Calculations of the development of an undular bore

TLDR

In shallow water, a long elevation wave steepens into a bore, which is undular when its surface rise is less than 0.28 of the water depth, and the governing equations are of the same order needed to derive solitary waves. The study investigates the growth of an undular bore originating from a long wave that creates a gentle transition between uniform flow and still water. The authors provide a physical account of the bore’s development and present numerical results obtained with finite‑difference approximations to the governing partial differential equations. The numerical results agree generally with experimental measurements.

Abstract

If a long wave of elevation travels in shallow water it steepens and forms a bore. The bore is undular if the change in surface elevation of the wave is less than 0·28 of the original depth of water. This paper describes the growth of an undular bore from a long wave which forms a gentle transition between a uniform flow and still water. A physical account of its development is followed by the results of numerical calculations. These use finite-difference approximations to the partial differential equations of motion. The equations of motion are of the same order of approximation as is necessary to derive the solitary wave. The results are in general agreement with the available experimental measurements.