MTH6129 Oscillations, Waves and Patterns)

MTH6129 Oscillations, Waves and Patterns

Module Announcements

LECTURE ARRANGEMENTS FOR WEEK 12: on 26 March, the 12-1pm lecture operates as normal in Maths 103. The 4-5pm lecture is at the same time but is moved to MATHS 103. There will be no lecture on Tuesday 27 March or tutorials on that day. The REVISION LECTURE is scheduled for April 23, 12-1pm, Maths 103.

14/3/2012 Notes for week 10 lectures are now available, together with some links to other websites with some cartoons of dispersion and group velocity effects. If you find any sites of interest to the module, do let me know of them. Enjoy!

13/3/2012 There was a printing error in question 1(a) of coursework 6 that was handed out and put on line yesterday. There correct version is now available on this site.

29/2/2012 Coursework 5 now available online together with solutions for courseworks 3 & 4.

IMPORTANT ANNOUNCEMENT: ON MONDAY 27/2/12, THE LECTURE SCHEDULED AT 4PM HAS BEEN MOVED TO ROOM BR 3.01

07/02/2012: On-line lecture notes for week 5 on the derivation and solutions of the wave equation are now available. Next week we will be using some of the topics from Calculus III, notably `Fourier analysis' and solving PDEs by the method of separation of variables.

26/01/2012: On-line lecture notes on Coupled Oscillators are now available. We will be solving ODEs for coupled oscillators using an eigenvalue problem next week (week 4), so if you need to revise this topic from your Geometry I or Differential Equations notes, now would be a good time to do so. Also we will be working out the determinants of three-by-three matrices and starting to look at partial differentiation, so again these can be revisited from Calculus II.

18/01/2012: On-line lecture notes now available for Part 1. Additional notes on ODE's and links to the video footage on resonance added.

17/01/2012: Due to clashes with other modules there are now revised tutorial timings (see below).

Lecturer

Professor James E. Lidsey

E-mail: J.E.Lidsey@qmul.ac.uk

Office: Room 455, School of Mathematical Sciences

Office Hour: Tues 11:00-12:00

Lectures

Time		Room
Mon	12:00-13:00	Maths 103
Mon	16:00-17:00	Maths 203
Tue	13:00-14:00	BR 4.02

Exercise Classes

Time		Room
Tue	10:00-11:00	Eng 216
Tue	12:00-13:00	Eng 371

General Information

Welcome to MTH6129 Oscillations, Waves and Patterns.

Waves and vibrations are present in almost all physical systems, from the vibrations in strings to the waves of the oceans and atmosphere. Waves and patterns are also seen in chemical and living systems. This module is an introduction to the mathematical theory of waves, dealing with the solution of differential equations describing, for example, vibrations on strings and waves in fluids. Elementary ideas about nonlinear waves, such as shock formation, are described. The material is illustrated with applications from a wide variety of different systems.

I hope you enjoy the module and wish you all well. Let's enjoy it together which for me means plenty of feedback from you during the term. This can be through questions in class, during the office hours or by email. Please ask questions - it's much more fun having some feedback!

For syllabus and other information, including recommended textbooks, etc., see Module information.

IT IS VITAL TO TAKE ALL COMPONENTS OF THIS COURSE SERIOUSLY. ATTENDANCE AT LECTURES AND EXERCISE CLASSES IS COMPULSORY AND RANDOM REGISTERS WILL BE TAKEN AT BOTH. REGISTRATION OF STUDENTS MISSING LECTURES OR EXERCISE CLASSES OR FAILING TO HAND IN COURSEWORKS WITHOUT VALID REASONS WILL BE TERMINATED.

Syllabus

Part 1. Uncoupled Oscillator: Review of restoring forces and SHM; damped oscillations, strong, weak and critical damping; forced damped oscillations, transient and steady state solutions; resonance.

Part 2. Coupled Oscillators: normal coordinates, normal modes of vibrations, derivation of wave equation as the limit of many coupled oscillators.

Part 3. Waves: derivation of classical wave equation for string; D'Alembert's solution; travelling plane wave solutions; transverse vibrations on a string: harmonic waves, normal modes for string fixed at ends, solution by separation of variables; initial conditions and Fourier sine series; examples, such as vibrations and musical sounds.

Part 4. Waves in Fluids: linear surface waves on deep and shallow water; dispersion relation, phase and group velocities; waves on inclined beds.

Part 5. Patterns: circular membranes (drums): modes of oscillation and their patterns; nonlinear waves and solitons; qualitative introduction to waves and pattern formation in other systems, e.g. biological and chemical systems.

