Flow, deformation, and transport - Physical Processes in Earth and Environmental Sciences - page 127

The waves move with a velocity
proportional to the square root
of the water depth, independent
of the wavelength or period:

Every water particle rotates
about a time-mean ellipsoidal
motion, the ellipses becoming
more elongated with depth

Depth, h ,
< λ /20

c = gh

Fig. 4.43 Shallow water waves and their ellipsoidal orbitals. Shallow water waves are sometimes called long waves because their wavelengths are
long compared to water depths. Note that the orbital motions flatten with depth but do not change in maximum elongation.
Note : All waves in similar water depths travel at the same speed and transmit their energy flux at this rate.

Fig. 4.44 Time-lapse photograph of shallow water wave orbitals visualized by tracer particle. This flow visualization of suspended particles was
photographed under a shallow water wave traversing one wavelength, . The

clockwise orbits are ellipses having increasing elongation toward the bottom. Some surface loops show slow near-surface drift to the right.
This is called Stokes drift and is due to the upper parts of orbitals having a greater velocity than the lower parts and to bottom friction. The
surface drift is accompanied by compensatory near-bed drift to the left, due to conservation of volume in the closed system of the experimental
wave tank. Stokes drift without the added effects of bottom friction also occurs in short, deep water waves.

, left to right. Wave amplitude is 0.04 and water depth is 0.22

boundary layer of the water as the water collapses or spills
down the wave front, thus markedly increasing the air-
to-sea-to-air transfer of momentum, thermal energy,
organo-chemical species, and mass. The production of
foam and bubble trains is also thought to feed back to the
atmospheric boundary layer itself, leading to a marked
reduction of boundary layer roughness and therefore fric-
tion in hurricane force winds (Section 6.2).

dependence on the square of wave amplitude. The energy
flux (or wave power) is the rate of energy transmitted in
the direction of wave propagation and is given by Ecn ,

where c is the local wave velocity, and the coefficients are
n

1 in shallow water. In deep
water the energy flux is related to the wave group velocity
rather than to the wave velocity. Because of the forward
energy flux, Ec , associated with waves approaching the
shoreline, there exists also a shoreward-directed momen-
tum flux or stress outside the zone of breaking waves. This
is termed radiation stress and is discussed in Section 6.6.

0.5 in deep water and n

4.9.3 Surface wave energy and radiation stresses

The energy in a wave is proportional to the square of its
height. Most wave energy (about 95 percent) is concen-
trated in the half wavelength or so depth below the mean
water surface. It is the rhythmic conversion of potential to
kinetic energy and back again that maintains the wave
motion; derivations of simple wave theory are dependent
upon this approach (Cookies 14 and 16). The displace-
ment of the wave surface from the horizontal provides
potential energy that is converted into kinetic energy by
the orbital motion of the water. The total wave energy per
unit area is given by E

4.9.4 Solitary waves

Especially interesting forms of solitary waves or bores may
occur in shallow water due to sudden disturbances affect-
ing the water column. These are very distinctive waves of
translation , so termed because they transport their con-
tained mass of water as a raised heap, as well as transporting
the energy they contain (Fig. 4.45). These amazing fea-
tures were first documented by J.S. Russell who came
across one in 1834 on the Edinburgh-Glasgow canal in
central Scotland. Here are Russell's own vivid words,

ga 2 , where a is wave ampli- 0.5

tude ( 0.5 wave height H ). Note carefully the energy