The course is an introduction to the rich field of interfacial waves. The first half of the course prepares the student for studying wave phenomena by introducing discrete mechanical analogues of wave phenomena in fluid systems. The basic principles of normal mode analysis are introduced through point-mass systems connected through springs. The exact solution to the (nonlinear) pendulum equation is used to introduce the notion of amplitude dependence on frequency of the oscillator. The Kapitza pendulum is introduced as a discrete analogue for Faraday waves. Basic perturbation techniques are then introduced for subsequent use. The second half of the course introduces basics of interfacial waves viz. shallow and deep-water approximations, phase and group velocity, frequency and amplitude dispersion etc.. Capillary as well as capillary-gravity waves in various base state geometries (rectilinear, spherical (drops and bubbles including volumetric oscillations of the latter), cylindrical (filaments) are taught and the corresponding dispersion relation derived. The Stokes travelling wave is derived using the Lindstedt-Poincare technique and the amplitude dependence in the dispersion relation is highlighted. Side-band instability of the Stokes wave is discussed. Fluid particle trajectories for linear water waves is derived and the Stokes drift expression is derived. Time permitting, the Kelvin ship wave pattern in deep water is derived using the method of stationary phase. Introductory ideas in resonant interactions among surface gravity waves are discussed. The fundamental aspects studied in the course will be related to various engineering applications continuously.

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-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-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

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

1. Vibrations and Waves, A. P. French, M.I.T. Introductory Physics Series

2. A modern introduction to the mathematical theory of water waves, R. S. Johnson, Cambridge Texts in Applied Mathematics

3. Waves in Fluids, J. Lighthill, Cambridge Mathematical Library

4. Linear and nonlinear waves, G. B. Whitham, Wiley Interscience

5. Fluid Mechanics – P. K. Kundu, Ira. M. Cohen, D. Dowling, Academic Press

6. Nonlinear water waves – L. Debnath, Academic Press

7. Perturbation Methods – Ali H. Nayfeh, Wiley-VCH

8. The acoustic bubble - T. G. Leighton, Academic Press

Dr. Ratul Dasgupta is an Associate Professor at the Chemical Engg. Department at IIT Bombay. He completed his Ph.D. at the Jawaharlal Nehru Centre for Advanced Scientific Research in Bangalore and was a postdoctoral fellow subsequently at the Weizmann Institute of Science in Israel. He has been on the faculty at IIT Bombay since 2014. He works on interfacial waves, hydrodynamic stability and the mechanics of amorphous materials employing a combination of theoretical and computational tools and occasionally simple experiments.

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams:**23 October 2021** Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

**CRITERIA TO GET A CERTIFICATE**

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

**YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.**

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Bombay.It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams:

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Bombay.It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

## DOWNLOAD APP

## FOLLOW US