514 



•+ + + 



n +(h + n)u_^ = o , 



X 



•+ * + 1 ^ + 



u +gn,+— hn 



+ 3 + + + 

 X X t t 



+ 



-2ari - a^n 



+ + + + + + 



X X t XXX 



(95) 



The first of (95) is of the same form as (88) i and 

 hence invariant but clearly the second of (95) dif- 

 fers from {88)2. This means that the character of 

 the solutions of (88) , (89) and (90) is substantially 

 altered by superposing a constant rigid body trans- 

 lational velocity on the fluid, which is contrary 

 to what happens if we use the full three-dimensional 

 equations of motion for an inviscid fluid. On the 

 other hand, the set of equations (87) is not subject 

 to this drawback, and the equations do have useful 

 steady state solutions. It may be argued that be- 

 cause of the nature of the approximation in obtain- 

 ing (88) to (90) from the three-dimensional theory 

 we should not expect these equations to be invariant 

 under a superposed constant translational velocity, 

 but this then leaves in doubt which version of any 

 of the sets (88) to (90) are to be chosen as basic. 

 The difficulty disappears if we linearize any of 

 the above sets since the resulting equations are 

 then invariant under a small superposed constant 

 translational velocity, as we would expect. 



From the above discussion, it might appear that 

 the equations (87) may be preferable to any of (88) 

 to (90) , but arguments are put forward by Whitham 

 (1974, p. 462) to suggest that the system (88) is to 

 be preferred to (87) . Although considerable use has 

 been made of some of the equations (87) to (90) , it 

 would seem that they all rest on a somewhat shaky 

 physical foundation. By contrast, the system of 

 equations (71) and (80) do not possess the undesir- 

 able features of the type noted above: they are 

 properly invariant under superposed rigid body mo- 

 tions, admit general steady state solutions, and are 

 free from anomalies mentioned earlier. 



For the purpose of providing a more explicit com- 

 parison with the system of equations (87) to (90) , 

 we specialize the system of equations (83) to that 

 for a horizontal bottom for which (85) j 2 3 k hold. 

 Then, denoting again the elevation of the stream by 

 h + n, the differential equations (83) j 2 '^^'^ ^^ ^^~ 

 corded in the form 



n + (h +ri)u = 

 x 



u + g n^ + J hn^^^ 



(96) 



that the nonlinear equations (96) j 2 ^^^ invariant 

 under a constant superposed rigid body translation 

 while (88)1 2 3^s not.^t Within the scope of the 

 nonlinear theory, it does not seem reasonable to 

 neglect the quantity, R, in (96)2 on the basis of 

 either physical considerations or mathematical argu- 

 ments. It may be, however, that in some special 

 circumstances the solution of (88) is a good approxi- 

 mation to the solution of (96) , but this is a dif- 

 ferent question than that discussed above. In this 

 connection, it is worth noting that a solution to a 

 system of differential equations, which results from 

 neglecting certain terms in a more general system of 

 equations, in general, will not be the same as a 

 solution obtained by approximation from a correspond- 

 ing solution of the more general system of equations. 

 We close this section by calling attention to some 

 available evidence of the relevance and applicability 

 of the direct formulation for fluid sheets. The sys- 

 tem of equations (71) and (81) , or a special case of 

 it, has already been employed in some detailed stud- 

 ies of a number of two-dimensional problems of in- 

 viscid fluid sheets, as well as in some comparisons 

 with known previous solutions on the subject. We 

 mention here some of these studies and refer the 

 reader to the papers cited for additional informa- 

 tion: (a) the nonlinear differential equations admit 

 a solitary wave solution [see Green et al. (1974c)] 

 which is the same as that attributed by Lamb (1932, 

 Section 252) to Boussinesq and Rayleigh; (b) this 

 solitary wave solution, as well as appropriate jump 

 conditions and certain results derived from the 

 energy balance for an inviscid fluid sheet at con- 

 stant temperature [Green and Naghdi (1976a, Appen- 

 dix) ] , has been used by Caulk (1976) to discuss the 

 flow of an inviscid incompressible fluid under a 

 sluice gate; (c) the steady motion of a class of 

 two-dimensional flows in a stream of finite depth 

 in which the bed of the stream may change from one 

 constant level to another, and the related problem 

 of hydraulic jumps, both for homogeneous and non- 

 homogeneous incompressible fluids [Green and Naghdi 

 (1976a, Section 7) and Green and Naghdi (1977)],- 

 and (d) a class of exact solutions [Green and Naghdi 

 (1976a, Section 9) ] which characterize the main fea- 

 tures of the time-dependent free surface flows in 

 the three-dimensional theory of incompressible in- 

 viscid fluids [Longuet-Higgins (1972) ] . 



ACKNOWLEDGMENT 



The results reported here were obtained in the course 

 of research supported by the U.S. Office of Naval 

 Research under Contract N00014-76-C-0474 , Project 

 NR 062-534, with the University of California, 

 Berkeley . 



where 



I ^\tt 



1 1 



'x^tt 



. , [(f)2(2u(t. . +u2<|) + u. (j) + uu ()) ) ] 

 3 $ xt XX t X XX X 



h + n . (97) 



REFERENCES 



Benjamin, T. B., J. L. Bona, and J. M. Mahony (1972). 



Model equations for nonlinear dispersive systems. 



Phil. Trans. Royal Soc . Land. A272, 47. 

 Bogy, D. B. (1978) . Use of one-dimensional Cosserat 



theory to study instability in a viscous liquid 



jet. Phys. Fluids 21, 190. 



Clearly if R on the right-hand side of (96)2 c^" ^^ 

 neglected, then (96) j 2 reduce to those of Boussinesq 

 given by (88)1 2- ^t should be emphasized, however. 



++ 



^^The difference between the two disappears, of course, upon 



linearization. 



