Series FOURIER SERIES - salfordphysics.com

1 SERIES FOURIER SERIES . Graham S McDonald A self-contained Tutorial Module for learning the technique of FOURIER SERIES analysis Table of contents Begin Tutorial c 2004 Table of contents 1. Theory 2. Exercises 3. Answers 4. Integrals 5. Useful trig results 6. Alternative notation 7. Tips on using solutions Full worked solutions Section 1: Theory 3. 1. Theory A graph of periodic function f (x) that has period L exhibits the same pattern every L units along the x-axis, so that f (x + L) = f (x). for every value of x. If we know what the function looks like over one complete period, we can thus sketch a graph of the function over a wider interval of x (that may contain many periods). f(x ). x P E R IO D = L. Toc JJ II J I Back Section 1: Theory 4.

2 This property of repetition defines a fundamental spatial fre- quency k = 2 L that can be used to give a first approximation to the periodic pattern f (x): f (x) ' c1 sin(kx + 1 ) = a1 cos(kx) + b1 sin(kx), where symbols with subscript 1 are constants that determine the am- plitude and phase of this first approximation A much better approximation of the periodic pattern f (x) can be built up by adding an appropriate combination of harmonics to this fundamental (sine-wave) pattern. For example, adding c2 sin(2kx + 2 ) = a2 cos(2kx) + b2 sin(2kx) (the 2nd harmonic). c3 sin(3kx + 3 ) = a3 cos(3kx) + b3 sin(3kx) (the 3rd harmonic). Here, symbols with subscripts are constants that determine the am- plitude and phase of each harmonic contribution Toc JJ II J I Back Section 1: Theory 5.

3 One can even approximate a square-wave pattern with a suitable sum that involves a fundamental sine-wave plus a combination of harmon- ics of this fundamental frequency. This sum is called a FOURIER SERIES F u n d a m e n ta l F u n d a m e n ta l + 2 h a rm o n ic s x F u n d a m e n ta l + 5 h a rm o n ic s P E R IO D = L. F u n d a m e n ta l + 2 0 h a rm o n ic s Toc JJ II J I Back Section 1: Theory 6. In this Tutorial, we consider working out FOURIER SERIES for func- tions f (x) with period L = 2 . Their fundamental frequency is then k = 2 . L = 1, and their FOURIER SERIES representations involve terms like a1 cos x , b1 sin x a2 cos 2x , b2 sin 2x a3 cos 3x , b3 sin 3x We also include a constant term a0 /2 in the FOURIER SERIES .

4 This allows us to represent functions that are, for example, entirely above the x axis. With a sufficient number of harmonics included, our ap- proximate SERIES can exactly represent a given function f (x). f (x) = a0 /2 + a1 cos x + a2 cos 2x + a3 cos 3x + .. + b1 sin x + b2 sin 2x + b3 sin 3x + .. Toc JJ II J I Back Section 1: Theory 7. A more compact way of writing the FOURIER SERIES of a function f (x), with period 2 , uses the variable subscript n = 1, 2, 3, .. a0 X. f (x) = + [an cos nx + bn sin nx]. 2 n=1. We need to work out the FOURIER coefficients (a0 , an and bn ) for given functions f (x). This process is broken down into three steps Z. STEP ONE 1. a0 = f (x) dx . 2 . Z. STEP TWO 1. an = f (x) cos nx dx.

5 2 . Z. STEP THREE 1. bn = f (x) sin nx dx . 2 . where integrations are over a single interval in x of L = 2 . Toc JJ II J I Back Section 1: Theory 8. Finally, specifying a particular value of x = x1 in a FOURIER SERIES , gives a SERIES of constants that should equal f (x1 ). However, if f (x). is discontinuous at this value of x, then the SERIES converges to a value that is half-way between the two possible function values " V e rtic a l ju m p " /d is c o n tin u ity in th e fu n c tio n re p re s e n te d f(x ). x F o u rie r s e rie s c o n v e rg e s to h a lf-w a y p o in t Toc JJ II J I Back Section 2: Exercises 9. 2. Exercises Click on Exercise links for full worked solutions (7 exercises in total). Exercise 1.

