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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/academic-staff/mr6

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PA214 Waves and Fields Fourier Methods Lecture notes www2.le.ac.uk/departments/physics/people/academic-staff/mr6 www2.le.ac.uk/departments/physics/people/academic-staff/mr6 214 course texts Blue book, new chapter 12 available on Blackboard Notes on Blackboard Notes on symmetry and on trigonometric identities Computing exercises Exam tips mock papers Books Mathematical Methods in the Physical Sciences (Mary L. Boas) Library! Resources

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PA214 Waves and Fields Fourier Methods The wave equation Introduction Superposition tells us that sums of such terms must also be solutions,

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods Fourier sine series (half-range)

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PA214 Waves and Fields Fourier Methods How does this work ? Need a standard integral (new chapter 12 – A.2)… Fourier sine series

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods Periodic extension of Fourier sine series and that We know that sine waves have odd symmetry,

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods The wave equation

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods

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PA214 Waves and Fields Fourier Methods Can go through the same procedure with the solutions to other PDEs e.g. Laplace equation (see workshop 1 exercise 3),

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