Week-1:Introduction to waves and oscillations, Normal modes of linear vibrating systems with finite degrees of freedom, Eigenmodes (shapes of oscillation) and frequencies
Week-2:Normal modes of a linear, N degree of freedom spring-mass system, continuum limit, linear wave equation and normal modes
Week-3:Nonlinear pendulum: exact solution using elliptic integrals, amplitude dependence of frequency, intro. to perturbation methods: regular and singular, Lindstedt-Poincare technique
Week-4:Damped harmonic oscillator, Duffing oscillator, method of multiple scales
Week-5:Parametric instability and the Kapitza Pendulum, Introduction to Floquet analysis; Capillary-gravity waves on a fluid interface: governing equations and boundary conditions, Normal mode analysis, Deep and shallow water approximations and dispersion relations.
Week-6:Phase and group velocity, Cauchy-Poisson problem for surface waves in deep water: 2D rectilinear and cylindrical geometry, Standing and travelling waves, kinematic interpretation of group velocity; Waves on a fluid cylinder, Rayleigh-Plateau instability, oscillations of a hollow filament.
Week-7:Normal modes of a liquid drop and bubble, Normal modes of compound drops
Week-8:Wind waves and the Kelvin-Helmholtz instability, KH instability as a model for wind wave generation, surface waves in an uniform flow due to an oscillatory pressure source at the surface
Week-9:Stokes wave in deep water, nonlinear travelling wave of constant form, stability of Stokes wave (sideband instability), solitary waves, KdV equation and solitons
Week-10:Faraday instability on a fluid interface, subharmonic response, Floquet analysis, atomization from Faraday waves
Week-11:Particle trajectories in water waves, Stokes drift, long surface gravity waves on inviscid shear flows: Burns dispersion relation
Week-12:Shape and volume oscillations of bubbles, Minnaert frequency, Rayleigh-Plesset equation. (If time permits) Kelvin wave pattern of ship wake in deep water and method of stationary phase, Resonant interactions among water waves