Description

Physicists and mathematicians have been interested in the modelling of musical instruments for a very long time but mostly since the 19th century and the work of Helmhotlz.

The simplest models of most instruments reduces to solving the Laplace equation using specific boundary conditions corresponding to the instruments being modelled. This lets one compute the normal vibration modes, i.e. the frequencies of the notes produced by the instrument.

Comparison between the predicted frequencies and the real ones are good but not perfect. This comes from the fact that the simplest models ignore the friction and other structural properties of the materials from which the instruments are made. Better models can then be constructed which lead to more realistic descriptions of the sound produced by the instruments.

The project will consist in looking at the modeling of some instruments like the kettle drum and the piano string, staring with the simplest model and then progressing towards more realistic ones.

I will accept up to 4 students on this project.

Prerequisites

• Analysis in many variable II (MATH2031)

• Taking Continuum Mechanics IV (MATH4081) is recommended but not compulsory

Resources

• Introduction to Continuum Mechanics David J. Raymond.
• Music: a Mathematical Offering (Chapter 3)
• The physics of musical instruments. N.H. Fletcher, T.D. Rossing. New York : Springer, 1998.
• The Theory opf sound Lord Rayleigh.
• A Treatise on the Mathematical Theory of Elasticity A.E.H. Love New York : Dover Publishing, 1944.
• Theory of Elasticity: Course of Theoretical Physics. L.D. Landau, E.M. Lifshitz, A.M. Kosevich and L.P. Pitaevskii Butterworth-Heinemann, 1984
• Mathematical Physics (Chapter 8). Butkov: Adisson Wesley (1968)
• Fundations of Mathematical Physics Sadi Hassan: Prentice Hall (1991)
• Several papers available from the web