2005/06 Undergraduate Module Catalogue

MATH3374 Linear and Non-Linear Waves

15 creditsClass Size: 100

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2005/06

Pre-requisites

MATH2360 or MATH2420 or equivalent.

This module is approved as an Elective

Objectives

To give students a clear understanding of the properties of linear and non-linear waves and the relevance of these to physical systems. On completion of this module, students should be able to: a) Calculate dispersion relations and group velocity for simple dispersive wave equations and explain their physical significance; b) Solve simple wave propagation problems in one, two and three dimensions; c) Analyse one dimensional non-linear wave equations in terms of simple waves and discontinuities and use this to solve one dimensional non-linear wave propagation problems; d) Solve quasi-one dimensional steady problems with sonic point conditions.

Syllabus

Waves are present all around us, the most obvious examples being sound, light and water waves. There are many other types of waves, all of which are can be described by the same mathematical theory. This module covers the fundamental theory of both linear and non-linear waves. The important distinction between these is that in non-linear systems not only are there interactions between waves of different frequencies, but there is also a tendency to form sharp fronts, such as shock waves in gases and tidal bores in shallow water. The general theory is illustrated by examples from fluid mechanics, electomagnetism and biological systems. The topics covered are: a) Linear wave equations; characteristics; wave propagation; dispersion relations; group velocity; wave energy; Fourier series and Fourier Transforms; applications to sound waves, water waves and electromagnetic waves. b) Nonlinear wave equations; characteristics; simple waves; discontinuities; shock relations; Riemann problems; dissipation and shock structures; steady solutions and sonic point conditions; applications to compressible flow, shallow water, chromatography, chemotaxis.

Teaching methods

Lectures: 26 hours. 7 examples classes.

Methods of assessment

One 3 hour examination at end of semester (100%).

Reading list

The reading list is available from the Library website

Last updated: 13/05/2005

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