Hum, unfortunately I am not familiar with the Open University course, so I am just making a guess based on the course description you linked to.

Insofar as Fourier Analysis is concerned, a decent text is Stein and Shakarchi's Fourier Analysis: an introduction. ( http://press.princeton.edu/titles/7562.html ) You will most likely only need chapters 1, 2, 4, and 5, with a bit of knowledge of 3. One thing good about the book is that it was written as a first course in an analysis sequence, so doesn't assume too much knowledge about real and complex analysis.

Once you have a bit of Fourier analysis under your belt, reading Korner's Fourier Analysis ( http://books.google.com/books?id=OcZ5iKsGrmoC&lpg=PP1&dq=korner%20fourier%20analysis&pg=PR8#v=onepage&q=korner%20fourier%20analysis&f=false ) can be enlightening and give you some feel about what one can do using the machinery.

For the applications to wave equations as mentioned in the course description, somehow I feel that a textbook in electromagnetism (Jackson or Griffiths) may contain more practical material (look at the sections on standing waves and wave-guides).