ON THE DERIVATION AND SOLUTION OF ONE DIMENSION WAVE EQUATION

Abstract

Di erential equation is an equation which involves derivatives of unknown function as well as the function it self.The two basic DEs are ODE and PDE. ODE is an equation that the unknown function depends only on a single independent variable,while a PDE is an equation involving one or more partial derivative of function u, that depends on two or more variables.The 1D wave equation is one of the PDE that can be modeled by vibrating string, and solved analytically through ODE by method of separation variables.The statement of the problem we consider in this thesis is to determine the vibrations of string at any point x and any time t> 0 to model 1D wave equation @ 2 u @t 2 = c 2 @ 2 u @x 2 . The objective of thesis is to provide the basic derivation of 1D wave equation and its solution by using method of data analysis gathered from secondary sources would be studied ,analyzed and interpreted using examples of real world.The solution of 1D wave equation solved using Fourier series separation method in three steps is u n (x; t) = P 1 n=1 (Cn cos n t + Dn sin n t) sin n x L n=1,2,... Finally the shape of a string of length L plucked in the middle that has initial shape given in f (x) = 8 > < > : 0:1x; if 0 x 1 0:1(2 x); if 1 x 2: would be drawn using matlab.