1. Lecture 7 Fourier Series Skim through notes before lecture Ask questions in lecture After lecture read notes and try some problems See me in E47 office if you have any questions Best way to handle course Remember homework 1 for submission 31/10/08 ALL tutorial, problems class, homework in notes and web Completed solutions version of notes at Phils Problems All.ppt presentations from lectures at Phils Problems Loads of questions with worked answers at Phils Problems Where can I find stuff? http://www.hep.shef.ac.uk/Phil/PHY226.htm

2. Fourier Series Last lecture we learned how a Fourier series formed from sine and cosine harmonics can represent any periodically repeating function. What’s a harmonic? Below are the first three harmonics of sine and cosine F(x) = a 1 cos x F(x) = b 3 sin 3x F(x) = b 2 sin 2x F(x) = b 1 sin x F(x) = a 3 cos 3x F(x) = a 2 cos 2x For these cases L is taken to be 2  to simplify expressions n = 1 n = 2 n = 3

3. Summary The Fourier series can be written with period L as The Fourier series coefficients can be found by:- Fourier Series http://www.falstad.com/fourier/ Sine terms Cosine terms http://www.univie.ac.at/future.media/moe/galerie/fourier/fourier.html A key point to notice is that the summed output will repeat with the period of the 1 st harmonic We must decide on the amplitude of each harmonic term. This amplitude may be zero, positive, negative, big or small……..

4. Fourier Series - QUIZ Team A questions in white 1. What is when n = 3 ? 2. What is when n = 52 ? 3. What is when n = 1 ? 4. What is when n = 17 ? 5. What is when n = 52 ? 6. What is when n = 1 ? 7. What is when n = 4 ? Team B questions in red

5. Fourier Series - QUIZ 8. Team B: What is ?

6. Fourier Series - QUIZ 9. Team A: What is ?

7. Fourier Series - QUIZ 10. Team B: Describe the following step function in terms of f(x) and x ?

8. Fourier Series - QUIZ 11. Team A: What is ?

9. Fourier Series - QUIZ 12. Team B: Describe the following step function over one period in terms of f(x) and x ?

10. Fourier Series - QUIZ 13. Team A: What is the integral of f(x) over one period ?

11. Fourier Series - QUIZ 14. Team B: Describe the following step function over one period in terms of f(x) and x ?

12. Fourier Series - QUIZ 15. Team A: What is the integral of f(x) over one period ?

13. Fourier Series - QUIZ 16. Team B: If we were to represent the function below as a Fourier series what could you say about the value of a 0 ? Fourier series a 0 is baseline shifter. Half way between 20 and 70 is 45. So a o = 90

14. Fourier Series - QUIZ 17. Team A: If we were to represent the function below as a Fourier series what could you say about the values of the a n terms ? Fourier series odd function so all a n terms are zero

15. Fourier Series - QUIZ 18. Team B: If we were to represent the function below as a Fourier series what could you say about the sign of the b 1 term ? Fourier series

16. Fourier Series - QUIZ 18. Team B: If we were to represent the function below as a Fourier series what could you say about the value of the b 1 term ? Fourier series It would have a negative amplitude

17. Finding coefficients of the Fourier Series Find Fourier series to represent this repeat pattern. 0  x 1 Steps to calculate coefficients of Fourier series 1. Write down the function f(x) in terms of x. What is period? 2. Use equation to find a 0 ? Period is 2  3. Use equation to find a n ? 4. Use equation to find b n ?

18. Finding coefficients of the Fourier Series 5. Write out values of b n for n = 1, 2, 3, 4, 5, …. 4. Use equation to find b n ? 6. Write out Fourier series with period L, a n, b n in the generic form replaced with values for our example

19. Finding coefficients of the Fourier Series So what does this Fourier series look like if we only use first few terms? Use Fourier_checker on Phils problems website 0  x 1

20. Finding coefficients of the Fourier Series - QUIZ Find Fourier series to represent this repeat pattern. Steps to calculate coefficients of Fourier series 1. Write down the function f(x) in terms of x. What is period? 2. Use equation to find a 0 ? 3. 4. Team A find coefficients a n ? Team B find coefficients b n ? Period is 2 

21. Finding coefficients of the Fourier Series - QUIZ Find Fourier series to represent this repeat pattern. 3. Team A find coefficients a n ? Period is 2  Integrate by partsso set u = x and cos (nx) dx = dv and du = dxn=1n=2n=3n=4n=5

22. Finding coefficients of the Fourier Series - QUIZ Find Fourier series to represent this repeat pattern. 4. Period is 2  Integrate by partsso set u = x and sin (nx) dx = dv du = dxn=1n=2n=3n=4n=5 Team B find coefficients b n ?

23. Finding coefficients of the Fourier Series - QUIZ Find Fourier series to represent this repeat pattern. 5. Write out the first few terms of Fourier seriesn=1n=2n=3n=4n=5 L is the period = 2 n=1n=2n=3n=4n=5 So

24. Finding coefficients of the Fourier Series - QUIZ Can we check our Fourier series using Fourier_checker.xls at Phils Problems ??? Yes!!