Ondas Estacionárias: corda vibrante de um violão

Abstract

The present undergraduate final work has as its theme the Partial Differential Equations (PDE’s), more precisely the wave equation, which can be used to model the vertical displacement of a guitar string, for example, which remains fixed in the extremities and experiences a change, in order to enter a stationary vibratory pattern. Thus, we aim to establish a rule that governs the displacement u(x, t) – at any point x and in an instant of time t > 0 – of a guitar string of length L, stretched, flexible and fixed at both ends when it goes into vibration mode. As a theoretical foundation for solving the problem, it was necessary to retake and expand the concepts of Differential and Integral Calculus, Linear Algebra, Ordinary Differential Equations and Fourier Series, in addition to the appropriation of new knowledge and methods inherent to partial differential equations. Methodologically, in order to solve the research problem, namely, a² uxx = utt u(0, t) = u(L, t) = 0, t > 0 u(x,0) = f (x), ut(x,0) = g(x), 0 ≤ x ≤ L. it was necessary to rewrite it in the form of two auxiliary problems, A and B, solving them separately, through the following 3 steps: transform the wave equation (Partial Differential Equation) into two Ordinary Differential Equations (ODE’s) using the Variable Separation method; use the boundary conditions and one of the initial conditions in order to get a regular Sturm-Liouville problem, obtaining solutions through eigenvalues and eigenfunctions; use the principle of superposition, coupled with the last initial condition, to recognize coefficients of a Fourier Sine Series, obtaining respective solutions to the problems A and B. The sum of the solutions to such problems is then recognized as the solution to the research problem, requiring some assumptions about f and g.