6 Let f (x) be a function of period 2 such that . 1, < x < 0. f (x) =. 0, 0 < x < . a) Sketch a graph of f (x) in the interval 2 < x < 2 . b) Show that the FOURIER SERIES for f (x) in the interval < x < is . 1 2 1 1. sin x + sin 3x + sin 5x + .. 2 3 5. c) By giving an appropriate value to x, show that 1 1 1. = 1 + + .. 4 3 5 7. Theory Answers Integrals Trig Notation Toc JJ II J I Back Section 2: Exercises 10. Exercise 2. Let f (x) be a function of period 2 such that . 0, < x < 0. f (x) =. x, 0 < x < . a) Sketch a graph of f (x) in the interval 3 < x < 3 . b) Show that the FOURIER SERIES for f (x) in the interval < x < is . 2 1 1. cos x + 2 cos 3x + 2 cos 5x + .. 4 3 5.. 1 1. + sin x sin 2x + sin 3x .. 2 3. c) By giving appropriate values to x, show that 2.

7 (i) 4 = 1 31 + 15 17 + .. and (ii) 8 = 1 + 312 + 512 + 712 + .. Theory Answers Integrals Trig Notation Toc JJ II J I Back Section 2: Exercises 11. Exercise 3. Let f (x) be a function of period 2 such that . x, 0 < x < . f (x) =. , < x < 2 . a) Sketch a graph of f (x) in the interval 2 < x < 2 . b) Show that the FOURIER SERIES for f (x) in the interval 0 < x < 2 is . 3 2 1 1. cos x + 2 cos 3x + 2 cos 5x + .. 4 3 5.. 1 1. sin x + sin 2x + sin 3x + .. 2 3. c) By giving appropriate values to x, show that 1 1 1 2 1 1 1. (i) 4 = 1 3 + 5 7 +.. and (ii) 8 = 1+ 32 + 52 + 72 +.. Theory Answers Integrals Trig Notation Toc JJ II J I Back Section 2: Exercises 12. Exercise 4. Let f (x) be a function of period 2 such that x f (x) = over the interval 0 < x < 2.

8 2. a) Sketch a graph of f (x) in the interval 0 < x < 4 . b) Show that the FOURIER SERIES for f (x) in the interval 0 < x < 2 is . 1 1. sin x + sin 2x + sin 3x + .. 2 2 3. c) By giving an appropriate value to x, show that 1 1 1 1. = 1 + + .. 4 3 5 7 9. Theory Answers Integrals Trig Notation Toc JJ II J I Back Section 2: Exercises 13. Exercise 5. Let f (x) be a function of period 2 such that . x, 0 < x < . f (x) =. 0, < x < 2 . a) Sketch a graph of f (x) in the interval 2 < x < 2 . b) Show that the FOURIER SERIES for f (x) in the interval 0 < x < 2 is . 2 1 1. + cos x + 2 cos 3x + 2 cos 5x + .. 4 3 5. 1 1 1. + sin x + sin 2x + sin 3x + sin 4x + .. 2 3 4. c) By giving an appropriate value to x, show that 2 1 1. = 1 + 2 + 2 +.

9 8 3 5. Theory Answers Integrals Trig Notation Toc JJ II J I Back Section 2: Exercises 14. Exercise 6. Let f (x) be a function of period 2 such that f (x) = x in the range < x < . a) Sketch a graph of f (x) in the interval 3 < x < 3 . b) Show that the FOURIER SERIES for f (x) in the interval < x < is . 1 1. 2 sin x sin 2x + sin 3x .. 2 3. c) By giving an appropriate value to x, show that 1 1 1. = 1 + + .. 4 3 5 7. Theory Answers Integrals Trig Notation Toc JJ II J I Back Section 2: Exercises 15. Exercise 7. Let f (x) be a function of period 2 such that f (x) = x2 over the interval < x < . a) Sketch a graph of f (x) in the interval 3 < x < 3 . b) Show that the FOURIER SERIES for f (x) in the interval < x < is 2.

10 1 1. 4 cos x 2 cos 2x + 2 cos 3x .. 3 2 3. c) By giving an appropriate value to x, show that 2 1 1 1. = 1 + 2 + 2 + 2 + .. 6 2 3 4. Theory Answers Integrals Trig Notation Toc JJ II J I Back Section 3: Answers 16. 3. Answers The sketches asked for in part (a) of each exercise are given within the full worked solutions click on the Exercise links to see these solutions The answers below are suggested values of x to get the SERIES of constants quoted in part (c) of each exercise . 1. x = 2, . 2. (i) x = 2, (ii) x = 0, . 3. (i) x = 2, (ii) x = 0, . 4. x = 2, 5. x = 0, . 6. x = 2, 7. x = . Toc JJ II J I Back Section 4: Integrals 17. 4. Integrals R b dv b Rb du Formula for integration by parts: a u dx dx = [uv]a a dx v dx R R.