Abstract
The paper deals with long water waves propagating in a straight canal of constant depth and variable section. In the formulation of this problem, a simplified, one-dimensional model is considered that is based on the assumption of a “columnar” fluid motion. To this end, a system of material coordinates is employed as independent variables in the description of this phenomenon. The main attention is focused on transient solutions corresponding to a fluid motion starting from rest.With respect to the initial value problem considered,we confine our attention to a finite domain fluid motion induced by a piston-type generator placed at the beginning of the canal. For a finite elapse of time, measured from the starting point, the solution in the finite fluid area mimics a solution within an infinite domain, inherent for wave propagation problems. The main goal of our investigations is to describe the evolution of the free surface (the wave height) at the smallest section of the canal. Numerical examples are provided to illustrate the model formulation developed in this paper. The accuracy of this approximate description is assessed by comparing its results with data obtained in hydraulic experiments performed in a laboratory flume.