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# 2020/21 Taught Postgraduate Module Catalogue

## MATH5373M Advanced Linear and Nonlinear Waves

### 20 creditsClass Size: 37

Module manager: Dr Oleg Chalykh
Email: O.Chalykh@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2020/21

### Pre-requisite qualifications

MATH2365 or equivalent.

### This module is mutually exclusive with

 MATH3374 Linear and Non-Linear Waves

This module is approved as an Elective

### Module summary

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 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 physical theories, such as fluid mechanics, gas dynamics and biological systems.

### 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) Solve the linear wave equation in one dimension.
b) Solve one dimensional non-linear wave propagation problems. Simple waves and discontinuities.
c) Calculate dispersion relations and group velocity for simple dispersive wave equations and explain their physical significance.
d) Use Bäcklund transformations to calculate soliton solutions.

### Syllabus

a) Linear wave equations; characteristics; causality; wave propagation; dispersion relations; group velocity; wave energy.
b) Hyperbolic systems; characteristics; shock relations. Simple waves and Riemann invariants. Applications to compressible flow and shallow water.
c) Travelling wave solutions of nonlinear dissipative and dispersive waves.
d) An additional topic on soliton solutions of some nonlinear dispersive wave equations. Bäcklund transformations and conservation laws.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 22 1.00 22.00 Private study hours 178.00 Total Contact hours 22.00 Total hours (100hr per 10 credits) 200.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular example sheets.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Open Book exam 3 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated