Fourier sine series

1
Fourier Methods

• 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

Dr Mervyn Roy (S6) www2.le.ac.uk/departments/physi
cs/people/academic-staff/mr6
2
Resources

• Lecture notes www2.le.ac.uk/departments/physics/p
  eople/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!

3
Introduction
The wave equation
 
 
 
Superposition tells us that sums of such terms
must also be solutions,
 
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Fourier sine series (half-range)
 
 
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Fourier sine series
How does this work ? Need a standard integral
(new chapter 12 A.2)
 
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Periodic extension of Fourier sine series
 
We know that sine waves have odd symmetry,
 
and that
 
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The wave equation
 
 
 
 
 
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Can go through the same procedure with the
solutions to other PDEs e.g. Laplace equation
(see workshop 1 exercise 3),