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Fluid Waves
Fluid Waves
Richard Manasseh
First edition published 2022
by CRC Press
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ISBN: 978-0-367-27164-0 (hbk)

ISBN: 978-1-032-11319-7 (pbk)
ISBN: 978-0-429-29526-3 (ebk)
DOI: 10.1201/9780429295263
Typeset in CMR10
by KnowledgeWorks Global Ltd.
To Irena and Dylan
Contents
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Preface xvii
I Theory and classical applications 1

1 Fundamentals 3
1.1 Summary of key points . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Basic fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Fluid mechanics and fluid dynamics . . . . . . . . . . 6
1.2.2 Constitutive relations for fluid continua . . . . . . . . 7
1.2.2.1 Stress . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2.2 Strain . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2.3 Relation between pressure and volumetric
strain . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2.4 Relation between shear stress and the rate of
shear strain . . . . . . . . . . . . . . . . . . . 12
1.2.2.5 Surface tension . . . . . . . . . . . . . . . . . 14
1.2.3 Conservation laws . . . . . . . . . . . . . . . . . . . . 15
1.2.3.1 Conservation of mass . . . . . . . . . . . . . 15
1.2.3.2 Conservation of momentum . . . . . . . . . . 17
1.2.4 Scaling of equations and dimensionless groups . . . . . 20
1.3 Flow descriptions . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Euler and Bernoulli equations . . . . . . . . . . . . . . . . . . 27
1.5 Wave tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.1 Complex exponentials . . . . . . . . . . . . . . . . . . 29
1.5.2 Wave equation notations . . . . . . . . . . . . . . . . . 31
1.5.3 Separation of variables and d’Alembert solutions . . . 31
1.5.4 Measuring a wave . . . . . . . . . . . . . . . . . . . . 36
1.5.5 Oscillators and resonance . . . . . . . . . . . . . . . . 37
1.5.6 Introduction to spectra and Fourier transforms . . . . 39
1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
vi
Contents vii
2 Water-surface waves 47
2.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 Linear water-wave theory . . . . . . . . . . . . . . . . . . . . 51
2.3.1 The waves we see . . . . . . . . . . . . . . . . . . . . . 51
2.3.2 Potential flow . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.2.1 Physical assumptions that lead to potential
flow . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.3 Laplace’s equation . . . . . . . . . . . . . . . . . . . . 56
2.3.4 Boundary conditions for water waves . . . . . . . . . . 56
2.3.5 Airy’s solution for surface gravity waves . . . . . . . . 59
2.3.5.1 Separation of variables solution . . . . . . . . 59
2.3.5.2 Applying the boundary conditions . . . . . . 60
2.3.5.3 Dispersion relation . . . . . . . . . . . . . . . 61
2.3.5.4 Ripples . . . . . . . . . . . . . . . . . . . . . 62
2.3.5.5 Phase speed . . . . . . . . . . . . . . . . . . 63
2.3.5.6 Velocity field . . . . . . . . . . . . . . . . . . 65
2.3.6 Surface elevation . . . . . . . . . . . . . . . . . . . . . 66
2.3.7 Particle trajectories . . . . . . . . . . . . . . . . . . . 69
2.3.8 Group velocity . . . . . . . . . . . . . . . . . . . . . . 70
2.3.9 Deep-water approximation . . . . . . . . . . . . . . . . 75
2.3.10 Consequences of deep water . . . . . . . . . . . . . . . 77
2.3.10.1 Maximum wavelength of ocean swell . . . . . 77
2.3.10.2 V-shaped wakes in deep water . . . . . . . . 77
2.3.10.3 Deep-water wave focusing . . . . . . . . . . . 80
2.3.11 Shallow-water approximation . . . . . . . . . . . . . . 80
2.3.12 Consequences of shallow water . . . . . . . . . . . . . 82
2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3 Sound waves 89
3.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.3 Linear sound-wave theory . . . . . . . . . . . . . . . . . . . . 92
3.3.1 Use and control of sound . . . . . . . . . . . . . . . . 92
3.3.2 The wave equation for sound waves . . . . . . . . . . . 94
3.3.3 Solution of the wave equation . . . . . . . . . . . . . . 98
3.3.4 Relation to shallow-water waves . . . . . . . . . . . . 100
3.3.5 Refraction . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3.6 Acoustic impedance . . . . . . . . . . . . . . . . . . . 102
3.3.7 Reflection, scattering and transmission . . . . . . . . . 103
3.3.8 Representation and measurement of sound . . . . . . . 105
3.3.8.1 Spectral representation of sound . . . . . . . 105
3.3.8.2 Sound-measurement instruments . . . . . . . 107
3.3.9 Geometrical spreading . . . . . . . . . . . . . . . . . . 110
3.3.10 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . 111
viii Contents
3.4 Building acoustics . . . . . . . . . . . . . . . . . . . . . . . . 113

