## Waves

### Basic Parameters and Definitions

A wave is a transfer of energy in the form of a disturbance, usually through a material substance or a medium.
All waves have a velocity, period, frequency, wavelength, and amplitude.
• Velocity is the propagation speed of a wave and is denoted by the letter c.
• Period is the time it takes to go through 1 cycle at a fixed point and is denoted by the letter T.
• Frequency is the number of cycles completed in a unit of time.
In mathematics, we often use the radian frequency where the units are radians per second.
It is denoted by w, the Greek letter omega.
• Wavelength is the distance between consecutive crests and is denoted by the Greek letter l, lambda.
• Amplitude is the distance from the origin (equilibrium position) to the maximum displacement and is denoted by the letter A.

### Fundamental Equations

 Fundamental Wave Equationy = Acos(2p/l (x - ct))or can be rewritten as: = Acos((2px / l) - (2pct / l)) The relationship between frequency and periodT = 1/frequency = 1/(w/2p) = 2p/w The relation between velocity, frequency, and wavelength Any two of the above parameters will result in the third because:c = lw/2psubstituting w = 2p/T, we get:c = l/T

### Scaling and Shifting

For the information on scaling and shifting functions refer to: Mathematics 309 - the elementary geometry of wave motion
Examples of shifting and scaling wave functions:
 Horizontal Scaling: changes in l y = cos(x) y = cos(.5x) y = cos(2x) **Notice that an increase in wavelength = a decrease in the frequency and vice versa.Therefore it is possible to change the frequency instead of the wavelength when scaling horizontally** Vertical Scaling: changes in A y = cos(x) y = 2cos(x) y = 1/2cos(x) Horizontal Shifting: Changes in c y = cos(x) y = cos(x - pt/2) y = cos(x + pt ) Vertical Shifting: adding a constant to the wave function y = cos(x) y = cos(x) - 2p y = cos(x) + p

Combining it all:

y = cos(x)
y = 1.5cos(x - p/2) - p/2

### Affine Equations

Recall affine functions in 2D and 3D
in 2D: Ax + By + C an affine function of 2 variable
in 3D: Ax + By + Cz + d an affine function of 3 variables

Consider in 2D the function
Ax + By + C
to start set C = 0
then Ax + By = 0 is just the figure below

If C does not = 0, then we can rewrite the equation as:
y = (C - Ax) / B

this means that the level lines (a curve where a function has a single value) would not be vertical lines, in fact they would look like this:
We want a uniform way to describe the level lines of
Ax + By = 0
[A, B] . [x, y] = 0
means that they are orthogonal, [A, B] perpendicular to [x, y]
Furthermore, level lines of Ax + By are all perpendicular to [A, B].

### Waves in Several Dimensions

What does affine equations have to do with Wave functions?
Notice that the wave height is just the equation:
A cos(ax + by)
where (a, b) is related to l, the wavelength vector.
How does it change with time?
A cos(ax + by + ct)
where ct = w, the radian frequency.

Page created by Julie Fun (worked individually) Math 309