the distance traveled in a second.

wave velocity = freq * wavelength

If you divide the distance between peaks by the time required per cycle you get the velocity with which peaks pass

wave velocity = wavelength / period

If you divide the distance traveled in a second by the number of peaks in a second you get the distance between peaks

wavelength = velocity / frequency

If you divide the distance traveled in a second by the distance between peak you get the number of peaks passing per second

frequency = velocity / wavelength

The tension in a string and its mass per unit length determine how fast a pulse travels in that string

Wave velocity = `sqrt( T / (m / L) )

For a standing wave in a string fixed at both ends we have a sequence of alternating nodes and  antinodes.  Since there are nodes at both ends we can get the following configurations:

NAN (fundamental, 1/2 wavelength)

NANAN (1st overtone, 1 wavelength)

NANANAN (2d overtone, 3/2 wavelength)

etc. to get 2, 5/2, 3, 7/2, 4, 9/2, ... wavelengths

If we know the string length and number of nodes and antinodes we therefore get the wavelength:

if 1/2 wavelength = L then wavelength = 2L

if 2/2 wavelength = L then wavelength = L

if 3/2 wavelength = L then wavelength = 2/3 L

If we know the frequency of the standing wave and the wavelength we can find wave velocity, which is the same for all standing waves in the string:

v = f * wavelength

for fundamental v = f1 * 2L

for first overtone v = f2 * L

for second overtone v = f3 * (2/3 L )

where f1, f2, f3 are frequencies of first, second, third harmonics

A holler lawg with both ends open has antinodes at both ends so has possible standing-wave configurations ANA, ANANA, ANANANA, corresponding to 1/2 wavelength, 2/2 wavelength, 3/2 wavelength etc. in the length of the lawg.  So open holler lawgs is just lak strangs what done swapped nodes and antinodes, but of course the speed of the wave is the speed of sound.  It the holler lawg is in the crick then you gotta use the speed of sound in a crick, which gits tricky if the crick's a-flowin through the lawg, but what otherwise ain't bad 'cause you jest gotta use the speed of sound in water.

If a holler lawg is closed at one end and open at the other then you have a node at one end and an antinode at the other.  So you can get

NA (1/4 wavelength) along lawg

NANA (3/4 wavelength--4 fenceposts give you three spains), 

NANANA (5/4 wavelength)

If you're walkin down the hall at velocity vSource chuckin' bowlin balls at the other end every `dt seconds and the bowlin balls travel at velocity vWave then

• in `dt seconds after chuckin the first ball you get vSource * `dt closer to the other end
• this gives the next ball a 'jump' of  `dtJump = vSource * `dt / vWave on the last ball
• so the two balls will be just `dt - `dtJump apart in time
• so balls hit end of hall at frequency f ' = 1 / (`dt - `dtJump) = 1 / (`dt - vSource * `dt / vWave) = 1/`dt * [ 1 / (1 - vSource / vWave) ] = f / (1 - vSource / vWave).

Doppler Shift with Source in Motion:  f ' = f / ( 1 - vSource/vWave )

The energy of a simple harmonic oscillator is .5 k A^2, where A is the amplitude of motion.  We can use this to figure out how much energy is stored in a traveling wave (gotta know about the sines and cosines) or a standing wave (gotta know sines, cosines and calculus).  But we can see that k is proportional to tension and A is the amplitude of the motion (max displacement from equilibrium) and also to velocity so

energy in a wave is proportional to the 3/2 power of the tension and to square of amplitude

Interference typically occurs when a wave is split and half of it moves further than the other half, or when waves driven by in-phase sources are observed at positions where the path distances from the two sources differ.

Constructive interference occurs when the path difference is a whole number of wavelengths, which results in the two signals being in phase:  path diff = n * `lambda, n = 1, 2, 3, ...

Destructive interference occurs when the path difference is a whole number of wavelengths, plus half a wavelength, which results in the two signals being half a cycle out of phase.  path diff = (n + 1/2) * `lambda, n = 1, 2, 3, ....

Interference can also occur when part of a wave is reflected from one surface and another part from another surface lying at some fixed distance beneath the first (e.g., when part of the light from a source is reflected from a gasoline film and most of the rest from a water surface on which the film is floating).

By considering how wavelets spread out during rock-chuckin' and other activities we can understand much of the behavior of waves--e.g., how they spread around obstacles, or how the direction of propagation changes when the speed of propogation changes at an interface between two materials.

When a 'ray' moves from one medium to another in which the velocity of propagation differs from that in the first,

Snell's Law tells us that the angle of incidence and angle of refraction of a wave