APPENDIX C
LAGRANGIAN DERIVATION OF THE GOVERNING WAVE EQUATION
The governing equation for wave motion can be derived through use of a variational
principle where a Lagragian is the function that is varied. Luke (1967) showed that a
Lagrangian for waves is given by
L Jh Z
dz
(C.l)
Kirby (1984) used a time averaged Lagrangian to derive a wave equation. For the following
derivation, equation (C.l) will be used. The third term in Eq. (C.l) is determined straight
forward, while the first two terms need to be expanded about fj the mean water line in a
Taylor series.
4>t +
(W)a
dz + [r¡- r))4>t |r, +(fl fj)
(C.2)
We assume that tj and can be expressed as
*) = V + *Vi
= o + efi (C.3)
where c is an ordering parameter and / is the depth dependency as defined in Eq. (3.3). For
the sake of simplicity we will assume that the gradient of fo represents the steady horizontal
currents. This means that fo is not a function of time and that W will not be included as
it was in the derivation in Chapter 3. This is because the W term is not expanded about
z = fj as it was in the Greens function derivation and is not needed in order to obtain the
(Vfc U)j£ term which as will be seen is obtained more directly in the present derivation.
Including W here will only produce terms that are of higher order in the mean current.
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