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Gourlay (1974) in which the shoreline in the lee of a breakwater was made to curve and
meet the breakwater as occurs with tombolo formation. The results of the model were in
close agreement with Gourlays experimental values.
5.1 Wave Set-up
Longuet-Higgins and Stewart (1964) gave a simple but elegant derivation for the wave
induced set-up on a plane beach assuming linear shallow water theory. For the steady state
case of normally incident waves on a plane beach without any y variations and without any
currents the depth integrated equations of motions reduce to
gdi + as~=0
(5.1)
dx pD dx
In shallow water for normally incident waves the radiation stress term from linear wave
theory, Eq. (2.28), becomes
Sxx = E(n + l) = lE=-^pgH2
(5.2)
A spilling breaker assumption is made so that the wave height if is a constant ratio k to
the water depth D = h0 + fj,
H = K(h0 + fj) (5.3)
Substituting equations (5.2) and (5.3) into Eq. (5.1) gives
3rj _ 3 dK2(h0 + fj)1 3 2 fdhQ dfj\
dx 16(h0 -I- fj) dx 8* V dx + dx)
Solving for gives the result
dfj |/c2 dh0
dx 1 + |/c2 dx
that the slope of the water surface inside the breaker line is a constant times the slope of
the beach.
Previous models are able to produce the results of the analytical expression but to
date none, including the present, have matched the experimental results. In figure 5.1 it
is seen that the slope of the mean water surface produced by the wave set-up is nearly