where the origin of *x *is at the wave crest. The volume of water within the wave above the still-water level

per unit crest width is

1

16 3

2

(II-1-90)

3

(i) An equal amount of water per unit crest length is transported forward past a vertical plane that is

perpendicular to the direction of wave advance. Several relations have been presented to determine the

celerity of a solitary wave; these equations differ depending on the degree of approximation. Laboratory

measurements suggest that the simple expression

(II-1-91)

gives a reasonably accurate approximation to the celerity of solitary wave.

(j)

The water particle velocities for a solitary wave (Munk 1949), are

1 % cos(*My*/*d*) cosh(*Mx*/*d*)

(II-1-92)

[cos(*My*/*d*) % cosh(*Mx*/*D*)]2

sin (*My*/*d*) sinh (*Mx*/*d*)

(II-1-93)

[cos (*My*/*d*) % cosh (*Mx*/*D*)]2

where *M *and *N *are the functions of *H/d *shown in Figure II-1-17, and *y *is measured from the bottom. The

expression for horizontal velocity *u *is often used to predict wave forces on marine structures situated in

shallow water. The maximum velocity *u*max occurs when *x *and *t *are both equal to zero; hence,

(II-1-94)

1 % cos(*My*/*d*)

(h) Total energy in a solitary wave is about evenly divided between kinetic and potential energy. Total

wave energy per unit crest width is

3

3

8

2

2

ρ*gH*

(II-1-95)

33

and the pressure beneath a solitary wave depends on the local fluid velocity, as does the pressure under a

cnoidal wave; however, it may be approximated by

(II-1-96)

(l)

Equation II-1-96 is identical to that used to approximate the pressure beneath a cnoidal wave.

(m) As a solitary wave moves into shoaling water it eventually becomes unstable and breaks. A solitary

wave breaks when the water particle velocity at the wave crest becomes equal to the wave celerity. This

occurs when (Miles 1980, 1981)

II-1-48

Water Wave Mechanics

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