Up: PHYS 20171 Maths of waves and fields

PHYS 20171 Maths of waves and fields

Lecture summaries

 

Below are links to scanned versions of my start-of-lecture summary slides. They are all in the form of (rather large) pdf files.

Lecture 1

  • Topics
    • 0 Ordinary differential equations
      • 0.1 First-order, linear
      • 0.2 Second-order, linear
  • Summary, examples of PDEs
  • Handout: Course overview
  • Textbook references: Boas 13.1, 8.5
  • Further reading: Riley 16.1, 13.1.1; Stephenson 1, 2

Lecture 2

  • Topics
      • 0.3 Simple harmonic motion
      • 0.4 Complex exponentials
    • 1 Wave problems in one dimension
      • 1.1 A stretched string
      • 1.2 Separation of variables
  • Summary
  • Textbook references: Boas 2.9, 2.11, 2.12, 13.1, 13.4
  • Further reading: Riley 2.3, 2.7.1, 17.1; Stephenson 3.1
  • Examples: Sheet 1, Q2

Lecture 3

  • Topics
      • 1.3 Eigenvalue problem
      • 1.4 Normal modes
      • 1.5 Separable solutions
      • 1.6 Superposition
      • 1.7 Initial conditions
  • Summary
  • Textbook references: Boas 13.4
  • Further reading: Riley 15 (intro); Stephenson 3.1
  • Examples: Sheet 1, Q1, Q2

Lecture 4

  • Topics
    • 2 Fourier series
      • 2.1 Sine series
  • Summary
  • Textbook references: Boas 7.1-7.5
  • Further reading: Riley 10; Stephenson 3.2
  • Examples: Sheet 2, Q1

Lecture 5

  • Topics
      • 2.2 General Fourier series
  • Summary
  • Textbook references: Boas 7.3, 7.5, 7.8, 7.10
  • Further reading: Riley 10.2
  • Applet (by Fu-Kwun Hwang)
  • Examples: Sheet 2, Q2

Lecture 6

  • Topics
      • 2.3 Completeness
      • 2.4 Symmetries
      • 2.5 Complex exponential series
  • Summary
  • Textbook references: Boas 7.6, 7.7, 7.11
  • Further reading: Riley 10.1, 10.3, 10.7
  • Examples: Sheet 3, Q1, Q2

Lecture 7

  • Topics
      • 2.6 Sets of orthogonal functions
      • 2.7 Orthogonality of eigenfunctions
      • 2.8 Initial conditions (1.7 continued)
  • Summary, review
  • Textbook references: Boas 7.11, 12.6
  • Further reading: Riley 15.3, 17.2; Stephenson 3.1-3.4
  • Applet (by Paul Falstad)
  • Examples: Sheet 4, Q1, Q2

Lecture 8

  • Topics
    • 3 Other PDE's
      • 3.1 Laplace's equation
  • Summary
  • Textbook references: Boas 13.2
  • Further reading: Riley 16.1.3, 17.2; Stephenson 4.2: Example 3
  • Examples: Sheet 5, Q1

Lecture 9

  • Topics
      • 3.2 The heat-flow equation
  • Summary
  • Textbook references: Boas 13.3
  • Further reading: Riley 16.1.2, 17.1; Stephenson 4.2: Examples 1, 2

Lecture 10

  • Topics
    • 4 Integral transforms
      • 4.1 Fourier transform
      • 4.2 Travelling waves
  • Summary
  • Textbook references: Boas (2nd ed) 15.4; (3rd ed) 7.12
  • Further reading: Riley 11.1
  • Examples: Sheet 5, Q2, Sheet 6, Q1

Lecture 11

  • Topics
      • 4.3 Wave packets
      • 4.4 Band width
      • 4.5 Group velocity
  • Summary
  • Further reading: Riley 11.2
  • Examples: Sheet 6, Q2

Lecture 12

  • Topics
      • 4.6 Convolutions
      • 4.7 The Dirac delta-function
  • Summary
  • Textbook references: Boas (2nd ed) 15.5, 15.7; (3rd ed) 8.10, 8.11
  • Further reading: Riley 11.1.7, 11.1.3, 11.1.4
  • Examples: Sheet 6, Q3

Lecture 13

  • Topics
    • 5 Series solution of ODEs
      • 5.1 Taylor series
      • 5.2 Convergence
      • 5.3 Linear independence
  • Summary
  • Textbook references: Boas 1.12, 1.6, 1.10
  • Further reading: Riley 3.6, 3.3
  • Examples: Sheet 7, Q1

Lecture 14

  • Topics
      • 5.4 SHM the hard way
      • 5.5 Legendre's equation
  • Summary
  • Textbook references: Boas 12.2
  • Further reading: Riley 14.2, 14.6
  • Examples: Sheet 7, Q2

Lecture 15

  • Topics
      • 5.6 Legendre polynomials
      • 5.7 Orthogonality
      • 5.8 Legendre series
  • Summary
  • Textbook references: Boas 12.2, 12.7, 12.8, 12.9
  • Further reading: Riley 14.6
  • Examples: Sheet 8, Q1

Lecture 16

Lecture 17

  • Topics
    • 6 Problems in 2 or 3 dimensions
      • 6.1 Rectangular membrane
  • Summary
  • Further reading: Riley 17.2 (heat-flow equation)
  • Applet (by Paul Falstad)
  • Examples: Sheet 8, Q2; Sheet 10, Q1

Lecture 18

  • Topics
      • 6.2 Waveguide
  • Summary
  • Examples: Sheet 9, Q1

Lecture 19

  • Topics
      • 6.3 The hot plate
  • Summary
  • Textbook references: Boas 10.9, 13.6 (circular drum)
  • Further reading: Riley 17.3: pages 552-559, 567-569; Stephenson 5.1-5.4
  • Applet (by Paul Falstad)
  • Examples: Sheet 9, Q2; Sheet 10, Q2

Lecture 20

  • Topics
      • 6.4 Waves on a sphere
  • Summary
  • Textbook references: Boas 10.9, 13.7 (Laplace's equation)
  • Further reading: Riley 17.3: pages 559-561 (Laplace's equation); Stephenson 5.4: Example 3

Lecture 21

  • Topics
      • 6.4 Waves on a sphere (continued)
      • 6.5 Waves inside a sphere
  • Summary
  • Textbook references: Boas 13.7 (Laplace's equation)
  • Further reading: Riley 17.3: pages 559-565 (Laplace's equation); Stephenson 5.4: Example 3
  • Examples: Sheet 10, Q3

Lecture 22

  • Topics
    • 7 Solution of the heat-flow equation in spherical polar coordinates with uniform initial conditions
  • This lecture cannot be summarised. Go boil yourself an egg!
  • Examples: Sheet 10, Christmas Turkey

12th December 2007

Up: PHYS 20171 Maths of waves and fields