1. PA214 Waves and Fields Fourier Methods Blue book New chapter 12 Fourier sine series Application to the wave equation Fourier cosine series Fourier full range series Complex form of Fourier series Introduction to Fourier transforms and the convolution theorem Fourier Methods Dr Mervyn Roy (S6)

2. PA214 Waves and Fields Fourier Methods Last time

3. PA214 Waves and Fields Fourier Methods “ One point … cannot be emphasized too strongly. To use mathematics effectively in problems, you need not just knowledge, but skill. Skill can be obtained only through practice. “ - Mary L. Boas

4. PA214 Waves and Fields Fourier Methods Use Fourier methods to find solutions to other PDEs e.g. Laplace equation (see workshop 1 exercise 3),

5. PA214 Waves and Fields Fourier Methods Obtaining series expressions for constants Expansion for  first discovered around 1400 by Indian mathematician, Madhava (found  to 11 d.p.). - see workshop 1 exercise 1

6. PA214 Waves and Fields Fourier Methods Fourier cosine series (half range)

7. PA214 Waves and Fields Fourier Methods Fourier cosine series

8. PA214 Waves and Fields Fourier Methods

9. PA214 Waves and Fields Fourier Methods

10. PA214 Waves and Fields Fourier Methods

11. PA214 Waves and Fields Fourier Methods

12. PA214 Waves and Fields Fourier Methods Periodic extension of Fourier cosine series and that We know that cosine waves have even symmetry,

13. PA214 Waves and Fields Fourier Methods

14. PA214 Waves and Fields Fourier Methods Comparison of sine and cosine series

15. PA214 Waves and Fields Fourier Methods Comparison of sine and cosine series sawtooth wave triangle wave

16. PA214 Waves and Fields Fourier Methods Full range Fourier series

17. PA214 Waves and Fields Fourier Methods

18. PA214 Waves and Fields Fourier Methods

19. PA214 Waves and Fields Fourier Methods Some physics! square wave n

20. PA214 Waves and Fields Fourier Methods Some physics! The Swift gamma ray burst mission lightcurve power spectrum

21. PA214 Waves and Fields Fourier Methods 5 nm Quantum dot Some physics! potential, V quantum states,  2

22. PA214 Waves and Fields Fourier Methods Complex form of Fourier series Could equally well write - sometimes the complex form is much more convenient

23. PA214 Waves and Fields Fourier Methods Complex form of Fourier series