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 在本論文中，在分析Shallow water equation 時，通常會先探討伴隨科氏力的三維Navier-Stokes Equation，接著會忽略垂直數度分布，將三維的Navier-Stokes Equation 推導成二維的Shallow water equation，並同乘以一個旋轉矩陣，使二維的Shallow water equation 消去source term。可在對稱空間中做座標轉換，進而使二維的Shallow water equation 成為一雙曲型系統的conservation laws。
 In this paper , we consider the following three dimension shallow water equations with Coriolis force then we derived the two dimension shallow water equation . And we transformed the two dimension shallow water equation 3by3 conservative system without source term . We can use the results in traditional hyperbolic systems of conservation laws to study the shock waves and rarefaction waves.
 1. Introduction.....................................12. Generalization to shallow water equations........43. Transformation of shallow water equations........73.1 The shallow water equations.....................73.2 Symmetric space.................................123.3 Traveling waves.................................184. The Rankine-Hugoniot condition...................22Reference...........................................23
 [1] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd edition.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemati-cal Sciences], 325. Springer-Verlag, Berlin, 2005.[2] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm.Pure Appl. Math. 18 (1965), pp. 697-715.[3] Jonathan Goodman, Zhouping Xin, Viscous Limits for Piecewise Smooth Solutions toSystem of Conservation Laws, Arch. Rational Mech. Anal. 121 (1992), pp. 235-265.[4] Benoit Cushman-Roisin,Jean-Marie Beckers, Introduction to Geophysical Fluid Dynamics,(2007).[5] J. M. Hong, C. H. Hsu, Y. C. Su, Global solutions for initial-boundary value problem ofquasilinear wave equations, J. Di. Equ. 245 (2008), pp. 223-248.[6] P. D. Lax, Hyperbolic system of conservation laws, II, Comm. Pure Appl. Math. 10 (1957),pp. 537-566.[7] J. Smoller, Shock Waves and Reaction Diusion Equations, Springer-Verlag, New York,Berlin (1983).[8] B. Whitham, Linear and nonlinear waves. New York, John Wiley, 1974.
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