513 



valid for incompressible, inviscid, and homogeneous 

 fluids. A more general derivation for propagation 

 of fairly long waves in a nonhomogeneous stream of 

 variable initial depth in which the mass density is 

 allowed to vary with depth is contained in a recent 

 paper of Green and Naghdi (1977) . 



In the case of incompressible inviscid fluid 

 sheets, the nonlinear equations for wave propagation 

 in water of variable depth can also be derived from 

 the three-dimensional theory: the procedure involves 

 the use of the (three-dimensional) equation for con- 

 servation of energy, the incompressibility condition, 

 invariance requirements under superposed rigid body 

 motions, along with a single approximation (21) for 

 the position vector. Then, by (6) and (21), the ap- 

 proximation for the (three-dimensional) velocity 

 field is given by 



= V + 



(86) 



where y and w in (86) have the same forms as those 

 in (67) . A derivation of this kind has been carried 

 out by Green and Naghdi (1976b). It is important, 

 however, to note that this derivation is limited to 

 incompressible inviscid fluids which do not require 

 constitutive equations. ^^ 



It is natural to ask what are the relationship and 

 advantages (if any) between the above system of equa- 

 tions and those which are currently employed by other 

 investigators. To provide a ready comparison, we 

 list below from Whitham (1974) alternative forms of 

 equations for water waves moving in the direction of 

 a fixed x-axis for a stream of initial constant depth, 

 h. Let the elevation of the stream be h + n • Then, 

 for unidirectional flow and in terms of n and the 

 horizontal velocity, u, we recall from Whitham 

 (1974, pp. 460-463) the system of equations 



nt + {u(h + n) } 







* 1 9 



J^ + uu + g n + -z c hn 

 t X X 3 XXX 



(87) 



and the pair of equations attributed to Boussinesq, 

 namely 



or an equation due to Benjamin et al. (1972) given 

 by 



t 2 h X 



1 9 



- ch^n ^ = 



5 xxt 



(90) 



As already remarked by Green and Naghdi (1977) , 

 it may immediately be verified that the set of equa- 

 tions (88) and (90) only have steady state solutions 

 if n and u are both constants. Also, although the 

 K.dV. Eq. (89) admits a solitary wave in which the 

 velocity at infinity is zero and the stream there is 

 at its undisturbed height, h, it does not admit a 

 steady state solution with u constant and n = at 

 infinity. This fact is related to another property 

 of (89) which is also shared by (88) and (90) : the 

 three sets of equations (88) to (90) are not invari- 

 ant in form under a constant superposed rigid body 

 motion of the whole fluid. To see this, suppose 

 that a constant superposed rigid body translational 

 velocity is imposed on the whole fluid so that the 

 particles at the place, x, are displaced to x"*" at 

 time, t , specified by 



t + a 



(91) 



where a and a are constants. The variables that oc- 

 cur in the differential equations (87) -(90) are n = 

 ri(x,t) and u = u(x,t). Let r]+ = n''"(x+,t+) and u+ = 

 u''"(x+,t"'") be the corresponding scalar quantities de- 

 fined over the region of space occupied by the fluid 

 after the imposition of the superposed rigid body 

 motion (91) j . Then, from (68) j and (8.6) we obtain 



u(x,t) = u (x , t ) - a 



u (x + at, t + a) 



(92) 



We expect the elevation, h + n , of the fluid to re- 

 main unaltered by superposed rigid body motions; and, 

 since h remains unaltered also, this leads us to re- 

 quire that 



n(x,t) = n (x ,t'*') = n*(x + at, t + a) . (93) 



ri + (h + ri)u = 

 x 



" ^ 'J \ + 3 ^^xtt = ° 



(88) 



where the notations in (87) and (88) are the same as 

 those in (70) , g is the acceleration due to gravity 

 introduced in Section 7 and c^ = g*h. Both systems 

 of equations (87) and (88) allow for wave propagation 

 in either direction along the x-axis. For waves mov- 

 ing along the positive x-direction only there is the 

 Korteweg-deVries (1895) equation — hereafter referred 

 to as the K.dV. equation — i.e.. 



From (92) and (93) , we calculate expressions of the 

 type 



xtt 



+ + 



1 + an 

 + + 



t X 



+ , + 9 + 



n + 2an + an 



+ + + + + + + + + 



Xtt xxt XXX 



n = n^ + un 

 t > 



+ + + 



n + u n 



(94) 



\ + -d + 2 



^) n + ^ ch^n = 

 h X 6 XXX 



(89) 



Recall that in the three-dimensional theory of incompress- 

 ible inviscid fluids the stress vector is specified in terms 

 of a pressure which is determined by the equations of motion 

 and the boundary conditions . 



with similar results for u^,Uj, and u in terms of u+ 

 and their derivatives. It was noted by Green and 

 Naghdi (1977) that if the independent variables, x, 

 t, in (88) to (90) are changed to (91) , the equations 

 for u,n in terms of x+,t"'" are different from those in 

 terms of x,t and this was illustrated explicitly with 

 reference to the K.dV. equation (89). Here, we con- 

 sider the pair of equations (88) j 2- After substi- 

 tuting (92) -(94), they become 



