Equação da onda unidimensional e bidimensional: um estudo das soluções analíticas

Abstract

Waves are disturbances in a medium, which move or propagate themselves in space, transporting energy from one point to another. They can be: mechanical waves, electromagnetic waves or matter waves. The theme involving waves is of great importance in the modern world, being present in civil construction, diagnostic medicine, geology, communications, music, among other areas. In the example of the music industry, the control of musical waves has a great influence on its economic growth. The analysis of the relationship between the materials that musical instruments are made and how the sound waves produced by these instruments behave provides the possibility of generating sounds with better quality. The basis of those analyzes is the behavior of the sound waves produced. In this sense, the study of waves is of great importance to obtain the best musical quality, in the same way, if we analyze other areas where the waves are present, we will reach the same conclusion regarding their importance. Considering these facts, this master's dissertation aims to bring the study of the Wave Equation starting from basic knowledge, indispensable to anyone wishing to develop the study of the analytical solutions of this equation. Therefore, this work studies the techniques of analytical solution of the One-Dimensional and Two-Dimensional Wave Equation, starting with the modeling of the Wave Equation performed from the application of Newton's law. After this modeling, it addresses some prerequisites for the resolution of the Wave Equation, the definition of EDP, some of its classifications, and then the resolutions are described by the Separation of Variables and Classical Integral Transform Technique methods. Finally, analyzes of the One-Dimensional Wave Equation model are performed based on the use of graphs and some considerations about the Two-Dimensional Wave Equation.