525 



The reduction of (13) on the hypothesis that Hx 

 O(ans) yields 



{d2/c3)n + 3(cd)" nn + 2n + (Abc)n = o, (i6) 



3 

 where 



A( ) s (d/dx)log( ) 



(17) 



(note that Ac = SjAd) . Equation (16) , which appears 

 to have been derived originally by Shuto (1974) , 

 reduces to the KdV equation if b and d are constant. 



The vertically averaged, horizontal velocity is 

 given by 



u = (gn/c) [l+0(a)] , 



(18) 



whilst the vertical velocity is 0(a'5u). The mass, 

 momentum, and energy of the wave therefore are given 

 by 



CO 00 CO 



M = pbc I nds, M = pbcd | uds = Mc, E = pgbc jn^ds, 



(19a, b,c) 



within 1+0 (a). The limits of integration may be 

 replaced by ±hT for a wave of period T. 



Multiplying (16) through by hihc)^ and ben, 

 respectively, and integrating over -" < s < ■» on 

 the assumption that ri , ri^, and riss vanish in the 

 limits, we obtain the integral invariants 



'"I 



I = (be) nds, J = be n ds. 



(20a, b) 



It follows that E = pgj is conserved. On the other 

 hand. 



h 1 h 



M = pl(bc) and M = pl(bc^) 



(21a, b) 



Y(x) = 2(cd/aoj)x'(x) 



(24a) 



such that the phase speed of the wave is given by 

 C- = -St/^x " c/n-hyU/d] = [g(d+Ya)]'^. (24b) 



Conservation of mass and energy imply the constraints 

 (see Section 3) 



<N> = 0, a^bcT<N2> = J, 



(25a, b) 



where < > implies an average over a 2tt interval of 

 8 and J is the integral invariant obtained through 

 the substitution of (23) into (20b) . 



A formal, asymptotic development of the descrip- 

 tion (23) may be obtained by expanding N(e,x) and 

 Y(x) in powers of an appropriate measure of the 

 slow variation of b and d and invoking (25a) and 

 the requirement that the period of 9 be 2v . The 

 first approximation, which is obtained by substit- 

 ing (23) into (16) and then neglecting all 

 derivatives with respect to the slow variable x, 

 corresponds to that for a cnoidal wave [Lamb (1932, 

 §253)]. It may be placed in the form 



N = cn2[ (K/iT)e|m] - <cn2> , <cn2> = [m-l+(E/K) ] /m, 



(26a, b) 



Y = [2-m-3(E/K)]/m, aL/d^ = (16/3)mK2 5 U(m),(26c,d) 



where cn(u|m) is an elliptic cosine of modulus nn 

 and K and E are complete elliptic integrals in the 

 notation of Abramowitz and Stegun (1955), and U(m) 

 is the local Ursell parameter. Substituting (26) 

 into (25b) , we obtain 



JL^/V(bdV2) = u2<i^2> = p{„ 



(27a) 



where 



[2(2-m) (E/K) - 3(E/K)' 



so that, except for special combinations of b and 

 d, M and M are conserved only if /"„nds = 0. Non- 

 conservation of momentum is acceptable in consequence 

 of the horizontal thrust exerted on the fluid by 

 the bottom and walls of the channel, but non- 

 conservation of mass is generally unacceptable. 



We remark that the neglect of both dispersion 

 and nonlinear ity, as represented by the first and 

 second terms, respectively, in (16), yields Green's 

 law, (bcj'^n = f(s), where f is an arbitrary function 

 of the characteristic coordinate, s. 



4. SLOWLY VARYING CNOIDAL WAVE 



Theory 



Kinematical and scaling considerations suggest that 

 an approximate solution of (16) for a wave of pre- 

 scribed period 



T = 2Tr/oj = (L/g)'^ (22) 



be posited in the form 



n(s,x) = a(x)N(e,x), e = us - xix) , (23a, b) 



where 6 and x are fast and slow variables, a(x) is 

 a slowly varying amplitude, and x^^) is a slowly 

 varying phase shift. It also is expedient to 

 introduce 



(l-m)]/(3m2) 



{27b) 



and 



.'* ,,3 



F = (4/3 )K2[2(2-m)EK - 3e2 - (l-m)K2]. (27c) 



It follows from (27), which determines m(x), that 

 m is constant if and only if bd ' ^ = constant, in 

 which special case (23) , (26) , and (27) constitute 

 an exact similarity solution of (16). 



The results (26a) and (27a) provide a parametric 

 relation between aL/d^ and JL^/^/bd^' ^ that may be 

 graphically represented as a plot of log F vs log (J 

 [see Miles (1978b) ] . The case of constant depth is 

 especially simple in that the plot of log F vs log 

 U is equivalent to -log b vs log a. The limiting 

 relations 



^2 , 



(8J) ^^(Ld)-^ (" ' °> 



and 



f ,4,,. 3/2 32/32/31 (U + 



F - (-U) , a - ^ b d 



(28a, b) 



(29a, b) 



intersect at U = 150 and provide rough approximations 

 for (J> 150. 



The preceding calculation is a generalization of 

 that of Svendsen and Brink-Kjaer (1972) , who consider 

 the one-dimensional (b = constant) shoaling problem; 

 however, they replace 9 + ut in (23b) by the 

 equivalent of [1 - '5Y(a/d] (x/c) , which is clearly 



