The Boussinesq Regime for waves 

 in a Gradually Varying Channel 



John Wilder Milas 

 University of California 

 San Diego, California 



ABSTRACT 



The Boussinesq equations for gravity waves of ampli- 

 tude a(x) and characteristic length l{x) in a 

 gradually varying channel of breath b(x) and depth 

 d(x) are derived from Hamilton's principle on the 

 assumptions that a/d E a << 1, (d/5,) ^ = 0(a), b'(x) 

 = 0(a3/2b/d) and d'(x) = 0(0^/2) (' e d/dx) . The 

 further assumption of unidirectional propagation 

 then leads to the Korteweg-deVries equation for a 

 gradually varying channel. It is shown that the 

 latter equation admits two integral invariants. 

 The second-order (in amplitude) invariant measures 

 energy, as expected, but the first-order invariant 

 measures mass divided by b'id'S ; accordingly, mass 

 is conserved only if either the first-order invariant 

 vanishes identically or bd'5 is constant, and only 

 the former possibility appears to be consistent 

 with conservation of energy. An approximate solution 

 for a cnoidal wave, which conserves both energy and 

 mass, is developed. The corresponding approximation 

 for a solitary wave (which may be regarded as a 

 limit of a cnoidal wave) does not conserve mass but 

 nevertheless provides an approximation to the evolu- 

 tion of the amplitude, a = b~^/^d~-^, that is in 

 agreement with experiments for gradual decrease of 

 depth or increase of breadth but not for decrease 

 of breadth. 



1. INTRODUCTION 



The Boussinesq regime for gravity waves of amplitude 

 a and characteristic length Si in water of depth d 

 is characterized by 



a = a/d << 1, B 



(d/J,)- 



<< 1, 



0(a), (la,b,c) 



where a and g are measures of nonlinearity and 

 dispersion, respectively, and (Ic) refers to the 

 asymptotic limit a ■!■ 0. The assumptions of one- 

 dimensional wave motion and uniform depth and the 



neglect of compressibility and viscosity then imply 

 Boussinesq' s equations for the free-surface displace- 

 ment and the depth-averaged velocity, ri(x,t) and 

 u(x,t). The further assumption of undirectional 

 propagation pe3nnits the elimination of u to obtain 

 the Korteweg-deVries (KdV) equation for n. The 

 classical derivations are given by Whitham (1974, 

 §13.11). An alternative derivation, starting from 

 the Luke-Whitham variational principle and using 5> 

 the velocity potential at the free surface, and ri 

 as dependent variables also has been given by Whit- 

 ham (1967) . 



I consider here the generalization of the 

 Boussinesq and KdV equations for a channel of grad- 

 ually varying breadth and depth b (x) and d(x) and 

 their approximate solution for slowly varying 

 cnoidal an(J solitary waves. I begin (in Section 2) 

 by deriving (what may be called) the Boussinesq chan- 

 nel equations directly from Hamilton's principle (to 

 which the Luke-Whitham variational principle is 

 equivalent in the present context) on the basis of 

 (1) and the further assumptions (which imply 

 gradually varying) 



b'(x) 



0(a /^b/d) 



d'(x) = 0(a /2) 



(2a, b) 



I then (in Section 3) invoke the hypothesis of uni- 

 directional propagation to obtain the KdV channel 

 equation, which was developed originally by Shuto 

 (1974) through a rather more involved procedure. 

 I then go on to consider cnoidal waves in Section 

 4 and the solitary wave in Section 5 on the basis 

 of the stronger assumptions 



<< a 



V2 



(b/d) 



(3a, b) 



A prominent feature of the KdV equation for a 

 uniform channel is the existence of an infinite 

 number of integral invariants (Whitham, 1974, §17.6). 

 The KdV equation for a slowly varying channel admits 

 only two such invariants, of first and second order 

 in the amplitude; the latter measures energy, as 



523 



