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

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http://dx.doi.org/10.1090/gsm/102/01