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) www2.le.ac.uk/departments/physics/people/mervynroy

2. PA214 Waves and Fields Fourier Methods So far…

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

4. PA214 Waves and Fields Fourier Methods Introduction to Fourier transforms But what about functions that are aperiodic? Use Fourier transforms… This is a Fourier integral – a representation of a non-periodic function in terms of simple harmonics

5. PA214 Waves and Fields Fourier Methods Introduction to Fourier transforms

6. PA214 Waves and Fields Fourier Methods

7. PA214 Waves and Fields Fourier Methods Convolutions output function response function input function response function output function Notes, figure 12.11

8. PA214 Waves and Fields Fourier Methods Wikipedia commons, Brian Amberg http://en.wikipedia.org/wiki/Convolution

9. PA214 Waves and Fields Fourier Methods Convolutions Convolution theorem

10. PA214 Waves and Fields Fourier Methods Real space Fourier transform Inverse transform Reciprocal space