524 



expected, but the former measures mass only if bd^ 

 = constant. This deficiency is presumbly a conse- 

 quence of the implicit neglect of the weak reflection 

 that accompanies the gradual variation of the channel: 

 the reflection coefficient for energy is second 

 order in some appropriate measure of the channel 

 variation and therefore has no cumulative effect, 

 whereas that for mass is first order and does have 

 a cumulative effect. The resulting difficulty may 

 be avoided for a wave that is either periodic or of 

 compact support simply by choosing a horizontal 

 reference plane such that the mean value of the 

 free-surface displacement vanishes identically (see 

 Section 4) , but the problem is more subtle for an 

 aperiodic disturbance of unlimited extent such as 

 a solitary wave (see Section 5) and remains unre- 

 solved. 



The primary goal , at least for practical applica- 

 tions, of the analysis of waves in a gradually 

 varying channel is the prediction of a as a function 

 of b and d. Green's law, which neglects both non- 

 linearity and dispersion, predicts [Lamb (1932, 

 §185)] 



a = b~ d 



(4) 



It is often used for practical shoaling calculations, 

 and Shuto (1973) finds that a = d'^S holds for 

 solitary waves on relatively steep slopes for a/d as 

 large as 2. On the other hand, the joint assumptions 

 of Boussinesq similarity (a/d = d /{. ) and conser- 

 vation of energy (which is proportional to a b£) 

 imply [Miles (1977a)] 



b-2/3d-l 



(5) 



Comparison with experiment (see Section 5) suggests 

 that (5) should be valid for a shoaling -or laterally 

 diverging channel if |6| < 0.1, where 



6 = (3a) 



-3/2 



[2d(b'/b) + 9d'] 



(6) 



but perhaps not for a laterally converging channel. 



The present results also have implications for 

 the approximate treatment of nonlinear wave propa- 

 gation along the lines initiated by Whitham (1974, 

 Ch. 8) in his treatment of shock-wave propagation 

 and since applied to solitary waves [Miles (1977a)], 



2. BOUSSINESQ CHANNEL EQUATIONS 



The boundary-value problem for gravity waves in an 

 ideal, homogeneous liquid may be deduced from 

 Hamilton's principle in the form [Broer (1974), 

 Miles (1977b)] 



6JJl{C, nldxdt = 0, L = £,^^ - jf{V<l,)^dy - |gn2,(7a,b) 



d 

 where x and y are horizontal and vertical coordinates; 

 6(x,t) and n(x,t) are the velocity potential at, and 

 the displacement of, the free surface; dx is an 

 element of area in the x space; d(x) is the quiescent 

 depth; and the velocity potential ((>(x,y,t) is 

 determined by 



(-d < y < n) 



(8) 



(t> + Vd-V(t. = (y = -d), ()> = C (y = n) . (9a, b) 



The solution of (8) and (9) is given by 



? - yV-(dVC) - |y2v2c + 0(B2c) 



(10) 



where P is defined by (lb) with d and i as scales 

 of y and x. The corresponding approximation to 

 the kinetic energy integral, after invoking n = 0(ad), 

 dVC = 0(6^50, e = 0(a), (2), and V-(AVB) = VA-VB + 

 AV^B, is 



n , 



/ (V(t>)2dy = (d+n) (VC)2 - id3(v2g)2 + £v [d3(v25)V5] 

 — d J J 



5-12 



+ 0(a3d 5 ) 



(11) 



Substituting (11) into (7) , invoking the further 

 approximation that (fi is independent of the transverse 

 coordinate in a channel of slowly varying breadth 

 b(x) and depth d(x), and integrating across the 

 channel, we obtain 



J-' t 2 x 6 XX 2 



bdxdt 



0. 



The corresponding Euler-Lagrange equations, 



^(bd^? ) + [b(d+n)C ] + bn, = 

 3 XX XX XX t 



and 



2 X 



+ gn 



(12) 



(13a) 



(13b) 



are counterparts of the Boussinesq equations [cf. 

 Whitham (1967)] . 



It is worth noting that the approximations to 

 this point are consistent with conservation of both 

 mass and energy: 



fnbdx = 0, 3 



i(d+n)C2 - |d352 + 1 ^2 

 2 x 6 XX 2 



bdx = 0, 

 (14a, b) 



where the integrals are over either (-■",") or a 

 periodic interval. The integral (14a) follows 

 directly from the integration of (13a) with respect 

 to X, subject to appropriate null or periodicity 

 conditions at the end points. The integral (14b) 

 may be similarly established or may be inferred 

 (through Noether's theorem) from the invariance of 

 the Lagrangian density in (12) under a translation 

 of t; it is an exact invariant of (13), but it 

 would be consistent with the antecedent approxima- 

 tions to approximate the specific energy in (14b) 

 by *5(d52 + gn2). 



3. KORTEWEG-DEVRIES CHANNEL EQUATION 



The Korteweg-deVries (KdV) equation for uni- 

 directional wave propagation in a uniform channel 

 may be deduced from the Boussinesq equations by 

 assuming that C and n are slowly varying functions 

 of t in a reference frame moving with the wave speed, 

 c. It is expedient in the present context to choose 

 X, rather than t, as the slow variable (since b and 

 d are prescribed as slowly varying functions of x) 

 and to introduce 



r dx _ 

 •J c(x) 



(c2 s gd) 



(15) 



as a characteristic variable. The direction of 

 propagation may be reversed by reversing the sign 

 of t in (15) . 



