Page 209 - Dynamic Loading and Design of Structures

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velocity equal to that at the Mean Water Level (MWL) in the crest. Linear extrapolation of the
kinematics above MWL is obtained by, for example, replacing eqn (5.8) by

(5.12)

A more sophisticated approach commonly used is the so-called Wheeler stretching (Wheeler,
1970). This modification introduces a new vertical co-ordinate which moves together with the
free surface. The velocity potential Φand the corresponding kinematics can then be obtained
by introducing a co-ordinate z instead of z in eqn (5.8); with . This
c
means that the kinematic quantities have the same size and vertical distribution, only now
with the free surface as starting point instead of the MWL, as shown in Figure 5.5b.
Gudmestad (1993) recently reviewed various engineering approximations to wave kinematics
and compared them with experimental results.
A deficiency of the original and modified Airy theory is that it provides symmetric waves
while extreme waves are known to be asymmetric (i.e. with a larger crest than trough). Higher
order wave theories have been proposed to better represent the shape and kinematics of the
waves (see e.g. Clauss et al., 1991).

Higher order wave theory
The linear wave theory represents a first order approximation of the free surface conditions,
which means that errors will become large as the waves become higher (i.e. as increases),
because of the neglected higher order terms. This deficiency can be improved by introducing
higher order terms. Commonly this is done by means of perturbation theory. Wave elevation
and velocity potential are then expanded into power series, with a being a small perturbation
parameter, so that the significance of additional terms decreases with their order (see e.g.
Sarpkaya and Isaacson, 1981)

(5.13)

(5.14)

(i)
Each individual potential Φ satisfies both the Laplace equation and the non-linear boundary
conditions with successive refinement. At the free surface, the velocity potential is expanded
as a Taylor series about the still water level to obtain successive approximations of higher
order wave theories:

(5.15)

The perturbation parameter (α turns out to be . The second order expansion
)