3.4.1 Reverberation . . . . . . . . . . . . . . . . . . . . . . . 113
3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4 Internal gravity waves 119

4.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3 Linear internal gravity-wave theory . . . . . . . . . . . . . . . 120
4.3.1 The influence of gravity within a fluid . . . . . . . . . 120
4.3.2 Two-layer rigid-lid interfacial waves . . . . . . . . . . 122
4.3.3 Waves in continuously stratified fluids . . . . . . . . . 125
4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Waves in rotating fluids 133

5.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 Linear inertia-wave theory . . . . . . . . . . . . . . . . . . . . 135
5.3.1 Coriolis force . . . . . . . . . . . . . . . . . . . . . . . 135
5.3.2 Inertial oscillations in an unbounded domain . . . . . 136
5.3.3 Relation to gravity waves . . . . . . . . . . . . . . . . 141
5.3.4 Inertial oscillations with boundary conditions . . . . . 142
5.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6 Introduction to some nonlinear wave theories 147

6.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.3 Nonlinearity in fluid waves . . . . . . . . . . . . . . . . . . . . 149
6.4 Stokes drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.4.1 Eulerian and Lagrangian displacements . . . . . . . . 149
6.4.2 Perturbation approach for the drift velocity of
1D waves . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.4.3 Perturbation approach for the drift velocity in
2D waves . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.5 Solitary waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.5.1 Balancing nonlinear momentum and dispersion . . . . 159
6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7 Nonlinear wave interactions 165

7.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.3 Mean flows driven by waves . . . . . . . . . . . . . . . . . . . 166
7.4 Nonlinearly coupled waves . . . . . . . . . . . . . . . . . . . . 173
7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Contents ix
II Further applications 179

8 Ocean wave energy conversion 181
8.2 Introduction to wave-energy conversion . . . . . . . . . . . . 183
8.2.1 The wave-energy resource . . . . . . . . . . . . . . . . 183
8.3 Issues with wave-energy conversion . . . . . . . . . . . . . . . 186
8.3.1 A plethora of inventions . . . . . . . . . . . . . . . . . 186
8.3.2 The need for resonance . . . . . . . . . . . . . . . . . 188
8.4 Wave-energy converter technologies . . . . . . . . . . . . . . . 189
8.4.1 Rigid pendulum . . . . . . . . . . . . . . . . . . . . . 189
8.4.2 Liquid pendulum (oscillating water column) . . . . . . 190
8.4.3 Heaving Buoy . . . . . . . . . . . . . . . . . . . . . . . 193
8.5 Analysis of a generic WEC . . . . . . . . . . . . . . . . . . . 195
8.5.1 Response of a generic WEC . . . . . . . . . . . . . . . 195
8.5.2 Useful power extracted from ocean waves . . . . . . . 196
9 Bubble acoustics 203

9.2 Volumetric oscillations of bubbles . . . . . . . . . . . . . . . . 205
9.2.1 The collapse of a spherical cavity . . . . . . . . . . . . 205
9.2.2 Natural frequencies of bubbles . . . . . . . . . . . . . 209
9.3 Rayleigh-Plesset equation . . . . . . . . . . . . . . . . . . . . 212
9.3.1 Surface tension, vapour and driving pressures . . . . . 212
9.3.2 Viscous dissipation . . . . . . . . . . . . . . . . . . . . 213
9.4 Linear bubble acoustics . . . . . . . . . . . . . . . . . . . . . 214
9.4.1 Linearised Rayleigh-Plesset equation . . . . . . . . . . 214
9.4.2 Thermal damping . . . . . . . . . . . . . . . . . . . . 215
9.4.3 Radiation damping . . . . . . . . . . . . . . . . . . . . 216
9.4.4 Linear damped bubble oscillator equation . . . . . . . 216
9.5 Applications of linear bubble acoustics . . . . . . . . . . . . . 219
9.5.1 Industrial measurements . . . . . . . . . . . . . . . . . 219
9.5.2 Sounds of ocean waves . . . . . . . . . . . . . . . . . . 220
9.5.3 Volcanic bubbles . . . . . . . . . . . . . . . . . . . . . 221
9.6 Nonlinear bubble acoustics and applications . . . . . . . . . . 222
9.6.1 Sonochemistry . . . . . . . . . . . . . . . . . . . . . . 222
9.6.2 Medical ultrasound diagnostics . . . . . . . . . . . . . 224
9.6.3 Medical ultrasound therapeutics . . . . . . . . . . . . 227
10 Surface-wave breaking in weather and climate 229