Lecture Notes

Lecture notes are provided 1-2 weeks after the topic is given in lectures. They are intended to ensure that notes from the lectures are correct. Additional reading from text books is strongly encouraged for a full understanding of the material.

Part 1. Uncoupled Oscillators: Notes on Uncoupled Oscillators are here. Summary notes on ODEs are here. A note on Barton's pendulums is here and a link to some footage is here . Links to the Tacoma_Narrows_Bridge disaster are here and here.

Part 2. Coupled Oscillators: Notes on Coupled Oscillators are here. Additional notes for the case where all 3 spring constants are equal can be found here. A short film showing the in- and out-of-phase normal modes is here.

Part 3a. The Wave Equation: Notes on the wave equation and the d'Alembert solution are here. Watch the derivation of the wave equation by clicking onto this site Derivation of the Wave Equation . Some background material on partial differential equations, and in particular the wave equation, can be found here . The method of Separation of Variables will be particulary useful for us.

Part 3b. Waves on Finite Strings: Notes on waves on finite strings and guitar strings can be found here.

Part 4a. Waves in Fluids: Notes on the derivations of the fluid equations and boundary conditions can be found here.

Part 4b. Applications of Linear Water Wave Theory: A summary of the important equations in linear water wave theory can be found here. Notes on the applications of the wave solution we derived and the dispersion relation can be found here. Some cartoons of wave superposition and beats can be found here. Further notes on dispersion and some more cartoons can be found here.

Part 4c. Ocean Waves and Tsunamis: Notes on Ocean waves and Tsunamis can be found here.

Part 5a. Vibrations and Waves on a Circular Membrane/Drum: Notes on Waves on a Circular Membrane/Drum can be found here. An excellent computer-generated simulation of the vibrations in terms of Bessel functions is here. (Scroll down to the bottom of that page.

Part 5b. Solitons: Notes on Solitons can be found here. A nice site on solitons with some background and history is here.

Courseworks

Courseworks are (usually) handed out in Lectures - email the module organiser if you are unable to collect due to absence with good reason, etc.

Starred coursework questions will be marked and returned within two weeks of hand-in. These must be collected during the exercise classes.

Coursework 1 . . . . . . . Solution to Coursework 1

Coursework 2 . . . . . . . Solution to Coursework 2

Coursework 3 . . . . . . . Solution to Coursework 3

Coursework 4 . . . . . . . Solution to Coursework 4

Coursework 5 . . . . . . . Solution to Coursework 5

Coursework 6 . . . . . . . Solution to Coursework 6

Coursework 7 . . . . . . . Solution to Coursework 7

Essential Prerequisites

The essential prerequisites for this module are MTH5106 Dynamics of Physical Systems and MTH5102 Calculus III . The former module will be very useful for parts 1 and 2 of the syllabus. The important material from Calculus III is Chapter 3 on vector differentiation, Chapter 6 on Fourier series, and Chapter 7 on Laplace's equation.

Useful Formulae and Background Material

Some useful formulae and integrals involving trigonometric functions can be found here Trigonometric functions

Some useful formulae and integrals involving hyperbolic functions can be found here Hyperbolic functions

Some useful formulae involving derivatives can be found here Derivatives

A really good online revision site is Just the Maths which covers the calculus and ordinary differential equations material required for this module.

Text Books

Vibrations and Waves: A P French (Chapman and Hall)

The Physics of Vibrations and Waves: H J Pain (John Wiley and Sons)

Vibrations and Waves in Physics: I G Main (Cambridge University Press)

Wave Motion: Billingham and A C King (Cambridge University Press)

Physics of Waves: W C Elmore and M A Heald (McGraw-Hill)

Waves: C A Coulson (Oliver and Boyd)

Wave Physics: S Nettel (Springer Verlag)

Online Videos

See the course tumblr at Oscillations Waves and Patterns for lots of interesting pictures and videos of oscillations, waves and patterns.

The Wave Equation and Partial Differential Equations

The wave equation is a linear, second-order, partial differential equation. It plays a central role in the `waves' section of the module. The solutions to this differential equation describe how waves travel in a wide variety of different physical settings. The wave equation also has lots of really neat mathematical properties, as we will see.

Watch the derivation of the wave equation by clicking onto this site Derivation of the Wave Equation

Some background material on partial differential equations, and in particular the wave equation, can be found here . The method of Separation of Variables will be particulary useful for us.

Exam Related Information

Homepage of James Lidsey