35
The first integral above is dropped since there are only steady current terms. The remaining
term, dropping the e, is
= r ft f**dz = -k2 r f2dz = -k2^-i (3.26)
J-h0 J-ho 9
The third term in Eq. (3.10) is
ft. W = ft. If,-ft. \-h0 (3.27)
The first term on the right hand side is equal to t* |/| since / 1^= 1 and has already been
determined. The second term on the right hand side of Eq. (3.27) is easily determined using
the bottom boundary condition Eq. (3.9)
-ft.\-ko=fVht-Vhh0 \-k. (3.28)
Using Eq. (3.20) this becomes
- ft. Uo= fU Vkh0 |_fco +e/2 |_fco Vht vhh0 + eftVkf Vfch0 |_fco (3.29)
The first term is dropped since it not order e and the third term is clearly higher order in
the slope. Thus, again dropping the t
-Mk=/!|-lVkfVA (3.30)
This term cancels the bottom boundary term resulting from the Leibnitz integration above.
The fourth term of Eq. (3.10) is also easily determined.
tf. |-*.= feBmcrah Jfch ^ l-*.= ~k tanh kh Y* ^ (331)
Evaluating t at the mean water surface and retaining only terms of order e yields
-t!.th=~i (3-32)
Collecting terms from (3.24),(3.26),(3.18),(3.30) and(3.32) yields the following linear
governing equation for t-
+ (V* )^ Vk(CCtVht) + (a2 k2CC)t = 0 (3.33)
This equation was first derived by Kirby (1984).