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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.
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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 ( 2 x/ - t ).
Exercises
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