kH
sinh [2k(z + d)/L] , [ 2irx
2Kt\
L
sinh (27rd/L)
\ L
T /
J
3
(kH?
sinh [4k(z 4- d)/L] . Mkx _ 4kt |.
4
\ L /
sinh4(27rd/L)
\ L
T /
Second-order équations for water-particle displacements from their
mean position for a finite amplitude wave are.
HgT2 cosh [2k(z + d)/L] . I2kx _ 2zrt\
kH______ 1-------* = " 4^L
cosh(27rd/L) “ Sm \ L
T /
8L sinh2 (27rd/L)
(2-53)
3 cosh [4k(z + d)/L]l . (4kx _ 4kî\
/ttHV Ct cosh [4k(z + d)/L]
1 " 2
sinh2(2Kd/L)
Sm \L
T /
\L ) 2
sinh2(2Kd/L)
and
HgT2 sinh [2k(z + d)/L]
/2kx
2Kt\
> _
--------- cos ----- — ——
4kL
cosh (2Kd/L)
\ L
T /
(2-54)
3 kH2 sinh [4k(z + d)/L]
/4kx
4kA
+--------- ------------------------- cos ------ — ---- 1 •
16 L
sinh4(2Kd/L)
\ L
T /
2.253 Mass Transport Velocity. The last term in Equation 2-53 is of
particular interest; it is not periodic, but is the product of time and
a constant depending on the given wave period and depth. The term predicts
a continuously increasing net particle displacement in the direction of
wave propagation. The distance a particle is displaced during one wave
period when divided by the wave period gives a mean drift velocity, U(z),
called the mass transport velocity. Thus,
U(z) =
kH\2 C cosh [4k(z + d)/L]
L ) 2
sinh2(2Kd/L)
(2-55)
Equation 2-53 indicates that there is a net transport of fluid by
waves in the direction of wave propagation. If the mass transport, indicated by Equation 2-55 leads to an accumulation of mass in any région, the
free surface must rise, thus generating a pressure gradient. A current,
formed in response to this pressure gradient, will reestablish the distribution of mass. Studies of mass transport, theoretical and experimental,
conducted bï Longuet-Higgins (1953, 1960), Mitchim (1940), Miche
(1944), Ursell (1953), and Russell and Osorio (1958). Their findings
indicate that the vertical distribution of the mass transport velocity is
modified so that the net transport of water across a vertical plane is
zéro.
2-38