10.2 Wave breaking and air-sea exchange . . . . . . . . . . . . . . 230
10.2.1 Criteria for wave breaking . . . . . . . . . . . . . . . . 230
10.2.2 Types of wave breaking . . . . . . . . . . . . . . . . . 233
10.3 Global climate consequences of ocean-wave breaking . . . . . 234
x Contents
10.3.1 Energy transfer from whitecapping to microscale

processes . . . . . . . . . . . . . . . . . . . . . . . . . 234
10.3.2 Outline of air-sea exchange . . . . . . . . . . . . . . . 236
10.3.3 Influence of sea-spray aerosols on climate . . . . . . . 238
10.3.4 Influence of bubbles from breaking waves on climate . 239
10.3.5 Outline of the oceanic part of the carbon cycle . . . . 240
11 Rotating-fluid waves in space and planetary systems 241

11.2 Rotating-fluid waves in stellar and planetary physics . . . . . 241
11.2.1 The origin of rotation . . . . . . . . . . . . . . . . . . 241
11.2.2 Inertia waves in stars . . . . . . . . . . . . . . . . . . . 244
11.2.3 Magnetism and life on planets and moons . . . . . . . 245
11.2.4 Geodynamo mechanisms . . . . . . . . . . . . . . . . . 247
11.3 Engineering of rotating spacecraft . . . . . . . . . . . . . . . . 249
11.3.1 Rotation for attitude control . . . . . . . . . . . . . . 249
11.3.2 Rotation for artificial gravity . . . . . . . . . . . . . . 250
12 Nonlinear environmental waves 253

12.2 Rogue waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
12.3 The tsunami: an ocean-surface soliton . . . . . . . . . . . . . 254
12.4 Internal solitons . . . . . . . . . . . . . . . . . . . . . . . . . . 256
12.4.1 Mesoscale atmospheric solitons . . . . . . . . . . . . . 256
12.4.2 Gravity currents . . . . . . . . . . . . . . . . . . . . . 259
12.4.3 Thunderstorm solitons and aviation . . . . . . . . . . 261
12.4.4 Oceanic internal solitons . . . . . . . . . . . . . . . . . 262
13 Streaming in medicine, industry and geophysics 265

13.2 Acoustic streaming in medicine . . . . . . . . . . . . . . . . . 266
13.3 Acoustic microstreaming . . . . . . . . . . . . . . . . . . . . . 268
13.3.1 Microstreaming principles . . . . . . . . . . . . . . . . 268
13.3.2 Microbubble microstreaming in medicine . . . . . . . . 269
13.4 Streaming in rotating fluids and planetary physics . . . . . . 271
13.4.1 Observations of streaming flow in rotating fluids . . . 271
13.4.2 Possible mechanisms for streaming flows in rotating
fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Bibliography 275
Index 287
Nomenclature xi
Nomenclature
Number type R: real; C: complex. MKS: metre-kilogram-second units; SI: standard
Système International units. To save space with multiple units, slashes are used
instead of the power notation used elsewhere in this book, so kg/m/s2 is kg m−1 s−2 .
Dashes (-) indicate dimensionless quantities. A few symbols, e.g. m and T , have two
meanings; if so, the meanings are quite different and should be apparent from the
context. Symbols only used briefly during a derivation or explanation are not below,
since their definition is on the same page or very close.
Roman letters
Type Units
MKS SI
• Used to denote any variable any any

