ABSTRACT
The subject of partial differential equations, and in particular the study of wave equation, had its beginning in eighteenth century. The method of images is used with D'Alembert's formula to solve several problems for the semi-infinite and finite strings, and to prove certain maximum principles which are needed to analyze the error of a solution due to an error in the initial conditions. This chapter derives D'Alembert's formula for the solution of initial value problems for the infinite string. It shows that under certain ideal conditions, a solution of the wave equation can be interpreted as the time—dependent profile (or amplitude) of a vibrating string. The chapter suggests that the basic method for solving the heat equation carries over to the wave equation. It show that, in conjunction with Newton's law, the assumption that the string admits nontrivial transverse vibrations actually implies that the string is linearly elastic for s in any range which is encountered during such a vibration.
