Linear and nonlinear (M)HD waves and oscillations (MAGIC045)
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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)
Spring 2009 (Monday, January 19 to Friday, March 27)
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