A Amplitude of a wave or oscillator C any
A× Area, cross-sectional, in general R m2
Ah Area, cross-sectional, of pipe or cylinder R m2
a Amplitude of an oscillator R m
ai Constant used in various derivations, e.g. a1 , a2 R any
aa Acoustic absorption co-efficient R -
au Amplitude of solution to KdV equation R -
a Acceleration vector C m/s2
B Parameter in solution to KdV equation R -
b Constant used in various derivations, e.g. b1 , b2 R any
c Speed of wave propagation R m/s
c0 Speed of wave propagation, linear approx. R m/s
D Length constant, typically a diameter R m
D Prefix indicating material (‘total’) derivative
d Length constant, typically engineering depth R m
Ev Bulk modulus R kg/m/s2 Pa
F Force C kg m/s2 N
F Force vector C kg m/s2 N
F Force amplitude per unit mass R m/s2
F̂ Displacement amplitude of forcing R m
F Function of azimuthal angle ϕ only C -
Fr Froude number R -
xii Nomenclature
f Frequency (in cycles per second) R s−1 Hz

f0 Frequency, natural (in cycles per second) R s−1 Hz
fd Frequency shift due to Doppler effect R s−1 Hz
f (•)Function of • any any
f⊕ Approx. to Coriolis parameter at given latitude R rad/s
g Acceleration due to gravity, surface of Earth R m/s2
g Acceleration due to gravity vector R m/s2
g0 Reduced gravity in a two-layer system R m/s2
gc0 Reduced gravity in a gravity current R m/s2
g(•) Function of • any any
H Height or depth constant, typically some
engineered object, such as a ship draught R m
Ĥ Wave height when the waves may be nonlinear R m
Hm0 Significant wave height R m
h Height or depth constant, typically some
natural feature such as depth of the sea R m
I Intensity of wave energy R kg/s3 W/m2
=(•) Imaginary part of •
Ir Iribarren number √ R -
i Imaginary number ( −1) C -
i Integer index R -
Jm Bessel function of the first kind, order m R -
j Integer index R -
K Stiffness of a spring R kg/s2
k Wavenumber, general, or vertical (z-direction) R m−1
k Wavenumber vector R m−1
L Length constant, or length scale R m
L⊕ Radius of deformation (or Rossby radius) R m
Lp Sound pressure level R - dB
Lv Length scale in the vertical direction R m
L Dimensionless wavelength R -
` Wavenumber in the y-direction R m−1
m Mass R kg
m Wavenumber in azimuthal direction R rad−1
Nomenclature xiii
n Integer number R -
N Buoyancy frequency R rad/s
N Integer number, usually maximum in series R -
P Pressure, total (or ‘absolute’) R kg/m/s2 Pa
P0 Pressure, total, in some initial steady state R kg/m/s2 Pa
P∅ Pressure, ambient R kg/m/s2 Pa
Patm Pressure, atmospheric R kg/m/s2 Pa
Ph Pressure, hydrostatic R kg/m/s2 Pa
Ps Pressure, engineering static R kg/m/s2 Pa
P Function of spatial variables (e.g. x, y, z) only R kg/m/s2 Pa
P̄η̂ Power, cycle-averaged, per unit crest length of
of single-frequency wave R kg m/s3 W/m
P̄H Power, cycle-averaged, per unit crest length of
of ocean-wave spectrum R kg m/s3 W/m
P̄P Power, cycle-averaged, engineering system R kg m2 /s3 W
p Pressure, dynamic C kg/m/s2 Pa
p̂ Pressure amplitude R kg/m/s2 Pa
pR Real part of p R kg/m/s2 Pa
p∅ Pressure, dynamic, reference (acoustics) R kg/m/s2 Pa
p∞ Pressure from external driver (acoustics) C kg/m/s2 Pa
pσ Pressure, surface tension R kg/m/s2 Pa
Q Pressure constant, usually a pressure scale R kg/m/s2 Pa
R Radius, time-varying, of an object R m
R0 Radius constant R m
R Function of r only R -
<(•) Real part of •
Re Reynolds number R -
r Distance in the radial direction R m
St Strouhal number R -
s Distance in an arbitrary direction R m
T Time constant, typically one period (cycle) R s
T Temperature, absolute R K K
T Function of t only C -
t Time R s
t Dimensionless time R -
xiv Nomenclature
U Speed constant, usually a velocity scale R m/s

