Mathematics 309
Part I - The elementary geometry of wave motion
2. Waves in 1D
Even in 1D it is a bit difficult to say what is and isn't a wave, but certain objects are clearly conforming. A simple wave is one which amounts to a graph similar to that of the cosine function that moves regularly in time.
The fundamental relation
The important parameters of a simple wave are amplitude, wave length, velocity, period, and frequency.- The amplitude A is the distance from the x-axis to the highest point in the graph.
-
The wavelength
is the distance between wave crests.
- The velocity c is the velocity at which the crests move.
- The period T is the amount of time between crests at a fixed point.
-
The frequency f is the number of crests
passing a fixed point in each unit of time.
It is normally measured in c.p.s. (cycles per second).
A slightly different frequency
is measured in radians per second.
The second is 2
times the first.
- f = 1/T. This follows immediately from the definitions.
-
In time T, one wavelength passes by
a fixed point, so that the original crest travels
distance . Therefore
c = /T.
The equation of the graph
The equation of a wave at time t=0, amplitude 1, and period 2 is
y = cos x. That with amplitude A
is y = A cos x. Changing the wave length from 2
to amounts to a horizontal scaling by
/2 , and therefore has
equation
y = A cos 2 x/.
Since it is shifted to the right by ct
in time t,
the equation at time
t is
y = A cos (2/) (x - ct)
= A cos ( 2x/ - t ).