the celerity relationship (Equation 2.115) reduces to
1/ 2
gλ
c=
(2.117)
2π
For shallow water waves (long waves)
yo 1
<
(2.118)
λ 20
Then Equation 2.115 reduces to
c = gy o
(2.119)
The time of travel of one water crest to another at a given point is called the period T and can be
defined from the celerity and wave length
c = λ/T
(2.120)
In Equation 2.117, the celerity is independent of depth and depends on gravity g and wave
length λ. This is the celerity of ocean waves. In Equation 2.119, the celerity is a function of
gravity and depth which describes small amplitude waves in open channels. These two
equations apply only to small amplitude waves; that is ao/λ << 1.
The celerity of finite amplitude shallow water waves has been determined both analytically using
Bernoulli's equation and experimentally, and is given by the expression
1/ 2
(y o + 2ao )2
c=
gy o
(2.121)
(y o + ao ) y o
When 2ao is small in comparison to yo
1/ 2
2a
c = 1 + o gy o
(2.122)
yo
Generally as 2ao/yo approaches unity the crest develops a sharp peak and breaks.
In the above equations, c is measured relative to the fluid. If the wave is moving opposite to the
flow then, when c > V, the waves move upstream; when c = V, the wave is stationary; and when c
< V, the wave moves downstream. When V = c for small amplitude flow,
V = c = gy o
(2.123)
The ratio of the flow velocity to the celerity of a shallow water wave of small amplitude is defined
by the Froude number:
2.33
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