EM 1110-2-1100 (Part II)
30 Apr 02
2π(z % d)
cosh
H
L
A'
(II-1-34)
2
2πd
sinh
L
2π(z % d)
sinh
H
L
B'
(II-1-35)
2
2πd
sinh
L
(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
2πz
H
d
1
L
e
>
(i.e., deepwater limit)
for
A'B'
(II-1-36)
2
L
2
(g) For shallow-water conditions (d/L < 1/25), the equations become
H L
A'
(II-1-37)
2 2πd
and
H
z
1%
B'
(II-1-38)
2
d
II-1-18
Water Wave Mechanics
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