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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