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School of Mathematics

# MATH45122 - 2011/2012

General Information
• Title: Non-Linear Waves and Granular Flow
• Unit code: MATH45122
• Credits: 15
• Prerequisites: None
• Co-requisite units: None
• School responsible: Mathematics
• Members of staff responsible: Prof. N. Gray
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Other Resources

 You can click on the adjacent image to see a video showing the formation of a granular shock wave. An avalanche of 100's and 1000's enters from the top right and flows down an inclined chute until the front hits the end wall. This brings the grains rapidly to rest and an upslope propagating shock wave forms at which the flow thickness increases by a factor of twelve. It is analogous to density stopping waves that one sometimes encounters on motorways, as well as shocks in gas dynamics. Further animations can be found on the course materials page.

## Specification

### Aims

The aim of the course is to introduce some ideas associated with nonlinear wave propagation such as wave steepening, shock formation, dispersion, and soliton properties, in the context of granular flows, such as avalanches and other natural hazards. Non-linear waves have an extremely broad range of applications from the formation of tsunamis in water to shock waves in gas dynamics, as well as density waves in traffic flows.

### Brief Description of the unit

Granular flows provide a simple setting in which to introduce the effects of nonlinearity in physical systems. In particular waves may steepen and break to form shocks (jumps in solutions). The formation of such discontinuities will be demonstrated using simple granular flow experiments during the lectures. The mathematical theory underlying such systems will be explained in detail, using many examples from applications ranging from particle segregation and shock wave formation in granular avalanches, to gas dynamics and shallow water flows such as tsunamis.

### Learning Outcomes

On successful completion of the course unit students will be able to

• understand the differences in the solution properties and physics of physical systems governed by the kinematic wave equation, the K - dV equation, Burger's equation and segregation equation, as well as the avalanche and the shallow water equations;
• solve ut + c(u)ux = 0 for given initial data and be able to identify the formation of shocks;
• understand how breaking waves in two-dimensions can be represented in terms of shock waves;
• solve the avalanche and shallow water equations using the method of characteristics for simple flows;
• perform a phase plane analysis for the K - dV and related equations to identify travelling wave solutions, solitary wave solutions.

None.

### Syllabus

1. The hyperbolic wave utt = c02Δ2u, ut + c0ux = 0; wave forms; Fourier synthesis; dispersion;
C(k) = dw/dk, group velocity; diffusion, e.g. Burger's linear equation ut + c0ux = vuxx.
2. First order wave equation ut + c(u)ux = 0; characteristics; conservation ideas; conservation forms; granular and traffic flow models. Waves in other physical systems.
3. First order equations in two-dimensions; breaking size segregation waves and their representation in terms of shocks.
4. Shallow water wave theory; the nonlinear equations; wave breaking, dam break problems, via characteristics; normal and oblique shocks in granular flows, linearisation and check against linear theory, and linear irrotational theory.
5. Irrotational water wave theory to obtain the Boussinesq equations; steady solutions of the Boussinesq equations; derivation of the Korteweg-de Vries equation from Boussinesq equations; conservation laws for K - dV; analytical solution of K - dV equation; the soliton.

### Textbooks

• P.G. Drazin and R.S. Johnson, Solitons, An Introduction, CUP 1989.
• G.B. Whitham, Linear and Non-linear Waves, Wiley 1974.
• J. Stoker, Water Waves, Wiley Interscience 1957.
• L. Debnath, Nonlinear Water Waves, Academic Press 1994.

### Teaching and learning methods

22 lectures and 11 examples classes. In addition students should expect to do at least seven hours private study each week for this course unit.

### Assessment

Mid-semester coursework: weighting 20%
End of semester examination: two and a half hours weighting 80%

## Arrangements

Course materials from lecturer.