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Using Mathematica to Enhance Learning of Oceanographic Process: Breaking of Waves and Burger's Equation
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| Organization: | U.S. Naval Academy |
| Department: | Department of Mathematics |
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| United States Naval Academy, Annapolis, MD |
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Traditionally, the undergraduate curriculum in Advanced Engineering Mathematics is heavily geared towards the applications of linear mathematics to the standard engineering and science models. This is especially the case in the treatment of the solutions of linear partial differential equations, where by using normal modes and Fourier series one takes advantage of the student's familiarities with eigenvalues and eigenvectors and the principles of superposition to build the solutions of such equations. Nonlinear partial differential equations are often shunned because the above analogies generally break down and one needs to start with different building blocks. A notable exception is the method of characteristics for hyperbolic differential equations whose effectiveness is documented for both linear and nonlinear equations. Because hyperbolic equations are the primary examples of equations that support wave propagation, these equations enjoy a special status in oceanography, especially in the context of underwater acoustics and wave formation in shallow coastal waters. In this paper we consider the Burger's equation as a prototypical hyperbolic partial differential equation and describe some aspects of its solutions and their behavior using Mathematica. We will outline some simple programs in the language of this software that demonstrate how waves break in this model and how one is able to predict the time of "blow-up" of a solution by measuring some of the geometric parameters in the initial perturbation that forms the wave.
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