CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD

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1 CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period. The periodic function is shown in the diagram below. The Fourier series is given by: f(x) a + ( cos nx + sin nx) () n [ ] [ ] a f( x)dx dx dx x x + + [() ( ) ] + [( ) ()] f ( x)cos nx d x cos nx d x cos nx d x + sin nx + sin nx n n f ( x)sin nx d x sin nx d x sin nx d x + cos nx cos nx cos cos n( ) cos n cos n + n n [ ] [ ] 57 4, John Bird

2 n When n is even, b n [ ] [ ] 8 n n n When n is odd, b n [ ] [ ] ( 4) 8 8 Hence, b, b3 3, 8 b 5 5 d so on Substituting onto equation () gives: f(x) sin x+ 8 sin 3x+ 8 sin 5 x i.e. f(x) 8 sin sin 3 x+ x+ sin 5 x For the Fourier series in Problem, deduce a series for 4 at the point where x When x, f(x), hence sin + sin + sin i.e d i.e For the waveform shown below determine (a) the Fourier series for the function d (b) the sum of the Fourier series at the points of discontinuity. (a) The Fourier series is given by: f(x) a + ( cos nx + sin nx) () n 58 4, John Bird

3 [ ] / / / / / / a f( x)dx dx+ dx+ dx x / / / f ( x) cos nx d x () cos nx d x cos nx d x () cos nx d x + + / When n, a ( ) cos nx d x / / sin nx sin n sin n / n n a a4 a6 d so on for all even values of n When n, ( ) 3 3 When n 3, a ( ) 5 5 When n 5, a5 ( ) Hence, a7 7, a 9 9 d so on / / / / f ( x)sin nx d x ()sin nx d x sin nx d x ()sin nx d x + + Whatever value of n is chosen, b n Substituting onto equation () gives: / / sin nx d x cos nx cos n cos / n n n f(x) + cox cos 3x + cos 5x cos 7 x i.e. f(x) cos cos3 + x x+ cos5 x (b) The sum of the Fourier series at the points of discontinuity (i.e. at /,, 3/,...) is: + 4. For Problem 3, draw graphs of the first three partial sums of the Fourier series d show that as the series is added together term by term the result approximates more d more closely to the 59 4, John Bird

4 function it represents. In the diagram below graphs of cos x, cos 3x d 3 5 cos 5x d f(x) are shown A graph of cos x cos 3x + cos 5x is also shown 3 5 Finally, a graph of f(x) + cos x cos 3x + cos 5x is sketched. If further harmonics 3 5 were added then the waveform would approach that shown in Problem 3 5 4, John Bird

5 5. Find the term representing the third harmonic for the periodic function of period given by: f(x), when x, when x The periodic function is shown in the diagram below. sin nx f ( x)cos nx d x cos nx d x n f ( x)sin nx d x sin nx d x The third harmonic is when n 3, cos nx n n n cos n cos cos n i.e. b3 cos 3 Since the Fourier series is given by: f(x) a + ( cos nx + sin nx), n the 3rd harmonic term is: sin 3 x 3 6. Determine the Fourier series for the periodic function of period defined by: f(t), when t, when t, when t The function has a period of 5 4, John Bird

6 The periodic function is shown in the diagram below / / [ ] [ ] / / a f( t)dt dt dt dt t t ( ) + ( ) / / f ( t)cos nt d t cos nt d t cos nx d x + When n is even, a n / sin nt sin nt sin n sin n sin n n n / n When n, a sin sin sin [ ] [ ] When n 3, a3 sin sin 3 sin [ ] [ ] When n 5, a5 sin sin 5 sin [ ] [ ] It follows that a7 7, a 9 9 d so on / / f ( t)sin nt dt sin nt dt sin nt dt + / cos nt cos nt cos n cos cos n cos n + + n n / n n cos n n cos n cos cos + + cos n n n n When n is odd, b n 5 4, John Bird

7 4 + When n is even, b ( ) b b 4 6 () ( ) Similarly, b 8, b 5, d so on Substituting into f(t) a + ( cos nt + sin nt) n gives: f(x) + cost cos 3t+ cos 5t cos 7 t sin t+ sin 6t+ sin t i.e. f(x) cos cos 3 cos 5... sin sin 6 t t+ t + t+ t+ sin t Show that the Fourier series for the periodic function of period defined by: f(θ) θ, when θ sin θ, when θ is given by: f(θ) cos θ cos 4θ cos 6θ... (3) (3)(5) (5)(7) The periodic function is shown in the diagram below a [ ] ( ) ( ) f( )d d sin d cos cos cos θ θ θ + θ θ θ ( cos ) ( ) 53 4, John Bird

8 cos θ dθ sinθ cos θ dθ n + n sin θ θ sin θ θ sinθ sinθ dθ ( + n ) + ( n ) ( + n) + ( n) from 6, page 76 ( n) ( n) ( n) ( n) of the textbook cosθ + θ cos + cos cos + n n + n n + n n When n is odd, a n n n + n n When n, cos 3 cos( ) 4 a When n 4, cos 5 cos( 3 ) a (3)(5) (3)(5) (3)(5) When n 6, cos 7 cos( 5 ) a (5)(7) (5)(7) (5)(7) sin θdθ sinθsin θdθ n + n sin θ( + n) sin θ( n) + + n n from 9, page cos( θ nθ) cos( θ nθ) Substituting into f(θ) a + ( cos nθ + sin nθ) n 76 of the textbook gives: f(θ) cos θ cos 4θ cos 6 θ (3)(5) (5)(7) i.e. f(θ) cos θ cos 4 θ cos 6 θ... (3) (3)(5) (5)(7) 54 4, John Bird