Bibliography: p. 181-187.
|Series||Lecture notes in engineering ;, 15|
|LC Classifications||TC172 .K56 1986|
|The Physical Object|
|Pagination||xxv, 187 p. :|
|Number of Pages||187|
|LC Control Number||85027662|
1. 1 AREAS OF APPLICATION FOR THE SHALLOW WATER EQUATIONS The shallow water equations describe conservation of mass and mo mentum in a fluid. They may be expressed in the primitive equation form Continuity Equation _ a, + V. (Hv) = 0 L(l;,v;h) at (1. 1) Non-Conservative Momentum Equations a. 1. 1 AREAS OF APPLICATION FOR THE SHALLOW WATER EQUATIONS The shallow water equations describe conservation of mass and mo mentum in a fluid. They may be expressed in the primitive equation form Continuity Equation _ a, + V. (Hv) = 0 L(l;,v;h) at (1. 1) Non-Conservative Momentum Equations a M("vjt,f,g,h,A) = at(v) + (v. Equations (Bpf2) and (Bpf4) constitute the planar, shallow water wave equations. This set of two, non-linear, partial diﬀerential equations need to be solved for the unknown functions, u(x,t)andH(x,t)given appropriate initial and/or boundary conditions. Commonly, the method of characteristics is used to solve these equations numerically. sidered phenomena. However, in many other cases, nonlinear wave equations emerge even in rst order approximations to more general sets of fundamental equations describing the dynamics of a given system. In this book, we focus on the shallow water problem, in particular on solutions to equations which go beyond the Korteweg-de Vries equation.
3 Specify boundary conditions for the Navier-Stokes equations for a water column. 4 Use the BCs to integrate the Navier-Stokes equations over depth. In our derivation, we follow the presentation given in  closely, but we also use ideas in . C. Mirabito The Shallow Water Equations. Waves in the shallow water system There are various diﬀerent types of small amplitude wave motions that are solutions to the shallow water equations under diﬀerent circumstances. These waves in the shallow water system behave in a similar manner to those that occur in . Water depth h trough Wave height H SWL l Must satisfy FSBC, substitute ` into equation (3): ¡!2 coshkh+gksinhkh = 0; which gives: Deep water waves Intermediate depth Shallow water waves or short waves or wavelength or long waves kh >> 1 Need to solve!2 = gktanhkh kh. The shallow water equations and one-dimensional wave propagation. Traditional discussions of hydraulic effects such as those found in engineering text books are often based on analyses of steady flows. At the same time, interpretation of these effects almost always involves waves and wave propagation. We therefore.
positive water depth. In this non-hydrostatic wave model development, the fractional time step method is adopted. The shallow water equations without non-hydrostatic pressure terms are solved for approximation of velocity; a tri-diagonal equation for non-hydrostatic pressure terms is then solved, and the approximate velocity is corrected by. in the horizontal plane during wave passage (for f>0). However, unlike the atmosphere, the shallow water system is two-dimensional, so propagation of Poincaré waves is purely horizontal. In the non-rotating case (f = 0), the wave phase speed is!=k = c, so non-rotatingwavesarenotdispersive. The shallow water system is our first example of a nonlinear hyperbolic system; solutions of the Riemann problem for this system consist of two waves (since it is a system of two equations), each of which may be a shock or rarefaction (since it is nonlinear). Shallow Water Solitary Wave Wave Solution Internal Wave Water Wave These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.