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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.
These are not independent.
  • 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 ).