Efficient Techniques for Mastering Fourier Series in Physics

1. Lecture 7Fourier Series Remember homework 1 for submission 31/10/08 Best way to handle course • 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 Where can I find stuff? • 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 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) = a1cos x F(x) = b1sin x n = 1 For these cases L is taken to be 2p to simplify expressions F(x) = a2cos 2x F(x) = b2sin 2x n = 2 F(x) = a3cos 3x F(x) = b3sin 3x n = 3

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

4. Fourier Series - QUIZ Team A questions in white Team B questions in red 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 ?

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 Fourier series 16. Team B: If we were to represent the function below as a Fourier series what could you say about the value of a0 ? a0 is baseline shifter. Half way between 20 and 70 is 45. So ao = 90

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

15. Fourier Series - QUIZ Fourier series 18. Team B: If we were to represent the function below as a Fourier series what could you say about the sign of the b1 term ?

16. Fourier Series - QUIZ Fourier series 18. Team B: If we were to represent the function below as a Fourier series what could you say about the value of the b1 term ? It would have a negative amplitude

17. 1 0 p 2p 3p x Finding coefficients of the Fourier Series 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? Period is 2p 2. Use equation to find a0? 3. Use equation to find an? 4. Use equation to find bn?

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

19. 1 0 p 2p 3p x 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

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? Period is 2p 2. Use equation to find a0? Team A find coefficients an? 3. 4. Team B find coefficients bn?

21. Finding coefficients of the Fourier Series - QUIZ Find Fourier series to represent this repeat pattern. Period is 2p Team A find coefficients an? 3. Integrate by parts so set u = x and cos (nx) dx = dv and du = dx

22. Finding coefficients of the Fourier Series - QUIZ Find Fourier series to represent this repeat pattern. Period is 2p Team B find coefficients bn? 4. Integrate by parts so set u = x and sin (nx) dx = dv du = dx

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

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