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Unformatted text preview: Chapter 16 Shallow Water Equations The shallow water equations model tsunamis and waves in bathtubs. This chapter is more advanced mathematically than earlier chapters, but you might still find it interesting even if you do not master the mathematical details. The shallow water equations model the propogation of disturbances in water and other incompressible fluids. The underlying assumption is that the depth of the fluid is small compared to the wave length of the disturbance. For example, we do not ordinary think of the Indian Ocean as being shallow. The depth is two or three kilometers. But the devastating tsunami in the Indian Ocean on December 26, 2004 involved waves that were dozens or hundred of kilometers long. So the shallow water approximation provides a reasonable model in this situation. The equations are derived from the principles of conservation of mass and conservation of momemtum. The independent variables are time, t , and two space coordinates, x and y . The dependent variables are the fluid height or depth, h , and the twodimensional fluid velocity field, u and v . With the proper choice of units, the conserved quantities are mass, which is proportional to h , and momentum, which is proportional to uh and vh . The force acting on the fluid is gravity, represented....
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 Spring '11
 Adams
 Advanced Math, Equations

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