sheltered waters. However, in the case of island berthing structures for large vessels, which
are sited in deep and relatively unsheltered waters, the wave forces may represent a signifi-
cant proportion of the total force required to be calculated. Also, piles supporting the
approach trestle to a jetty are not required to withstand berthing impact forces. Thus wave
forces, even in fairly sheltered waters, when combined with wind pressures on the super-
structure and current drag on the piles, may produce substantial loading transverse to the
axis of the trestle.
A simple approach to the calculation of wave forces on fixed structures is to assume that
the maximum wave force can be expressed as the equivalent static force caused by a solitary
wave of the shape shown in Figure 8.11. This shape is representative of a breaking wave. An
oscillatory wave has a different shape but the factors given in Figure 8.12 and Table 8.1 for use
with equations 8.8 and 8.9 are applicable only to breaking wave conditions. Drag and inertial
forces are exerted on the structure by the water particles which move in an elliptical path as
shown. From the work of Wiegel et al.(8.3), Reid and Bretschneider(8.4), Dailey and Stephen(8.5),
and Bretschneider(8.6), it is possible to calculate the water particle velocity uat any point
having co-ordinates xhorizontally from the wave crest and zvertically above the sea bed.
The water particle velocity can be related to the velocity of advance of the wave crest (the
wave celerity c) and expressed in terms of (u/c)^2 and 1/g du/dt for various ratios of xand zto
the height hof the trough of the wave above the sea bed.
The solitary-wave theory is limited in its application to a range of conditions defined
by the ratio of the wave period to the water depth. Because the equations given below are
applicable only to breaking wave conditions they represent the maximum force which can
be applied to a structure. Breaking wave conditions are unlikely to occur in deep water
berths for large tankers, and these conditions are likely to be found only in fairly shallow
water on exposed jetty sites, for example along the line of the approach structure from the
Piling for marine structures 409Table 8.1Surface elevations, velocities, and accelerations for solitary breaking wave
Distance Surface Values of Values of
from crest elevation
x/h zs/h At At At Average Height to At At At Average Height to
surface z h bottom value centroid surface z h bottom value centroid
0 1.78 1.000 0.176 0.109 0.226 1.19 0 0 0 0
0.2 1.67 0.430 0.170 0.106 0.181 1.03 0.242 0.073 0.031 0.081 1.14
0.4 1.57 0.276 0.156 0.099 0.150 0.92 0.347 0.137 0.060 0.133 1.02
0.6 1.48 0.201 0.133 0.092 0.123 0.83 0.380 0.184 0.087 0.164 0.93
0.8 1.41 0.138 0.106 0.078 0.097 0.80 0.357 0.214 0.110 0.180 0.88
1.0 1.35 0.092 0.082 0.070 0.077 0.70 0.321 0.225 0.127 0.186 0.78
1.2 1.29 0.062 0.063 0.058 0.061 0.65 0.280 0.225 0.140 0.187 0.73
1.4 1.25 0.041 0.046 0.048 0.047 0.61 0.243 0.209 0.146 0.182 0.68
1.6 1.21 0.029 0.032 0.038 0.035 0.59 0.209 0.192 0.148 0.173 0.65
1.8 1.18 0.020 0.023 0.029 0.027 0.56 0.174 0.171 0.145 0.159 0.62
2.2 1.13 0.009 0.011 0.018 0.014 0.50 0.122 0.128 0.130 0.130 0.57
2.6 1.08 0.004 0.005 0.009 0.007 0.50 0.088 0.091 0.109 0.102 0.53
3.0 1.05 0.002 0.002 0.004 0.003 0.50 0.065 0.067 0.084 0.078 0.51
3.4 1.03 0.001 0.001 0.002 0.002 0.50 0.049 0.049 0.062 0.058 0.50
5.0 1.01 0.000 0.000 0.000 0.000 0.50 0.012 0.012 0.017 0.016 0.50
(u/c)^2 1/g· (du/dt)