By Karlheinz Spindler
A accomplished presentation of summary algebra and an in-depth therapy of the functions of algebraic thoughts and the connection of algebra to different disciplines, comparable to quantity thought, combinatorics, geometry, topology, differential equations, and Markov chains.
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Additional resources for Abstract algebra with applications: in two volumes
Barre´ de Saint-Venant2). 1, is the equation for conservation of momentum, derived from Newton’s3 second law of motion, stating that the acceleration of an object is equal to the balance of forces, in this case the component of gravity in the direction of flow and friction. 1 is the Eulerian4 acceleration term while the second term is the convective acceleration term. The coefficient S accounts for the shape of the channel. The more irregular a cross section and the more the variation in flow velocity over the cross section, the larger the S.
Hence the dynamic equilibrium that is reached as a balance between erosion and sedimentation lies nearer to the maximum eroded profile than to the minimum (silted up) profile. Thus the width of a stream mainly reflects the situation of its maximum eroding capacity. In a river the width is determined by the bankfull flow, in an estuary by the peak spring-tidal discharge. Observations in excavated tidal canals in Indonesia (at Karang Agung, in the Banyuasin estuary, South Sumatra where the author carried out field surveys in 1989) illustrate this process.
Analytical solutions not only provide insight into the processes at play, more importantly, they provide a means for verification or falsification. This chapter describes the hydraulic equations of alluvial estuaries where there is a close interaction between geometry and flow, mutually influencing each other in continuous feedback. As a result, a regular topography appears in which mathematical laws can be discerned that can be described by surprisingly simple analytical equations. In combining the conservation of mass and momentum equations with the topography of an alluvial estuary, a number of analytical equations are derived for: 1) tidal propagation, 2) tidal damping, 3) tidal amplification, 4) wave celerity, 5) phase lag, and 6) the influence of river flow on tidal damping.