Module Details
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title WAVES, MATHEMATICAL MODELLING
Code MATH427
Coordinator Professor N Movchan
Mathematical Sciences
Nvm@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2024-25 Level 7 FHEQ Second Semester 15

Aims

This module gives an introduction to the mathematical theory of linear and non-linear waves. Illustrative applications involve problems of acoustics, gas dynamics and examples of solitary waves.

Learning Outcomes

(LO1) Apply essential modelling techniques in problems of wave propagation.

(LO2) Analyse how mathematical models of the same type can be successfully used to describe different physical phenomena.

(LO3) Master background mathematical theory used in models of acoustics, gas dynamics and water  waves.

(S1) Problem solving skills

(S2) Numeracy

Syllabus
 

Hyperbolic PDEs. Definitions. Characteristics. Formulation of problems. D''Alembert''s formula.

Outline of inviscid fluid dynamics. Linear theory. Planewaves. Reflection and transmission at a plane interface. Spherical waves.

Dipole fields. Scattering by a solid sphere.

Vibration of an infinite volume. Poisson''s formula.

Introduction to non-linear theory. Systems of quasi-linear first-order partial differential equations. Characteristics. Riemann invariants. Model examples.

Conservation laws, weak solutions and shocks. Definitions and model examples.

Water wave theory. Governing equations. Linearised model. Dispersive waves. Model examples. Non-linear equations, solitary waves.

Recommended Texts
Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):
MATH101 Calculus I; MATH103 Introduction to Linear Algebra; MATH102 CALCULUS II 

Co-requisite modules:
 

Modules for which this module is a pre-requisite:
 

Programme(s) (including Year of Study) to which this module is available on a required basis:
 

Programme(s) (including Year of Study) to which this module is available on an optional basis:
 

Assessment
EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Final Exam  120    60       
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Homework 1  0    10       
Homework 2  0    10       
Homework 3  0    10       
Homework 4  0    10