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