Journal of Applied Mathematics
Volume 2005 (2005), Issue 3, Pages 259-271

On the convective nature of roll waves instability

C. Di Cristo1 and A. Vacca2

1Dipartimento di Ingegneria Idraulica ed Ambientale, Universitá di Napoli Federico II, Via Claudio 21, Napoli 80125, Italy
2Dipartimento di Ingegneria Civile, Seconda Universitá di Napoli, Via Roma 29, Aversa (Ce) 81031, Italy

Received 17 November 2004; Revised 25 January 2005

Copyright © 2005 C. Di Cristo and A. Vacca. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A theoretical analysis of the Saint-Venant one-dimensional flow model is performed in order to define the nature of its instability. Following the Brigg criterion, the investigation is carried out by examining the branch points singularities of dispersion relation in the complex ω and k planes, where ω and k are the complex pulsation and wave number of the disturbance, respectively. The nature of the linearly unstable conditions of flow is shown to be of convective type, independently of the Froude number value. Starting from this result a linear spatial stability analysis of the one-dimensional flow model is performed, in terms of time asymptotic response to a pointwise time periodic disturbance. The study reveals an influence of the disturbance frequency on the perturbation spatial growth rate, which constitutes the theoretical foundation of semiempirical criteria commonly employed for predicting roll waves occurrence.