Journal of Fluid Mechanics

Extensions of the mild-slope equation

D.  Porter a1 and D. J.  Staziker a1
a1 Department of Mathematics, The University of Reading, PO Box 220, Whiteknights, Reading, RG6 6AF, UK

Article author query
porter d   [Google Scholar] 
staziker dj   [Google Scholar] 


The use of the mild-slope approximation, which is invoked to simplify the problem of linear water wave diffraction-refraction by bed undulations, is reassessed by using a variational method. It is found that smooth approximations to the free surface elevation obtained by using the long-standing mild-slope equation are not consistent with the continuity of mass flow at locations where the bed slope is discontinuous. The use of interfacial jump conditions at such locations significantly improves the accuracy of approximations generated by the mild-slope equation and by the recently derived modified mild-slope equation. The variational principle is also used to produce a generalization of these equations and of the associated jump condition. Numerical results are presented to illustrate the main points of the theory.

(Published Online April 26 2006)
(Received April 5 1995)
(Revised June 27 1995)