DESCRIPTION OF PLANE WAVES
63
In équation (3.1), A defines the amplitude of the wave and k is the wave
number. In radian measure, kx = 2ttz/A; or x/X is the multiple of the angle 2tt,
where A is the particular value of x for which cos kx = 1, yielding the next crest
of the wave in advance of the first at x = 0. The wave length A is expressed in
terms of wave number by
fc=T
(3.2)
The general expression for r) in a progressive sinusoidal wave moving at velocity
c in the positive x direction is
r/(x, t) = A cos k(x ~ et)
(3.3)
Now, if we let x = 0 and consider the variation of the instantaneous surface
élévation with time as the wave passes the origin 0, we hâve from équation (3.3)
77x=o = A cos(—/cet) = A cos ket
(3-4)
which is depicted in Figure 3.3. Again, the significance of kc is that of converting
ket to radian measure, or
ket — 2tt—
(3.5)
where T is the particular value of t that makes the cosine term unity, giving the
next crest of the wave in succession to the first at t = 0. The quantity T is the
period of the wave and
kc = — = h) r ad/sec
(3.6)
Figure 3.3 Variation of instantaneous surface élévation with time.