Synopsis for B6b: Waves and Compressible Flow

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2B45)

Recommended Prerequisites:

The Part A (second-year) course `Fluid Dynamics and Waves'. Though two half-units are intended to stand alone, they will complement each other. This course combines well with B5 Differential Equations and Applications. The introductory Michaelmas Term course B568a is a prerequisite for both parts of the course, and the material in that course will be assumed to be known.

Overview

Propagating disturbances, or waves, occur frequently in applied mathematics. This course will be centred on some prototypical examples from fluid dynamics, the two most familiar being surface gravity waves and waves in gases. The models for compressible flow will be derived and then analysed for small amplitude motion. This will shed light on the important phenomena of dispersion, group velocity and resonance, and the differences between supersonic and subsonic flow, as well as revealing the crucial dependence of the waves on the number of space dimensions.

Larger amplitude motion of liquids and gases will be described by incorporating non-linear effects, and the theory of characteristics for partial differential equations will be applied to understand the shock waves associated with supersonic flight.

Learning Outcomes

Students will have developed a sound knowledge of a range of mathematical models used to study waves (both linear and non-linear), will be able to describe examples of waves from fluid dynamics and will have analysed a model for compressible flow. They will have an awareness of shock waves and how the theory of characteristics for PDEs can be applied to study those associated with supersonic flight.

Synopsis

1––2 Equations of inviscid compressible flow including flow relative to rotating axes.

3––6 Models for linear wave propagation including Stokes waves, internal gravity waves, inertial waves in a rotating fluid, and simple solutions.

7––10 Theories for Linear Waves: Fourier Series, Fourier integrals, method of stationary phase, dispersion and group velocity. Flow past thin wings.

11––12 Nonlinear Waves: method of characteristics, simple wave flows applied to one-dimensional unsteady gas flow and shallow water theory.

13––16 Shock Waves: weak solutions, Rankine–Hugoniot relations, oblique shocks, bores and hydraulic jumps.

1. H. Ockendon and J. R. Ockendon, Waves and Compressible Flow (Springer, 2004).
2. J. R. Ockendon, S. D. Howison, A. A. Lacey and A. B. Movchan, Applied Partial Differential Equations (revised edition, Oxford University Press, Oxford, 2003). Chapters 2.5, 4.5–7.
3. D. J. Acheson, Elementary Fluid Dynamics (Oxford University Press, 1990). Chapter 3
4. J. Billingham and A. C. King, Wave Motion (Cambridge University Press, 2000). Chapters 1–4, 7,8.

Background Reading

1. M. J. Lighthill, Waves in Fluids (Cambridge University Press, 1978).
2. G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1973).