Surface gravity water waves 145
5. Assume a constant water depth (h o = 5 m) and wave amplitude (a o =
0.5 m). Neglect the friction and turbulence effects. For the open-sea
boundary apply the expression ζ(x,t) = 2a o sin (kL − σt) cos [k(x − L)]
and for the coastal boundary use a no-flux condition (closed-end)
∂
∂
=
ζ
x end
0. Conduct the simulation and explain the physical significance of the results.
Example 6.2
This exercise deals with the effects of a submerged breakwater on the
propagation of one-dimensional linear long wave. The data used are
as follows:
Wave amplitude of the incident wave = 1 m
Wave period = 8 s
Bed friction coefficient = 0.001
Water depth = 5 m
Height of the breakwater = 3.33 m
Length of the breakwater = 20 m
Location of the breakwater = Mid-point of the longitudinal length
Longitudinal length = 400 m
The discretization steps for the simulation were taken as Δx = 2 m
and Δt = 0.1 s. The governing equation was Equation 6.23 but the
turbulent energy dissipation was neglected. The boundary condition
for the open-sea boundary was a combination of incident and radiated
wave, while for the coastline boundary it was free radiation.
The simulation was conducted for a period of 3200 seconds (40 wave
periods) and the results are presented in Figure 6.3. From this figure,
the partial wave reflection and partial wave attenuation (absorption)
due to the presence of the submerged breakwater is evident. Similar
conclusions can be derived from the sudden reduction of the wave
height root-mean-square value H
N
H
rms
m
m
N
2
2
1
1
=
=
∑ at the breakwater
location (Figure 6.4).
Computer code 6.2
% Example 6.2 One Dimensional Linear Long Wave Over a
Submerged Breakwater
% ho = Water depth [m];
% xf = Factor for sizing the height of the submerged
breakwater;
% fb = bed friction;
% a = Incident wave amplitude [m];