f x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y

Transcription

1 Fourier Series When the French mathematician Joseph Fourier (768 83) was tring to solve a problem in heat conduction, he needed to epress a function f as an infinite series of sine and cosine functions: f a n a n cos n b n sin n a a cos a cos a 3 cos 3 b sin b sin b 3 sin 3 Earlier, Daniel Bernoulli and eonard Euler had used such series while investigating problems concerning vibrating strings and astronom. The series in Equation is called a trigonometric series or Fourier series and it turns out that epressing a function as a Fourier series is sometimes more advantageous than epanding it as a power series. In particular, astronomical phenomena are usuall periodic, as are heartbeats, tides, and vibrating strings, so it makes sense to epress them in terms of periodic functions. We start b assuming that the trigonometric series converges and has a continuous function f as its sum on the interval,, that is, f a n a n cos n b n sin n Our aim is to find formulas for the coefficients a n and b n in terms of f. Recall that for a power series f c n a n we found a formula for the coefficients in terms of derivatives: c n f n an!. Here we use integrals. If we integrate both sides of Equation and assume that it s permissible to integrate the series term-b-term, we get But f d a d a n a n n a n cos n b n sin n d cos n d cos n d n sin n n sin n sinn n b n sin n d because n is an integer. Similarl,. So sin n d f d a

2 FOURIER SERIES Notice that a is the average value of f over the interval,. and solving for 3 a gives a f d To determine a n for n we multipl both sides of Equation b cos m (where m is an integer and m ) and integrate term-b-term from to : f cos m d a n cos n b n sin n cos m d a n a cos m d n a n cos n cos m d n b n sin n cos m d We ve seen that the first integral is. With the help of Formulas 8, 8, and 6 in the Table of Integrals, it s not hard to show that sin n cos m d cos n cos m d for all n and m for n m for n m So the onl nonzero term in () is a m and we get Solving for, and then replacing m b n, we have a m f cos m d a m 5 a n f cos n d n,, 3,... Similarl, if we multipl both sides of Equation b sin m and integrate from to, we get 6 b n f sin n d n,, 3,... We have derived Formulas 3, 5, and 6 assuming f is a continuous function such that Equation holds and for which the term-b-term integration is legitimate. But we can still consider the Fourier series of a wider class of functions: A piecewise continuous function on a, b is continuous ecept perhaps for a finite number of removable or jump discontinuities. (In other words, the function has no infinite discontinuities. See Section.5 for a discussion of the different tpes of discontinuities.)

3 FOURIER SERIES 3 7 Definition et f be a piecewise continuous function on,. Then the Fourier series of f is the series a a n cos n b n sin n where the coefficients and in this series are defined b a n b n n a f d a n f cos n d b n f sin n d and are called the Fourier coefficients of f. Notice in Definition 7 that we are not saing f is equal to its Fourier series. ater we will discuss conditions under which that is actuall true. For now we are just saing that associated with an piecewise continuous function f on, is a certain series called a Fourier series. EXAMPE Find the Fourier coefficients and Fourier series of the square-wave function f defined b f if if and f f So f is periodic with period and its graph is shown in Figure. Engineers use the square-wave function in describing forces acting on a mechanical sstem and electromotive forces in an electric circuit (when a switch is turned on and off repeatedl). Strictl speaking, the graph of f is as shown in Figure (a), but it s often represented as in Figure (b), where ou can see wh it s called a square wave. _ (a) FIGURE Square-wave function _ (b) SOUTION Using the formulas for the Fourier coefficients in Definition 7, we have a f d d d

4 FOURIER SERIES and, for n, a n f cos n d sin n n n d sin n sin cos n d b n f sin n d cos n n n d cos n cos sin d Note that cos n equals if n is even and if n is odd. n if n is even if n is odd The Fourier series of f is therefore a a cos a cos a 3 cos 3 b sin b sin b 3 sin 3 sin sin sin 3 Since odd integers can be written as n k, where k is an integer, we can write the Fourier series in sigma notation as k sin sin 5 7 sink k sin 3 sin 5 sin 7 sin 5 In Eample we found the Fourier series of the square-wave function, but we don t know et whether this function is equal to its Fourier series. et s investigate this question graphicall. Figure shows the graphs of some of the partial sums S n sin when n is odd, together with the graph of the square-wave function. 3 sin 3 n sin n

5 FOURIER SERIES 5 S S S _ S S S _ FIGURE Partial sums of the Fourier series for the square-wave function We see that, as n increases, S n becomes a better approimation to the square-wave function. It appears that the graph of S n is approaching the graph of f, ecept where or is an integer multiple of. In other words, it looks as if f is equal to the sum of its Fourier series ecept at the points where f is discontinuous. The following theorem, which we state without proof, sas that this is tpical of the Fourier series of piecewise continuous functions. Recall that a piecewise continuous function has onl a finite number of jump discontinuities on,. At a number a where f has a jump discontinuit, the one-sided limits eist and we use the notation f a lim l a f f a lim l a f 8 Fourier Convergence Theorem If f is a periodic function with period and f and f are piecewise continuous on,, then the Fourier series (7) is convergent. The sum of the Fourier series is equal to f at all numbers where f is continuous. At the numbers where f is discontinuous, the sum of the Fourier series is the average of the right and left limits, that is f f If we appl the Fourier Convergence Theorem to the square-wave function Eample, we get what we guessed from the graphs. Observe that f in f lim l f and f lim l f and similarl for the other points at which f is discontinuous. The average of these left and right limits is, so for an integer n the Fourier Convergence Theorem sas that k k sink f if n n if n (Of course, this equation is obvious for n.)

