3. Sédiment Transport
45
force to the water through which the waves propagate. Such forces, applied
to a given volume of water, can be a non-zero résultant. In the new balance
of momentum, these wave-induced forces hâve to be balanced which can be
achieved by a pressure gradient (slope in the mean water level) or a few
velocity (longshore current) with counteracting bottom shear stress.
In reality, the radiation stress is neither a true stress (force per area,
N/m2), nor a true force (N), but a force (force per unit length, N/m) over
the entire depth (resulting from the intégration of the force per unit area
over the water depth).
Unlike hydrostatic pressure, the radiation stress is not isotropie; indeed,
just as with stresses, it is associated with a given direction or plane. In this
discussion, these planes are vertical and perpendicular to the two horizontal
axes, X oriented in the direction of wave propagation and Y along the wave
crest. This will yield the principal stresses; Sxx and Syy • (Note: when the
radiation stress is related to the longshore current, the axes will become
parallel to the coastline.)
The radiation stresses were derived from the linear wave theory équations by integrating the dynamic pressure over the total depth under a
wave and over a wave period, and subtracting from this the intégral static
pressure below the still-water depth for unit width:
o
(3-2)
where d: water depth, 77: instantaneous water level, p: pressure, p: water
density and V: velocity
This intégral is équivalent to the X-component of the force acting in
this plane; its unit, in the SLsystem, is kgm/s2, or N (Newton). The waveinduced contribution to the time average value of the intégral per unit width
is, by définition, the XX-component of the radiation stress, written as Sxx'rd+r)
r-d
Sxx = /
(p + pV2>)dz - / Podz
(3.3)
Jo
Jo
in which the hydrostatic contribution is equal to:
[ Podz = f pgzdz = (l/2)p#d2
(3.4)
Jo
Jo
In the notation Sxx, one subscript (X) stands for the direction of
transfer (through a plane X = constant) and the other for the component
of momentum being transferred (X) (Fig. 3.6).
45
force to the water through which the waves propagate. Such forces, applied
to a given volume of water, can be a non-zero résultant. In the new balance
of momentum, these wave-induced forces hâve to be balanced which can be
achieved by a pressure gradient (slope in the mean water level) or a few
velocity (longshore current) with counteracting bottom shear stress.
In reality, the radiation stress is neither a true stress (force per area,
N/m2), nor a true force (N), but a force (force per unit length, N/m) over
the entire depth (resulting from the intégration of the force per unit area
over the water depth).
Unlike hydrostatic pressure, the radiation stress is not isotropie; indeed,
just as with stresses, it is associated with a given direction or plane. In this
discussion, these planes are vertical and perpendicular to the two horizontal
axes, X oriented in the direction of wave propagation and Y along the wave
crest. This will yield the principal stresses; Sxx and Syy • (Note: when the
radiation stress is related to the longshore current, the axes will become
parallel to the coastline.)
The radiation stresses were derived from the linear wave theory équations by integrating the dynamic pressure over the total depth under a
wave and over a wave period, and subtracting from this the intégral static
pressure below the still-water depth for unit width:
o
(3-2)
where d: water depth, 77: instantaneous water level, p: pressure, p: water
density and V: velocity
This intégral is équivalent to the X-component of the force acting in
this plane; its unit, in the SLsystem, is kgm/s2, or N (Newton). The waveinduced contribution to the time average value of the intégral per unit width
is, by définition, the XX-component of the radiation stress, written as Sxx'rd+r)
r-d
Sxx = /
(p + pV2>)dz - / Podz
(3.3)
Jo
Jo
in which the hydrostatic contribution is equal to:
[ Podz = f pgzdz = (l/2)p#d2
(3.4)
Jo
Jo
In the notation Sxx, one subscript (X) stands for the direction of
transfer (through a plane X = constant) and the other for the component
of momentum being transferred (X) (Fig. 3.6).