CHAPTE R 1 FOURIE R SERIE S O N TH E CIRCL E 1.1 MOTIVATION AND HEURISTICS 1.1.1 Motivation from Physics Two major sources of Fourier series are the mathematical models for (i) the vibrating string and (ii) heat flow in solids. /. 7. /. 7 The vibrating string The first systematic use of trigonometric series can be found in the work of Daniel Bernoulli (1753) on the vibrating string. A simple harmonic motion of a string of length n is defined by the formula (1.1.1) f{x, t) =Asinnxcosto a) for suitable constants A, a, and n = 1, 2,.... A is the amplitude, n is the angular frequency, and a is the phase shift. The simple harmonic motion is a solution of the differential equation ftt = fxx, which is supposed to describe the small transverse displacement fix,t) of a tightly stretched string whose ends are fixed at x = 0 and x = n. More complex, multiple harmonic motions are obtained by linear superposition N (1.1.2) fix, t) = Y^Arc smnxcosint an). n=\ Functions of this form can be used to satisfy a variety of initial conditions, if we are given the values of / and the partial derivative df/dt when t = 0. This is possible whenever fix, 0) and df/dtix, 0) are expressed as finite linear com- binations ^2n=lansinnx. This may be less obvious in other cases for example 1
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