Journal of Engineering Mathematics 40: 17–42, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands. A derivation of the Green-Naghdi equations for irrotational flows J. W. KIM, K. J. BAI 1 , R. C. ERTEKIN and W. C. WEBSTER 2 Department of Ocean and Resources Engineering, SOEST, University of Hawaii, Honolulu, HI 96822, USA (e-mail: kjw@oceaneng.eng.hawaii.edu; ertekin@hawaii.edu) 1 Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul, 151-742, Korea (e-mail: kjbai@gong.snu.ac.kr) 2 Department of Civil Engineering, University of California, Berkeley, CA 94720, USA (e-mail: wwebster@socrates.berkeley.edu) Received 23 September 1998; accepted in revised form 30 June 2000 Abstract. A new derivation of the Green-Naghdi (GN) equations for ‘sheet-like’ flows is made by use of the principle of virtual work. Divergence-free virtual displacements are used to formulate the momentum equations weakly. This results in the elimination of the internal pressure from the GN equations. As is well-known in particle dynamics, the principle of virtual work can be integrated to obtain Hamilton’s principle. These integrations can be performed in a straightforward manner when the Lagrangian description of fluid motion is adopted. When Hamilton’s principle is written in an Eulerian reference frame, terms must be added to the Lagrangian to impose the Lin constraint to account for the difference between the Lagrangian and Eulerian variables (Lin). If, however, the Lin constraint is omitted, the scope of Hamilton’s principle is confined to irrotational flows (Bretherton). This restricted Hamilton’s principle is used to derive the new GN equations for irrotational flows with the same kine- matic approximation as in the original derivation of the GN equations. The resulting new hierarchy of governing equations for irrotational flows (referred to herein as the IGN equations) has a considerably simpler structure than the corresponding hierarchy of the original GN governing equations that were not limited to irrotational flows. Finally, it will be shown that the conservation of both the in-sheet and cross-sheet circulation is satisfied more strongly by the IGN equations than by the original GN equations. Key words: IGN equations, irrotational flow, Hamilton’s principle. 1. Introduction Since the discovery of solitary waves by Russell [1] there has been an increasing number of theories each using different approaches to predict the evolution of long nonlinear waves. Long waves can be considered a subset of more general sheet-like or ‘thin’ flows where one characteristic dimension of the physical problem is considerably smaller than the other two in a three-dimensional space. There are basically three major approaches to the derivation of long-wave equations. In the classical approach, conservation of mass and conservation of linear momentum equations form the equations of motion for all fluid particles throughout the continuum. These equations (Navier–Stokes) are approximated by use of several basic assumptions such as the inviscid-fluid and irrotational-flow assumptions. At this point two major scales, namely the dominant-length scale and the dominant-amplitude scale, are introduced. These scales or perturbation parameters are then introduced into approximate equations of motion and into the solid- and free-surface boundary conditions. The velocities and the free-surface elevation, both being unknown, are expanded into a perturbation series ordered in terms of the above scales. Then, one decides which terms in this expansion are retained and which ones are