General Prerequisites: Familiarity is assumed with A10 Fluids and Waves and ASO Integral Transforms. This course combines well with B5.2 Applied Partial Differential Equations and B5.3 Viscous Flow.

Course Overview: In this course, we derive, analyse and solve models for linear and nonlinear wave propagation, focusing on examples in fluid dynamics. The governing equations derived in Part A Waves and Fluids are extended to incorporate energy conservation and further physical effects, including stratification and rotation. Linearised models for wave propagation are analysed using normal modes and transform methods, to explain physical phenomena such as dispersion, group velocity, and the transition from subsonic to supersonic flow. Nonlinear wave problems in gas dynamics and shallow water theory are solved using the theory of characteristics. Models for the propagation of shock waves and bores are derived and solved using conservation principles.

Learning Outcomes: By the end of this course, students will be able:

- to derive the governing equations for inviscid compressible flow from first principles;
- to derive linearised models for small-amplitude perturbations, and to solve the resulting problems using Fourier analysis and the method of characteristics;
- to analyse Fourier integrals using the method of stationary phase;
- to solve nonlinear hyperbolic models for one-dimensional gas dynamics and shallow water theory using the method of characteristics;
- to derive and solve Rankine-Hugoniot relations and entropy conditions governing the propagation of shock waves and hydraulic jumps.

Course Synopsis: Equations of motion: Conservation of mass, momentum and energy for an inviscid compressible fluid. Entropy and the Second Law of Thermodynamics. Flow relative to a rotating frame.

Models for linear wave propagation: Acoustic waves, Stokes waves, internal gravity waves, and inertial waves in a rotating fluid.

Theories for Linear Waves: Normal modes, travelling waves. Fourier integrals, method of stationary phase, dispersion and group velocity. Sub- and supersonic flow past a thin wing.

Nonlinear Waves: method of characteristics, for one-dimensional unsteady gas dynamics and shallow water theory.

Shock Waves: Rankine-Hugoniot relations and entropy conditions for one-dimensional unsteady shock waves, bores and hydraulic jumps, and steady oblique shocks.