Fourier Series By : Fahriyatie Imilda Fitriana
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
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)
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)
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)
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)
By definition, the function f(x) is periodic iff (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
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)
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.
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.
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)
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.