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.

(LO1) Solve routine problems for hyperbolic equations, including Cauchy problems for a vibrating one-dimensional infinite string, using the method of characteristics.

(LO2) Solve routine problems for time-harmonic plane waves in three dimensions.

(LO3) Solve routine problems for the velocity potential describing spherical waves.

(LO4) Analyse and solve problems of vibration of an infinite volume in three dimensions using Poisson’s formula, including solving a Cauchy problem for a one-dimensional wave equation on a semi-line.

(LO5) Apply the method of characteristics to quasi-linear equations and systems; 1D piston problem.

(LO6) Apply the concept of weak solutions and shocks to quasi-linear equations.

(LO7) Derive the equations of linear water waves, analyse their solution and dispersion

(LO8) Analyse non-linear equations describing solitary waves.

(LO9) Derive solutions for 1D and 3D linearised models and analyse the equations for the velocity potential.

(LO10) Perform analysis of the piston problem with the use of Riemann invariants.

(S1) Problem solving skills

(S2) Numeracy

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.