Fourier series are used to represent periodic functions as the sum of simple oscillating functions like sines and cosines. This allows periodic functions, including discontinuous ones, to be broken down into their constituent frequencies or harmonics. Applications include representing sound waves, light waves, radio signals, and other physical phenomena involving wave motion or vibration. The Fourier coefficients determine the relative importance of each harmonic in the overall signal.

1. Fourier Series
By :
 Fahriyatie
 Imilda Fitriana

2. SIMPLE HARMONIC MOTION AND WAVE MOTION ;
PERIODIC FUNCTIONS
 We shall need much of the
 (2.1)
notation and terminology used
in discussing simple harmonic
motion and wave motion.
 Let particle P (figure 2.1)
move at constant speed around
a circle of radius A.At the same
time,let particle Q move up and
down along the straight line
segment RS in such a way that
the y coordinates of P and Q are
always equal.If is the anguler
velocity of P in radians per
second, and (figure 2.1) = 0 when
t = 0 ,then at a later time t

3.  The x and y coordinates of particle P in figure 2.1 are
 (2.3) ,
 If we think of P as the point z = x + iy in the complex
plane, we could replace (2.3) by a single equation to
describe the motion of P :
 (2.4)

4.  It is useful to draw a graph of x or y in (2.2) and (2.3) as a function
of t. Figure 2.2 represents any of the functions
 if we choose the origin correctly.The number A called the amplitude
of the vibration or the amplitude of the function.Physically it is the
maximum diplacement of Q from its equilibrium position.The period
of the simple harmonic motion or the period of the function is the
time for one complete oscillation , that is , (see figure 2.2)

5.  We could write the velocity of Q from (2.5) as
 (2.6)
 Here B is the maximum value of the velocity and is called the velocity amplitude.Note
that the velocity has the same period as the displacement.If tha mass of the particle Q is
m , its kinetic energy is :
 (2.7) Kinetic energy =
 We are considering an idealized harmonic oscillator which does not lose energy.The the
total energy ( kinetic plus potential) must be equal to the largest value of the kinetic
energy , that is ,.Thus we have :
 (2.8) Total energy =
 Notice that the energy is proportional to the square of the (velocity) amplitude ; we
shall be interested in this result later when we discuss sound.
 Waves are another important example of an oscillatory phenomenon.The mathematical
ideas of wave motion are useful in many fields ; for example , we talk about water
waves, sound waves , and radio waves.Let us consider, as a simple example, water
waves in which the shape of the water urface is ( unrealistically !) a sine curve.Then if we
take a photograph ( at the instant t = 0) of the water surface , the equation of this
picture could be written (relative to appropriate axes)
 (2.9)

6.  Where x represents horizontal distance and is the
distance between wave crests. Usually is called the
wavelength, but mathematically it is the same as the
period of this function of x. Now suppose we take another
photograph when the waves have moved forward a
distance (v is the velocity of the waves and t is the time
between photographs). Figure 2.3 shows the two
photographs superimposed.Observe that the value of y at
the point x on the graph labeled t, is just the same as the
value of y at the point on the graph labeled t =
0.if (2.9) is the equation representing the waves at t =
0, then
 (2.10)

8. By definition, the function f(x) is periodic if
f (x+p)= f(x) for every x; the number p is periode.
The period of sin x is 2π since sin (x+2 π)= sin x ;
similarly, the period sin 2π x is 1 since

9. Applications of Fourier Series
 We have said that the vibration of a tuning fork is an
example of simple harmonic motion. When we hear the
musical note produced, we say that a sound wave has
passed through the air from the tuning fork to our ears.
As the tunning fork vibrates it pushes against the air
molecules, creating alternately regions of high an low
pressure (figure 3.1)

10.  Now suppose that several pure tones are heard simultaneously.
In the resultant sound wave, the pressure will not be a single sines
function but a sum of several sine function.
 Higher frequencies mean shorter periods. If sin ωt and cos ωt
correspond to the fundamental frequency, then sin n ωt and cos nωt
correspond to the higher harmonics. The combination of the
fundamental and the harmonics is complicated periodic function
with the period of the fundamental.
 Given the complicated function, we could ask how to write it as a
sum of terms correspondening to the various harmonic. In general it
might require all the harmonic, that is, an infinite series of term. This
called a Fourier series. Expanding a function in a Forier series then
amounts to breaking it down into its various harmonics. In fact, this
process is sometimes called harmonic analysis.

11.  There are applications to other fields besides sound.
Radio waves, visible light , and x rays are all examples of
a kind of wave motion in which the “wave” correspend to
rayying strengths of electric and magnetic fields. Exactly
the same math equations apply as for water waves and
sound wave. We could then ask what light frequencies
(these correspend to the color) are in a given light beam
and in what proportions.


12.  This is a periodec function, but so are the functions shown in
figure 3.2. Then we could ask what AC frequencies (harmonics)
make up a given signal and in what proportions.
 When an electric signal is passed through a network (say a
radio), some of the harmonics may be lost. If most of the
important ones get through with their relative intensitas
preserved, we say that the radio processes “high fidelity”. To find
out which harmonics are the important ones in given signal, we
expand it in a Fourier series. The terms of the series with large
coefficients then represent the important harmonics (frequencies)

13.  Since sines and cosines are themselves periodic, it seems
rather natural to use series of them, rather than power
series, to represent periodic fuctions. There is another
important reason. The coefficients of a power series are
obtained. Many periodic functions is practice are not
continous or not differentiable (figure 3.2).
 Forier series (unlike power series) can represent
discontinous functions or functions whose graphs have
corners. On the other hand, forier series do not ussually
converge as rapidly as power series and much more care
is needed in manipulating them.