# MAGIC045: Linear and nonlinear (M)HD waves and oscillations

## Course details

### Semester

Spring 2010
Monday, January 11th to Friday, March 19th

### Hours

Live lecture hours
20
Recorded lecture hours
0
0

## Description

Part 1: Linear and nonlinear waves in fluids

L1: Examples of waves in nature; Waves on a stretched string; derivation of governing PDE; kinetic, potential energy; D'Alembert's GS, solution for strings of infinite length Heaviside fnc, 2 examples

L2: Standing waves on a string on a finite length, standing waves, normal modes, method of separation of variables, plucked string (example: triangle initial profile); Mode energy, Fourier transform, 2D wave equation, Bessel equation, Bessel's solution

L3: Plane waves, sound waves (3D wave equation), eq of continuity, velocity potential; Acoustic waveguides: Reflection at rigid wall A planar waveguide A cylindrical waveguide Energy transmission along waveguides (transmission, reflection, amplification)

L4: Linear inviscid/viscous water waves, incompressible fluids, governing equations (Laplace eq, Bernoulli eq), kinematic BC, monochromatic surface waves, DR, limits (shallow and deep water) concept of group velocity, wavepacket, particle path in surface waves

L5: Quasi-linear 1st order PDEs, associated equation, characteristics, 2 examples, properties of characteristics, discontinuities (weak, strong), shocks, jump condition

L6: Modelling traffic flow (example -> break down time), kinematic wave, Riemann problem, Burger's equation, Hopf-Cole tr-> diffusion eq.

Part 2: Linear MHD waves

L1: MHD equations (ideal), limits of MHD, MHD equilibria, force free field, potential field

L2: linear MHD waves in homogeneous media: Alfven waves (circularly polarised), slow and fast MHD waves, Fridrich's diagram, characteristics in ideal MHD

L3: Internal gravity waves, acoustic-gravity waves MHD waves at a single magnetic interface

L4: MHD waves in magnetic slabs, gov. eq., DR, classification of modes MHD waves in magnetic flux tubes (infinite), gov. eq., DR, modes

L5: MHD waves in thin flux tubes (gravitatinal startification), Klein-Gordon equation (sound, slow and Alfvenic)

L6: Observations of MHD waves and oscillations

Part 3: Nonlinear waves in fluids

L1: Surface waves, Korteweg - de Vries equation for shallow water (including derivation); Elementary solution (travelling wave) of the KdV equation, cnoidal waves, solitons

L2: The scattering problem; solitons and inverse scattering Examples: the delta function, the \sech2 function; Inverse scattering: The solution of the Marchenko equation; Examples: reflection coefficient with one pole, zero relflection coefficient

L3: The initial-value problem for the KdV equation; inverse scattering and the KdV equation; time evolution of scattering data, continuous and discrete spectra

L4: Reflectionless potentials, examples: solitary wave, two-soliton solution, N-solitons; [description of solution when b(k) ≠ 0: delta-fnc initial profile, ±\sech2 initial profile]

L5: Properties of the KdV equation: conservation laws, infinite set of conservation laws; Lax KdV hierarchy; Hirota's method: biliniear form; Backlund tansformation

L6: General inverse methods: AKNS method, ZS methods; Painleve conjecture

Part 4: Weakly nonlinear waves in MHD

L1: Nonlinear MHD surface waves in thin flux tubes (Leibovich-Pritchard-Roberts equation)

L2: Nonlinear surface and speudo-body waves in thin flux tubes (Molotovshchikov-Ruderman equation)

### Prerequisites

No prerequisites information is available yet.

### Syllabus

No syllabus information is available yet.

## Lecturer

• Rv

### Prof Robertus von Fay Siebenburgen

University
University of Sheffield

## Bibliography

No bibliography has been specified for this course.

## Assessment

### Attention needed

Assessment information will be available nearer the time.