A = The maximum distance from the rest position (0).
ג = The spatial distance between two crests of the wave.
C = The velocity at which crests travel.
T = The time require for one compete up/down cycles or the time to complete one wavelength.
ω = The number of cycles per second or radians per second.
T = 1 / ω (cycles)
= 1 / ( ω / 2∏ ) (radians)
C = Distance/ Time
= ג / T
= ω / ג
= ג ω / 2∏
This equation holds in any number of dimensions if C = speed as opposed to velocity, which has direction.
For example :
|C| = |ג| ω / 2∏ holds in any number of dimension.
At time 0:
Y = Cos (X)
At time t:
The graph is shifted to the right by the amount ct.
According to the basic idea of shifting and scaling, the new function is
Y = Cost (X - Ct)
We are going to scale 2∏ ---> ג .
Scaling by ג / 2∏.
The new function is Y = Cos (X / (ג /2∏ ))
= Cos ( 2∏X/ג )