Description

All objects, when hit or knocked, vibrate and these vibrations are transmitted to the ambient air as noise that we can hear. Some noise are dull, like a knock on a table top and others are pleasant, like piano strings, after they have been hit by a soft hammer, or the sound of a bell. What is the origin of this difference? What makes a sound nice and others dull or unpleasant?

To study the vibrations of objects, one uses the theory of elasticity and one assumes, for simplicity, that the vibrations are small. The problem then leads to some linear partial differential equations that one needs to solve.

One first tries to find simple solutions corresponding to a single frequency of vibration. Such solutions are called normal modes and it turns out that the general solution is always given by a superposition (a sum) of such solutions. The amount that each normal mode contributes to the general solution is determined by how the object is hit or knocked.

The shape that the object assumes when it vibrates is also related to the frequency of excitation. Check the followings for some good illustrations: sand on vibrating plate 1 , sand on vibrating plate 2 and vibrating drum.

The subjective quality of a sounds lies in its spectral of compositions. When all vibration frequencies are multiple of each other, the sound is considered as pleasant, like a guitar or harpsichord string for example. When the frequencies are nearly multiple of each other, like a piano string, the sound is even richer. When there is no simple relation between the different normal frequencies, the sound is dull, like a plain door or a stone.

The project will consist in looking at the vibrations of simple objects, starting with a simple string. We will then consider more complex systems like metal bars, plates ... and relate them to some musical instruments.

The work can be exclusively analytical, but students who enjoy doing numerical work can also solve some equations numerically using a programs like Python or any programming language of their choice.

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

Prerequisites

• Analysis in many variable II (MATH2031)

• Topics in Applied Maths IV (MATH4381) is recommended but not compulsory

Resources

• 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
• Music: a Mathematical Offering (Chapter 3)
• The Theory of sound Lord Rayleigh.
• Introduction to Continuum Mechanics David J. Raymond.
• Mathematical Physics (Chapter 8). Butkov: Adisson Wesley (1968)
• The physics of musical instruments. N.H. Fletcher, T.D. Rossing. New York : Springer, 1998.
• Fundations of Mathematical Physics Sadi Hassan: Prentice Hall (1991)
• Several papers available from the web