Dally and Osiecki (1994) generalized the wave energy balance equation
for the roller introduced by Dally and Brown (1995) to obliquely incident
waves,
d 1
A
A
PD +
ρRC 2 cos2 α = ρR gβ D
(31)
dx 2
T
T
where
PD = loss in organized wave energy flux through wave
breaking (obtained from Equation 13)
ρR = density of the roller
C = roller speed (taken to be proportional to the wave speed,
that is, C = βRCr, where βR is a coefficient)
α = wave angle
A
roller cross-sectional area,
T = wave period
βD = dissipation coefficient (about 0.1)
By defining the period-averaged mass flux (mR = ρrA/T), Equation 31 can be
solved conveniently for this quantity yielding:
d 1
2
PD +
mRCr cos α = gβ D mR
2
(32)
dx 2
where βR = 1.0 was assumed, and T = Tr is employed in the definition of mR.
The momentum flux in the roller is then obtained as MR = mRCr in the
direction of wave propagation. The additional terms in the longshore and
cross-shore momentum equations due to the roller are MRl = mRCr
sin (α) cos (α) and MRc = mRCr cos2(α), respectively, bearing in mind that