- Basic Parameters and Definitions
- Fundamental Equations
- Scaling and Shifting
- Affine Equations
- Waves in Several Dimensions

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, the Greek letter omega.**w** - Wavelength is the distance between consecutive crests and is denoted by the Greek letter
, lambda.**l** - Amplitude is the distance from the origin (equilibrium position) to the maximum displacement and is denoted by the letter
.**A**

Fundamental Wave Equation |

The relationship between frequency and periodT = 1/frequency = 1/(w/2p) = 2p/w |

The relation between velocity, frequency, and wavelengthAny two of the above parameters will result in the third because: substituting w = 2p/T, we get: |

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 - pt/2) y = cos(x + pt ) |
Vertical Shifting: adding a constant to the wave function
y = cos(x) - 2p y = cos(x) + p |

y = cos(x)

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

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:

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:

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].

Notice that the wave height is just the equation:

How does it change with time?