6 6 FOURIER SERIES Functions with Period If a function f has period other than, we can find its Fourier series b making a change of variable. Suppose f has period, that is f f for all. If we let t and tt f f t then, as ou can verif, t has period and corresponds to t. The Fourier series of t is a a n cos nt b n sin nt n where a tt dt a n tt cos nt dt b n tt sin nt dt If we now use the Substitution Rule with t, then t, dt d, and we have the following Notice that when these formulas are the same as those in (7). 9 If f is a piecewise continuous function on,, its Fourier series is where a na n cos n b n sin n a f d and, for n, a n f cos n d b n f sin n d Of course, the Fourier Convergence Theorem (8) is also valid for functions with period. EXAMPE Find the Fourier series of the triangular wave function defined b for and f f for all. (The graph of f is shown in Figure 3.) For which values of is f equal to the sum of its Fourier series? f FIGURE 3 The triangular wave function _

7 FOURIER SERIES 7 SOUTION We find the Fourier coefficients b putting in (9): Notice that a is more easil calculated as an area. a d ] ] d d and for n, a n cosn d cosn d because cosn is an even function. Here we integrate b parts with u and dv cosn d. Thus, Since sinn sinn d sinn cosn n cos n a n n n is an odd function, we see that n n We could therefore write the series as b n sinn d n cos n n cosn But cos n if n is even and cos n if n is odd, so a n n cos n n if n is even if n is odd Therefore, the Fourier series is cos cos3 n 9 k cosk The triangular wave function is continuous everwhere and so, according to the Fourier Convergence Theorem, we have 5 cos5 f n k cosk for all

8 8 FOURIER SERIES In particular, k k cosk for Fourier Series and Music One of the main uses of Fourier series is in solving some of the differential equations that arise in mathematical phsics, such as the wave equation and the heat equation. (This is covered in more advanced courses.) Here we eplain briefl how Fourier series pla a role in the analsis and snthesis of musical sounds. We hear a sound when our eardrums vibrate because of variations in air pressure. If a guitar string is plucked, or a bow is drawn across a violin string, or a piano string is struck, the string starts to vibrate. These vibrations are amplified and transmitted to the air. The resulting air pressure fluctuations arrive at our eardrums and are converted into electrical impulses that are processed b the brain. How is it, then, that we can distinguish between a note of a given pitch produced b two different musical instruments? The graphs in Figure show these fluctuations (deviations from average air pressure) for a flute and a violin plaing the same sustained note D (9 vibrations per second) as functions of time. Such graphs are called waveforms and we see that the variations in air pressure are quite different from each other. In particular, the violin waveform is more comple than that of the flute. t t FIGURE Waveforms (a) Flute (b) Violin We gain insight into the differences between waveforms if we epress them as sums of Fourier series: Pt a a cos t b sin t a cos t b sin t In doing so, we are epressing the sound as a sum of simple pure sounds. The difference in sounds between two instruments can be attributed to the relative sizes of the Fourier coefficients of the respective waveforms. The nth term of the Fourier series, that is, a n cos nt b n nt is called the nth harmonic of P. The amplitude of the nth harmonic is A n sa n b n and its square, A n a n b n, is sometimes called energ of the nth harmonic. (Notice that

9 FOURIER SERIES 9 A n b n for a Fourier series with onl sine terms, as in Eample, the amplitude is and the energ is A n b n.) The graph of the sequence A n is called the energ spectrum of P and shows at a glance the relative sizes of the harmonics. Figure 5 shows the energ spectra for the flute and violin waveforms in Figure. Notice that, for the flute, A n tends to diminish rapidl as n increases whereas, for the violin, the higher harmonics are fairl strong. This accounts for the relative simplicit of the flute waveform in Figure and the fact that the flute produces relativel pure sounds when compared with the more comple violin tones. A@ n A@ n FIGURE 5 Energ spectra n 6 8 (a) Flute 6 8 (b) Violin n In addition to analzing the sounds of conventional musical instruments, Fourier series enable us to snthesize sounds. The idea behind music snthesizers is that we can combine various pure tones (harmonics) to create a richer sound through emphasizing certain harmonics b assigning larger Fourier coefficients (and therefore higher corresponding energies). Eercises 6 A function f is given on the interval, and f is periodic with period. (a) Find the Fourier coefficients of f. (b) Find the Fourier series of f. For what values of is f equal to its Fourier series? ; (c) Graph f and the partial sums S, S, and S 6 of the Fourier series f f 3. f. f f cos f if if if if if if if if if 7 Find the Fourier series of the function f f f. f,. f t sin3t, t. A voltage E sin t, where t represents time, is passed through a so-called half-wave rectifier that clips the negative part of the wave. Find the Fourier series of the resulting periodic function f t E sin t if if if if if if if if if t t f f f f f 8 f f f f t f t

10 FOURIER SERIES 3 6 Sketch the graph of the sum of the Fourier series of without actuall calculating the Fourier series. 3.. f 3 f 5. f 3, 6. f e, if if if if 7. (a) Show that, if, then f 8. Use the result of Eample to show that Use the result of Eample to show that Use the given graph of f and Simpson s Rule with n 8 to estimate the Fourier coefficients a, a, a, b, and b. Then use them to graph the second partial sum of the Fourier series and compare with the graph of f. 8 3 n (b) B substituting a specific value of, show that n n n cosn n 6.5