U Spatial part of velocity vector C m/s
u Horizontal (x) or radial (r) velocity component C m/s
u Velocity vector (u, v, w) C m/s
u Dimensionless speed in x-direction R -
û Amplitude of x-direction velocity component R m/s
u0 Linear solution for fluid velocity (x-direction),
or zeroth eigenmode of the linear solution C m/s
ui ith eigenmode of of the linear solution C m/s
ũ Particle-location velocity (x-direction) C m/s
uR Real part of u R m/s
uS Drift speed (Stokes drift) R m/s
V Volume R m3
V Volume per unit mass R m3 /kg
V0 Volume per unit mass when fluid at rest R m3 /kg
v Horiz. (y) or azimuthal (ϕ) velocity component C m/s
w Vertical (z) or axial velocity component C m/s
x Distance in x-direction, usually horizontal R m
x Distance vector (x, y, z) R m
x̃ Location of a particle in the x-direction R m
X Function of x only (in 3D) or x only (in 1D) C -
y Distance in y-direction, usually horizontal R m
Y Function of y only C -
z Distance in z-direction, usually vertically up;
for rotating-fluids, z is along the axis of rotation R m
z̃ Location of a particle in the z-direction C m
Ẑ Acoustic impedance, specific characteristic R kg/m2 /s
Z Function of z only R m
Nomenclature xv
Greek letters
α Angle R rad
αT Thermal diffusivity R m2 /s
β Angle R rad
γ Adiabatic index (ratio of specific heats) R -
∆ Prefix indicating a difference in a variable
δ Perturbation in bubble radius C m
δω Stokes-layer thickness R m
Dimensionless small parameter in derivations R -
ζ Damping ratio R -
ζP Damping ratio due to useful-power extraction R -
ζr Damping ratio due to wave radiation R -
ζT Damping ratio due to thermal losses (bubbles) R -
ζµ Damping ratio, linearised fluid-dynamical
(wave-energy conversion) or viscosity (bubbles) R -
η Surface elevation of surface gravity waves C m
ηR Real part of η R m
η̂ Amplitude of surface gravity waves; if waves
are nonlinear, defined as Ĥ/2 R m
θ Angle, or latitude of the Earth R rad
κ Wavenumber in x-direction (Cartesian);
in r-direction (spherical polars) R m−1
κp Polytropic index R -
Λ Wavenumber in r-direction (cylindrical polars) R m−1
λ Wavelength R m
µ Dynamic viscosity R kg/m/s Pa s
ν Kinematic viscosity R m2 /s
ξ Displacement of a fluid particle (real) or of
mechanical oscillator in a fluid (complex) C m
ρ Density R kg/m3
ρ0 Time-averaged, constant density (may vary R kg/m3
slowly in z)
% Small variation in density from ρ0 C kg/m3
σ Surface tension coefficient R kg/s2 N/m
τ Stress component C kg/m/s2 Pa
τ Stress tensor C kg/m/s2 Pa
Φ Phase angle R rad
φ Velocity potential C m2 /s
ϕ Azimuthal angle R rad
Ω Angular rotation rate R rad/s
Ω⊕ Angular rotation rate of the Earth R rad/s
ω Frequency R rad/s
ω0 Frequency, natural (undamped) R rad/s
ω0ζ Damped natural frequency R rad/s
xvi Nomenclature
Operators
∇ Vector differential operator, ‘del’ or ‘nabla’ R m−1

∇2 ∇ · ∇, ‘Laplacian’ R m−2
∇⊥2 ∇ · ∇ in horizontal only R m−2
• Time-mean of • R any
Preface
Aims and motivation

The aims of this book are:
1. to list useful key results for selected fluid-wave problems;
2. to show full derivations of selected key results;

3. to illustrate diverse fluid-wave applications in engineering and nature.
An immense variety of phenomena involving mechanical waves occur in

fluids. Fluid waves animate the ocean, provide us with music and affect the
climate, while many vital new technologies for medicine, industry and in-
frastructure all involve fluid waves. Calculations are needed to pursue these
applications, and it is hoped this book may be a convenient entry-point both
for the calculations and for an appreciation of the applications.
This book is written for students undertaking the later years of a university
degree, or post-graduate and professional readers. The main prerequisite to
understanding this text is differential and integral calculus. Students should
ideally also have had some exposure to vector calculus and the principle of
perturbation methods. However, readers whose mathematical knowledge is
limited to algebra should still be able to use this book
How to use this book

