Waves: a mathematical approach to the common types of wave motion

Preface v
1: The wave equation

§1 Introduction 1
§2 General form of progressive waves 1 
§3 Harmonic waves 2
§4 Plane waves 4 
§5 The wave equation 5
§6 Principle of superposition 6
§7 Special types of solution 7
§8 List of solutions 13 
§9 Equation of telegraphy 1 5
§10 Exponential form of harmonic waves 16
§11 D\' Alembert\'s formula 18
§12 Inhomogeneous wave equation 23
§13 Boundary conditions and mixed problems 25 
§14 Extension of solutions by reflection 27 
§15 A solved example 29
§16 Examples 30

2: Waves on strings

§17 The governing differential equation 35
§18 Kinetic and potential energies 37
§19 Inclusion of initial conditions 38
§20 Reflection at a change of density 39
§21 Reflection at a concentrated load 41
§22 Alternative solutions 43
§23 Strings of finite length, normal modes 43
§24 String plucked at mid-point 45
§25 Energies of normal modes 47
§26 Normal coordinates 49 
§27 String with load at its mid-point 50
§28 Damped vibrations 52
§29 Method of reduction to a steady wave 53
§30 Uniqueness of motion by the energy integral method 54
§31 Examples 56

3: Waves in membranes

§32 The governing differential equation 59
§33 Solution for a rectangular membrane 60
§34 Normal coordinates for a rectangular membrane 62
§35 Circular membrane 63
§36 Uniqueness of solutions 65
§37 Examples 65

4: Longitudinal waves in bars and springs

§38 Differential equation for waves along a bar 67
§39 Free vibrations of a finite bar 68
§40 Vibrations of a clamped bar 69
§41 Normal coordinates 69
§42 Case of a bar in a state of tension 70
§43 Vibrations of a loaded spring 71
§44 Waves in an anharmonic lattice 74
§45 Examples 77

5: Waves in liquids

§46 Summary of hydrodynamical formulae 79
§47 Tidal waves and surface waves 8 1
§48 Tidal waves, general conditions 8 1
§49 Tidal waves in a straight channel 82
§50 Tidal waves on lakes and tanks 85
§51 Tidal waves on rectangular and circular tanks 87
§52 Paths of particles 88
§53 Method of reduction to a steady wave 88
§54 Surface waves, the velocity potential 90
§55 Surface waves on a long rectangular tank 91
§56 Surface waves in two dimensions 92
§57 Paths of the particles 94
§58 The kinetic and potential energies 95
§59 Rate of transmission of energy 96
§60 Inclusion of surface tension. General formulae 98
§61 Capillary waves in one dimension 99
§62 Examples 101

6 : Sound waves

§63 Relation between pressure and density 1 04
§64 The governing differential equation 1 04
§65 Solutions for a pipe of finite length 1 07
§66 Normal modes 107
§67 Normal modes in a tube with moveable boundary 1 08
§68 The velocity potential. General formulae 109
§69 The differential equation of wave motion 1 10
§70 An example 1 1 1
§71 Spherical symmetry 1 1 2
§72 The kinetic and potential energies 1 1 3
§73 Progressive waves in a tube of varying section 1 1 4
§74 Examples 1 1 6

7: Electric waves

§75 Maxwell\'s equations 1 1 8
§76 Non-conducting media and the wave equation 1 20 
§77 Electric and magnetic potentials 122
§78 Plane polarised waves in a dielectric medium 1 24
§79 Rate of transmission of energy in plane waves 1 26
§80 Reflection and refraction of light waves 1 27
§81 Internal reflection 132
§82 Partially conducting media, plane waves 1 33
§83 Reflection from a metal 136
§84 Radiation pressure 1 38 §85 Skin effect 1 38
§86 Propagation in waveguides 1 39
§87 Examples 145

8: General considerations

§88 Doppler effect 149
§89 Beats 1 5 1
§90 Amplitude modulation 1 52
§91 Group velocity 1 5 2
§92 Motion of wave ·packets 155
§93 Kirchhoff\'s solution of the wave equation 1 58
§94 Fresnel\'s principle 162
§95 Diffraction at a pin hole 1 64
§96 Fraunhofer diffraction theory 167
§97 Retarded potential theory 169
§98 Wave propagation in an inhomogeneous medium 1 7 1
§99 Examples 1 74

9: Nonlinear waves

§ 100 Nonlinearity and quasilinear equations 1 78
§ 101 Conservation equations 1 80
§ 102 General effect of nonlinearity 182
§ 103 Characteristics 1 84
§ 104 Wavefronts bounding a constant state 1 9 1
§ 105 Riemann invariants 1 93
§ 106 Simple waves 200
§ 107 The piston problem 202
§ 108 Discontinuous solutions and shock waves 207
§ 109 Examples 2 1 6

Answers 222

Index 226