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Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case
Author(s) -
Sachdev P. L.,
Vaganan B. Mayil
Publication year - 1994
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1994933251
Subject(s) - free surface , mathematics , mathematical analysis , constant (computer programming) , flow (mathematics) , plane (geometry) , shear flow , nonlinear system , surface (topology) , compressibility , classical mechanics , mechanics , physics , geometry , quantum mechanics , computer science , programming language
A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well‐known self‐similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.