The reader may consult first the Summary of key points that begins each
Chapter, where it is possible that a formula or result of use to the reader
may be found immediately. The page on which each key formula was derived
is indicated. It is highly recommended that at least the section leading up
to the key formula is read, to ensure the formula is not applied unawares of
the assumptions behind it, which may lead to erroneous results. Important
reference texts are also indicated in each Summary.
References are collected at the end of the book rather than after each
chapter. This is because some references are relevant to multiple chapters.
A few problems follow each chapter in the first part of the book. These
may be useful in the teaching of a subject based on this book. They include
very elementary questions that experience has shown can trip up students
undertaking calculations under time pressure, some multiple-choice questions
xvii
xviii Preface
to test students’ comprehension and a few harder questions. The majority of

questions should not take longer than a few minutes. The focus is on problems
placed in a practical-applications context.
Structure
Part I addresses Aims 1 and 2. It comprises selected introductory fundamental
topics and results on fluid waves. These are topics thought to be of perpetual
relevance, such as surface gravity waves and sound waves. It is not compre-
hensive, omitting some significant topics such as waves due to instabilities in
fluid flows, many classes of geophysical waves affected by stratification and
rotation, and shock waves in compressible flows. Within each chapter, the
introductory nature of this book has also meant that many sub-topics are
omitted. The references provided cover the missing material. Meanwhile, a
few topics are subjected to full, if rather lengthy, mathematical derivations so
that there are few gaps left as ‘exercises for the student’. Thus, the reader is
made aware of the many steps required to address a typical problem in this
field and is fully equipped to tackle similar problems. Inevitably, however,
many detailed derivations are not undertaken.
Part II addresses Aim 3. It is intended to showcase the breadth of rele-
vance of wave phenomena in fluids. Thus, Part II is rather like a collection
of specialised review papers, which in most cases refer to sources right up to
the date of publication of this book. Some chapters involve significant deriva-
tions, but most are explanatory. In some cases, the applications, which are as
diverse as renewable energy, cancer treatment, climate change, and the search
for habitable exoplanets, are pursued to a depth such that the relevance to
fluid waves may appear tangential; nevertheless, without the ability to cal-
culate fluid-wave problems, rigorous pursuit of the applications would not be
possible.
Approach and acknowledgements

Where references are made to significant equations in other chapters, equa-
tions are generally repeated, rather than forcing the reader to turn back
through many pages and lose one’s train of thought. A clear exception is
the two-layer internal wave derivation, for which many pages of derivation
would be identical to that of surface gravity waves.
The disadvantage of this ‘stand-alone’ approach, apart from verbosity, is
that some principles common to many types of wave are introduced in the
chapter where they may be of most benefit, possibly obscuring their relevance
to other types of wave. For example, refraction and reflection are introduced in
the context of sound waves, but these are relevant to all the waves in this book;
and conversely, beats, admittedly important in music, are instead derived for
water waves. Wherever possible, cross-references are made to counter this
issue.
Preface xix
I am very grateful to Filippo Nelli, Danica Tothova, Elissa Goodrich, Justin

Leontini and Shaung Zhu for helpful and insightful comments on the text, and
finally, to Gagandeep Singh and his team at Taylor & Francis for being patient
and helpful publishers.
Part I
Theory and classical

applications
1
Fundamentals
1.1 Summary of key points

• The ambient pressure in a fluid, P∅ , is given by (1.2) on page 8,
P∅ = Patm + Ph + Ps ,
where Patm is the atmospheric pressure, conventionally taken to be

101 325 Pa, Ps is some static pressure that may be applied by an engi-
neering system, and Ph is the hydrostatic pressure, given by (1.1) on page
8,
Ph = ρ0 gh ,
where ρ0 is the assumed-constant density of the fluid of depth h above
the point of interest and g is the acceleration due to gravity, which is
defined as 9.80665 m s−2 (Bureau international des poids et measures,
Paris, 2006), with more precise values of g calculable as a function of
latitude (Moritz, 2000) and height above sea level (Li and Götze, 2001).
For water, ρ0 = 998 kg m−3 and for air, ρ0 = 1.2 kg m−3 (Streeter and
Wylie, 1979). The total pressure P in the fluid, including any dynamic
pressure p due to fluid motion, is given by (1.3) on page 9,
P = P∅ + p .
• The Ideal Gas Law, relating the absolute pressure P and the volume
per unit mass, V, to their initial values, P0 and V0 , is given by (1.5) on
page 10,
P V κp = P0 V0 κp ,
where κp is the polytropic index (κp = γ for an adiabatic or isentropic
process, where the adiabatic index γ is very close to 1.4 for air).
• The Viscosity Equation (Newton’s Law of Viscosity), (1.8) on page 13,
relates the shear stress τ to the gradient in velocity by
du
τ =µ ,
dy
DOI: 10.1201/9780429295263-1 3
4 Fundamentals
where µ is the dynamic viscosity, and u is the speed of flow in the x-

