IV. Retreating Shorelines
3.1.6
Profile Nourishment
237
A bottom profile follows usually the equation :
3?/2 = 0.05 x
(8)
in which y is depth and x is distance from shoreline to point with depth y (Bruun
and Schwartz, 1985). The theory has been summarized as follows :
a) The profile is formed by the shear stress due to the wave action and is at
right angles to the shoreline. The material detached by the oscillating water
is removed by long-shore currents. As the shear stress due to wave action in
general - and particularly during storms - is far greater than the shear stress
originating from the longshore currents, this assumption cf.ms logical.
b) In the equilibrium profile the shear stress per unit bottom area (r) may be
assumed to be constant, i.e. the 'tzondition" at the bottom is the same (dr/dx
= dz/dt = 0). Confirmation of this assumption can only be attained by
experiments. One obtains 7" = ka u 2 .... where o is the density, k the
resistance coefficient and u the bottom water velocity. If k is assumed a
constant, then U~v, --- HTr/T sinh 27ry/L is also constant where T is the wave
period ; H, the wave height ; L, the wave length ; and y, the water depth.
c) dE~/dx = constant, where E~ is the transported wave energy per unit area of
the wave, and x is the distance from the shoreline. The loss of energy is
made up of a loss by spilling of the wave and a (very small) loss by internal
friction. The correctness of this assumption can only be proven by
experiments. Calculations give :
x = Lo 2~I22~-]+1
22L~-o]2 +43 2~--~--]2... 1
(9)
tLo ] 3
180tLo ]
where y is the water depth and Lo the deep water wave length. The series is
convergent for y < Lot8, i.e. for storm waves on the Danish west coast out to
depths of about 12 m (37 feet) where Lo = 100 m (328 feet). Since y << Lo,
the equation may be reduced to :
y3~ = px
(10)
where p is a constant.
If it now is assumed that the loss of energy is due only to bottom friction and
that this loss per unit area is constant, then :
3.1.6
Profile Nourishment
237
A bottom profile follows usually the equation :
3?/2 = 0.05 x
(8)
in which y is depth and x is distance from shoreline to point with depth y (Bruun
and Schwartz, 1985). The theory has been summarized as follows :
a) The profile is formed by the shear stress due to the wave action and is at
right angles to the shoreline. The material detached by the oscillating water
is removed by long-shore currents. As the shear stress due to wave action in
general - and particularly during storms - is far greater than the shear stress
originating from the longshore currents, this assumption cf.ms logical.
b) In the equilibrium profile the shear stress per unit bottom area (r) may be
assumed to be constant, i.e. the 'tzondition" at the bottom is the same (dr/dx
= dz/dt = 0). Confirmation of this assumption can only be attained by
experiments. One obtains 7" = ka u 2 .... where o is the density, k the
resistance coefficient and u the bottom water velocity. If k is assumed a
constant, then U~v, --- HTr/T sinh 27ry/L is also constant where T is the wave
period ; H, the wave height ; L, the wave length ; and y, the water depth.
c) dE~/dx = constant, where E~ is the transported wave energy per unit area of
the wave, and x is the distance from the shoreline. The loss of energy is
made up of a loss by spilling of the wave and a (very small) loss by internal
friction. The correctness of this assumption can only be proven by
experiments. Calculations give :
x = Lo 2~I22~-]+1
22L~-o]2 +43 2~--~--]2... 1
(9)
tLo ] 3
180tLo ]
where y is the water depth and Lo the deep water wave length. The series is
convergent for y < Lot8, i.e. for storm waves on the Danish west coast out to
depths of about 12 m (37 feet) where Lo = 100 m (328 feet). Since y << Lo,
the equation may be reduced to :
y3~ = px
(10)
where p is a constant.
If it now is assumed that the loss of energy is due only to bottom friction and
that this loss per unit area is constant, then :