sionless parameters. It is found that the problem of obtaining a steady solution in a coupled-mode model of shallow-water wave evolution is related more to the.
INTERCOMPARISON O F TRUNCATED SERIES SOLUTIONS FOR SHALLOW W A T E R W A V E S By James T. Kirby, 1 Member, ASCE ABSTRACT: The relationship between truncated Fourier series solutions of several different long-wave evolution equations is explored. The models are all designed to represent the same physics in the asymptote of small nonlinearity and frequency dispersion, and yet give different numerical solutions for given values of dimensionless parameters. It is found that the problem of obtaining a steady solution in a coupled-mode model of shallow-water wave evolution is related more to the problem of properly choosing the corresponding time-dependent evolution equation than to the problem of truncating the infinite series representation of the solution to that equation. In the process, a particular lowest-order coupled-mode model is related to a particular modified form of the Korteweg-deVries equation. It is shown that the existing lowest-order model may be inherently inaccurate in the case of relatively high waves, due to the nonphysical behavior of the form of the nonlinear term inherent in that model. Finally, solitary and cnoidal wave solutions are given for the entire family of modified Korteweg-deVries equations, and their properties are compared. INTRODUCTION
There has recently been a great deal of interest in solving the Boussinesq equations for weakly dispersive, weakly nonlinear shallow-water waves in the frequency domain, as a means of describing the shoreward evolution of complex nonlinear wave fields. In related studies, Freilich and Guza (1984) modeled the evolution of a broad spectrum of waves with vanishingly small directional distribution using a set of coupled-mode, ordinary differential equation (ODE) evolution equations, while Liu et al. (1985) used the parabolic approximation to model the two-dimensional spatial evolution of the spectrum over topography. Liu et al. applications were restricted to simple periodic waves with finite resolution of higher harmonics. When applying models of the form described here to laboratory data, it is desirable to choose initial conditions for the model integration that lead to the maintenance of steady wave forms in regions before the waves interact with topography or obstacles. Yoon and Liu (1989) have discussed this problem in a recent paper describing the Mach reflection of a cnoidal wave at a vertical wall. Yoon and Liu used a truncated spectral model similar to that described in Liu et al. (1985). In order to develop these models, we assume that the surface displacement r\(x, i) may be written in the form r\(x,t) = \ 2 An{x)ein{kx-M) + c.c
(1)
where the function A„(x) = the complex Fourier amplitude of the nth harmonic or spectral component of a periodic-in-time wave form. Further, h is 'Assoc. Prof., Ctr. for Appl. Coastal Res., Dept. of Civ. Engrg., Univ. of Delaware, Newark, DE 19716. Note. Discussion open until August 1, 1991. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on April 3, 1990. This paper is part of the Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 117, No. 2, March/April, 1991. ©ASCE, ISSN 0733-950X/91/00020143/S1.00 + $.15 per page. Paper No. 25660. 143
the still water depth; w is the wave angular frequency; and k is the linear, nondispersive wave number given by co2 = g^li
(2)
where g = the gravitational acceleration. Restricting attention to one-dimensional propagation, we write a model that is equivalent to the one given by Yoon and Liu, but in dimensional form, as
— - - nWA, + — S A>A-< + 2 7=1 S A*A«+
