Hydraulic Structures: Fourth Edition

(Fig. 14.2(c)). The profile of the solitary wave relative to the axis moving

with the celerity of the wave is

/Hsech^2 [(3H/d)1/2(x/2d)]. (14.39)

As a result of energy conservation, the energy contained in the solitary

wave is equal to that in the wave under deep-water conditions and to twice

the potential energy (consistent with the earlier finding that the kinetic

and potential energies in a wave are equal). The energy in a solitary wave

per unit length along the crest is therefore (Morris, 1963)

Es 2 

∞

∞



1

2

^ g^2 dx

 2  2 

∞

0

sech^4 

1/2

dx

 (Hd)3/2.

Using the criterion of equation (14.36), the above equation becomes

Es gH^3. (14.40)

As the energy of the oncoming waves at deep water within a single wave-

length is

Es(1/8) gH^20 L 0

this results in

(Hb/H 0 )0.38(H 0 /L 0 )^ 1/3 (14.41)

and

(db/H 0 )0.49(H 0 /L 0 )^ 1/3. (14.42)

Equations (14.41) and (14.42) are useful in determining the wave height

Hband the shallow-water depth dbat which the wave breaks, given the

propertiesH 0 andL 0 in deep water.

Coastal Engineering Manual (US Army, 2002) suggests a criterion

for breaking of irregular waves on horizontal bed in shallow water as

Hsb≈0.6db. Alternate expressions similar to Miche’s criterion using Hrmsat

breaking point have also been proposed. For the definition of Hrms, refer to

equation (14.47).

11.6



(^33)
8 g

(^33)
x

2 d

3 H



d

gH^2



2

590 WAVES AND OFFSHORE ENGINEERING