If you took the second-year MA20223 Vectors & PDEs course, you might remember discussing the applications of Fourier series to studying music (or more generally signal processing). There are a lot of beautiful applications of spectral and harmonic analysis to sound and music, and this project will delve into those applications.

It will be divided into two parts:

1. In Part 1, the student will study the basics of sound processing using Fourier transforms. One of the interesting applications we would like to look into is the design of a music visualiser, that converts between synthesising audio and converting this into equivalent wave-forms—for example, modelling the type of motion on a drum that would have resulted in making that sound.
2. In Part 2, the student will look into a famous mathematical problem about whether it is possible to infer the shape of a drum based on the sound it makes (i.e. is it possible to determine geometry based on eigenvalues). The student will apply the visualisations developed in Part 1 to dig into the research on this issue.

Additional reading:

[1] https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum

[2] http://www.math.udel.edu/~driscoll/research/drums.html

Pre-requisites: MA20223 is recommended but not essential. Because Part 1 will involve wrestling with some fairly heavy signal and sound processing modules, the student should be quite capable with programming and numerical algorithms. Optimally, a student who is not afraid of working with Python is desirable.