2π(*z *% *d*)

cosh

(II-1-34)

2

2π*d*

sinh

2π(*z *% *d*)

sinh

(II-1-35)

2

2π*d*

sinh

(e) Equation II-1-33 is the equation of an ellipse with a major- (horizontal) semi-axis equal to *A *and a

minor (vertical) semi-axis equal to *B*. The lengths of *A *and *B *are measures of the horizontal and vertical

displacements of the water particles (see Figure II-1-4). Thus, the water particles are predicted to move in

closed orbits by linear wave theory; i.e., a fluid particle returns to its initial position after each wave cycle.

Comparing laboratory measurements of particle orbits with this theory shows that particle orbits are not

completely closed. This difference between linear theory and observations is due to the mass transport

phenomenon, which is discussed later in this chapter. It shows that linear theory is inadequate to explain

wave motion completely.

(f) Examination of Equations II-1-34 and II-1-35 shows that for deepwater conditions, *A *and *B *are equal

and particle paths are circular (Figure II-1-4). These equations become

2π*z*

d

1

>

(i.e., deepwater limit)

for

(II-1-36)

2

L

2

(g) For shallow-water conditions (d/L < 1/25), the equations become

(II-1-37)

2 2π*d*

and

1%

(II-1-38)

2

II-1-18

Water Wave Mechanics

Integrated Publishing, Inc. |