direction with the y-direction at right angles. For water, µ = 1.0016 ×
10−3 Pa s at 20◦ C and for air, µ = 1.81 × 10−5 Pa s at 20◦ C (Streeter and
Wylie, 1979).
• The increased pressure on the concave side of a surface due to surface
tension is given by (1.11) on page 15,

1 1
pσ = σ + ,
r1 r2
where σ is the surface tension coefficient which takes a value of 0.07275 ±

0.00036 N m−1 for a water-air surface at 20◦ C (Vargaftik et al., 1983) and
r1 and r2 are the radii of curvature of the surface in the two planes at
right angles to the surface.
• The Law of Conservation of Mass, or Continuity Equation, is given by
(1.15) on page 17,
∂ρ
= −∇ · (ρu) ,
∂t
where ρ is the density that could vary in space and time, t is time, ∇
represents gradients in space and u is the velocity vector.
• The Law of Conservation of Momentum, or Momentum Equation, for a
fluid (the Navier-Stokes momentum equation), is given by (1.19) on page
20,
D(ρu)
= ∇ · τ + ρg ,
Dt
where D/Dt denotes the material (or ‘total’) derivative, τ is the stress
tensor and g is the gravitational acceleration vector; for an incompressible
fluid, (1.19) reduces to (1.20) on page 20,
Du 1
= − ∇P + ν∇2 u + g, ,
Dt ρ0
where the kinematic viscosity is given by ν = µ/ρ0 .

• For all waves, radian frequency, frequency, period, wavenumber
and wavelength are given by three relations, firstly (1.68) on page 36,
ω
f= ,
2π
where f is the frequency in Hertz (or cycles per second) and ω is the fre-
quency in radians per second (often just called the ‘frequency’); secondly,
Summary of key points 5
the period of the waves, T , where T = 1/f , is given by (1.69) on page 36

2π
T = ;
ω
and, according to (1.70) on page 36,
2π
λ= ,
k
where k is the wavenumber in m−1 and λ is the wavelength in metres.
• The relations between wave speed, frequency, wavenumber and wave-
length are given by (1.71) and (1.72) on page 36,
ω
c= ,
k
where c is the wave speed in metres per second, or equivalently by
c = fλ .
• Dimensionless numbers relevant to fluid waves include those derived in

§1.2.4 and listed below, for a length scale of L, velocity scale U and time
scale p
T (for oscillations, T = 2π/ω), and Stokes boundary-layer thickness
δω = 2ν/ω,
Traditional Abbreviation Function Ratio of forces

name of
number
UL
Reynolds Re Nonlinear-inertia to
ν viscous
U
Froude Fr √ Nonlinear-inertia to
gL gravity
L
Strouhal St Oscillatory- to
UT nonlinear-inertia
UT
Keulegan– KC Nonlinear-inertia to
Carpenter L oscillatory
p
Womersley Wo L √ω/ν Oscillatory-inertia to
or 2 L/δω viscous
U
Mach Ma Nonlinear-inertia to
c bulk-stiffness (flow
speed to sound speed)
6 Fundamentals
• The forced response of any linear oscillator is given by (1.83) on page

38,
a 1
=q ,
F̂ 0 2 2
(1 − ω ) + (2ζω ) 0 2
.
2ζω 0

−1
Φ = −tan
1 − ω0 2
where a is the real amplitude of the response, Φ is the phase, F̂ is the

real forcing amplitude in the same units as a, the damping ratio is ζ,
and ω 0 = ω/ω0 where ω0 is the natural frequency and ω is the forcing
frequency.
• The additional pressure due to surface tension, pσ , is given by (1.11)
on page 15,

1 1
pσ = σ + ,
r1 r2
where, for a water-air surface, σ = 0.07275 ± 0.00036 N m−1 at 20◦ C

