PHY1106 Waves and Oscillators

2009-2010

Aims

The concept of wave propagation permeates the whole of physics with many examples arising from many different physical phenomena; complete books have been written about acoustic waves, electromagnetic waves and de Broglie waves. Even so, there are common underlying principles which make it possible to understand many apparently unrelated systems. The primary aim of the module is to identify and make use of these concepts at an elementary level and also to introduce a wide range of physical phenomena as examples. The module starts by considering the characteristic parameters of a forced, damped harmonic oscillator, and relates them to the characteristic parameters of wave propagation. Later stages discuss the propagation and reflection of waves, using waves on a stretched string as the model system. Longitudinal waves in solids, sound waves in gases, and waves in periodic structures are also discussed, the last of these being the key to much of solid-state physics. The concepts introduced in this module will be developed later in the programme, e.g. in electromagnetism (PHY3143), quantum mechanics (PHY2002) and solid-state physics (PHY2009).

Intended Learning Outcomes

Students should be able to:

Module Specific Skills

• set up the equations associated with simple-harmonic motion, solve them for different physical conditions and recognise situations where they are applicable;
• construct the relevant expressions for alternating current and voltage (using complex-number representation) in an electrical circuit and derive basic quantities, e.g. power dissipation;
• manipulate the wave equation and its solution, explain the concepts of wave number, phase velocity, group velocity and dispersion; apply these concepts to waves in periodic structures and to longitudinal waves in solids and gases;

Discipline Specific Skills

• make a Fourier-series expansion of a simple periodic function;
• to take notes in lectures and then refine them later thereby developing skills in the efficient summarising of material;

Personal and Key Skills

• undertake guided self-study successfully;
• manage their time appropriately in order to meet the weekly-homework assignment deadlines.

Learning and Teaching Methods

Lectures, guided self-study, worksheets, tutorials and problems classes; e-learning resources. Additional problems will be given during lectures (and discussed in subsequent lectures) for homework.

Assignments

Four self-study packs. Ten homework problem sheets. Problem assignments are issued for problem classes.

Assessment

Problems-class assignments (10%), 30-minute mid-term test in Week L5 (20%), 30-minute mid-term test in L8 (20%) and one 90-minute examination (50%).

Syllabus Plan and Content

1. The Physics of Simple and Damped Harmonic Motion
   1. Simple harmonic motion - mass on a spring, equation of motion
   2. Phase angle, displacement, velocity, acceleration
   3. Energy of simple harmonic motion
   4. Damped simple harmonic motion (mechanical system) - oscillatory and logarithmic decrement (exponential notation)
   5. Quality factor Q- energy dissipation
   6. Critical and overdamped mechanical system
2. Forced Oscillator
   1. Steady-state solution for mass on a spring plus driving force
   2. Mechanical impedance (complex impedance, amplitude, phase factor)
   3. Amplitude resonance
   4. Power supplied by the driving force, Q-value
3. Alternating Electrical Currents (Steady State)
   1. Alternating voltage, phasor diagram, amplitude, phase, period
   2. Resistance, inductance and capacitance in an AC circuit: current-voltage relationships
   3. Complex impedance in AC circuits
   4. Power in AC circuits
   5. Series and parallel resonance
4. Introduction to Waves
   1. Definition and examples of wave motion. Transverse and longitudinal waves. Polarization. Plane and spherical waves
   2. Basic wave concepts: Amplitude and phase; Wave number k and angular frequency ω Phase velocity.
   3. The wave equation and its solutions.
   4. The Doppler Effect.
   5. Example: transverse waves on a string.
   6. Energy transfer in wave motion.
5. Superposition of Waves
   1. Standing waves and normal modes.
   2. Partial standing waves.
   3. Fourier series.
   4. Wave packets, dispersion and group velocity.
   5. Example: dispersed wave on a string.
6. Reflection and Transmission of Waves
   1. Characteristic impedance. Reflection and transmission coefficients of amplitude and energy.
   2. Example: Reflection and transmission of transverse waves on a string.
   3. Impedance matching and the quarter-wave transformer.
7. Waves on Periodic Structures
   1. Transverse waves on a one-dimensional periodic structure: dispersion relation, low-pass characteristic, first Brillouin zone.
   2. Normal modes on a one-dimensional periodic structure.
8. Other Examples of Waves
   1. Longitudinal waves in a solid.
   2. Sound waves in a gas.

Core Text

Young H.D. and Freedman R.A. (2007), University Physics (with Modern Physics) (12^th edition), Addison-Wesley, ISBN 0-321-50131-4 (UL: 530 YOU)

Supplementary Text(s)

Pain H.J. (2005), The Physics of Vibrations and Waves (6^th edition), Wiley, ISBN 0-470-01296-X (UL: 531.32 PAI)

Formative Mechanisms

This module is supported by problems classes and tutorials. Students are able to monitor their own progress by attempting problems sheets provided in the lectures, using the on-line self-assessment tests, homework marks, mid-semester tests and tutorial work. The graded mid-semester test scripts are discussed by tutors. Students with specific problems should first approach their tutor, and if the problem is not resolved, the lecturer.

Evaluation Mechanisms

The module will be evaluated using information gathered via the student representation mechanisms, the staff peer appraisal scheme, and measures of student attainment based on summative assessment.