## MATH2610 Oscillations and Waves

### 10 creditsClass Size: 200

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

MATH1382 or equivalent.

### Co-requisites

MATH2200, MATH2360 or equivalent

This module is approved as an Elective

### Objectives

This module provides an introduction to the theory of small-amplitude linear oscillations of systems of finitely many particles and of a continuous system. Both propagating and standing waves are analysed, the latter by using Fourier series. On completion of this module, students should be able to: (a) formulate and solve dynamical problems for systems of particles in terms of Lagrange?s equations; (b) solve wave problems associated with the 1-dimensional wave equation; (c) apply the technique of separation of variables to solve initial boundary value problems for the wave equation; (d) use Fourier series to solve wave problems; (e) have an appreciation of wave energy, and the concepts of reflection, transmission, dispersion and group velocity.

### Syllabus

This module provides an introduction to the theory of small-amplitude (linear) oscillations of systems of finitely many particles or of a continuous system such as a stretched string (a paradigm for more practical waves such as sound, water or electromagnetic waves). Both propagating waves and standing waves are analysed, the latter requiring the widely applied theory of Fourier Series. Normal modes of oscillation are introduced, as are Lagrange?s equations which lead on in subsequent modules to Hamiltonian Dynamics and Quantum Mechanics. Topics covered include: 1. Revision of simple harmonic motion (SHM) and damped harmonic motion. 2. Lagrange?s equations for systems of finitely many particles. 3. Normal modes of oscillation about stable equilibrium. 4. Derivation of wave equation for stretched string. 5. D?Alembert?s solution for an infinite string. 6. Standing waves on a finite string; separation of variables, eigenvalues. 7. Fourier series. 8. Energy in waves; reflection and transmission. 9. Dispersive waves; group velocity.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures (22 hours) and examples classes (11 hours).

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

15% coursework, 85% 2 hour written examination at end of semester.