ABSTRACT
This chapter deals with some of the basic elements of Fourier analysis. It provides a short historical introduction to the early developments in this field and a discussion of the mathematical analysis of sound. The chapter presents Fourier representations of functions and a number of examples. It discusses the Gibbs oscillation phenomenon. Daniel Bernoulli work led to considerable controversy at that time, and many of his contemporaries, including Euler and d'Alembert, viewed as untenable his claim to have found a general solution to the vibrating string problem. Their objections stemmed from the obvious fact that if one were to accept Bernoulli's arguments, then it would necessarily follow that the initial position of the string could be represented as an infinite sum of sine functions, which, as Euler observed, would have to be periodic and odd. The chapter examines a number of Fourier series representations of functions, and describes graphically how the lower order partial sums approximate the given functions.