(Vargaftik et al., 1983) and r1 and r2 are the radii of curvature of the
surface.
• Useful textbooks include Batchelor (1973) for an applied-mathematical

approach to fluid dynamics and Lamb (1932) for many classical fluid
dynamics derivations. Batchelor (1973) includes an appendix in which
the equations of motion for a fluid are given in cylindrical and spheri-
cal co-ordinates. Detailed derivations of many fluid-wave problems are in
Lighthill (1978). Streeter and Wylie (1979) provide an engineering ap-
proach to fluid mechanics, which includes detailed calculations on flow
in pipes and ducts, and turbomachinery and aerospace design. There are
very many other engineering textbooks on fluid mechanics, most of which
follow the same pattern. Useful relations for the density of the atmosphere
and of seawater are given in Gill (1982).
1.2 Basic fluid dynamics

1.2.1 Fluid mechanics and fluid dynamics
A fluid is a substance that flows. This statement, while easily grasped, is im-
precise. The formal definition of a fluid requires an understanding of stresses,
which will be outlined shortly. A fluid is defined as a substance that deforms
indefinitely in response to shear stress. Both liquids and gases are fluids. The
difference between a liquid and a gas is that a given mass of liquid forms a
Basic fluid dynamics 7
distinct surface bounding a finite volume, whereas a given mass of gas ex-
pands to fill whatever container it is housed in. Despite this difference, the
same fluid-dynamical laws - and almost all of the same fluid-wave phenomena
- apply equally to liquids and to gases.
Fluid mechanics is the study of forces and motions in fluids, and of the
forces and motions caused by fluids on solid objects and vice versa. Together
with solid mechanics, fluid mechanics forms the branch of classical physics
called continuum mechanics. The term fluid dynamics is often used inter-
changeably with ‘fluid mechanics’, but is usually reserved for those fluid me-
chanical situations where there is some motion. For example, the calculation
of the force on a dam wall due to the pressure of motionless water on the wall
is a fluid mechanics problem, but not a fluid dynamics problem. Since fluid
waves inherently involve motion, they are fluid-dynamical phenomena.
Continuum mechanics, and thus fluid mechanics and fluid dynamics, is
only valid in a continuum. This is a substance that is definitely composed
of indivisible ‘particles’, such as molecules or atoms, but whose behaviour
only matters to us for volumes very much larger than the individual particles.
Thus, it is possible to average over the very many particles inside the smallest
volume of interest to us and ignore the reality that the particles exist. Thus,
in fluid dynamics, individual molecules or atoms, which move in a statistical
fashion, confer their averaged quantities such as pressure, velocity, density or
temperature to continuum equations, allowing us to use the rules of calculus
to determine the behaviour of the continuum. It is worth noting that if the
smallest volume of interest is large enough, a continuum can also be considered
to be composed of ‘particles’ that are much larger than atoms or molecules.
For example, sand or grains of wheat can be observed to flow rather like a fluid,
and subject to suitable approximations, the flow of such granular materials
could be treated as a fluid dynamics problem. Meanwhile, the particles of
dust that compose the accretion disk of a forming solar system (discussed in
§11.2.1), even though there are great distances between the particles, can be
treated as a continuum when the smallest volume of interest is the size of a
planet, so there are still very many particles in the smallest volume.
1.2.2 Constitutive relations for fluid continua

1.2.2.1 Stress
Before introducing what is sometimes called the ‘Three Laws of Fluid Dy-
namics’, it is important to consider what we might call the ‘Zeroth Law of
Fluid Dynamics’. This is a set of relations, the constants in which depend on
the particular fluid being considered, so that the constants are different, for
example, for water, molten steel and blood, and different for cold air and hot
exhaust gas. The relations mostly connect stress to strain or rate of strain,
and to appreciate these relations, it is first necessary to be clear about stress
and about strain.
8 Fundamentals
Stress is force per unit area and has units of kg m−1 s−2 , with the SI name
Pascal, abbreviated Pa. It is divided into two types, normal stress and shear
stress. The normal stresses create the pressure in the fluid, though the normal
stresses are not uniquely related to the pressure unless the fluid is motionless
or incompressible. Stress, in general, is usually denoted with the symbol τ .
Since there are three dimensions of space (in Cartesian coordinates, the x,
y and z directions), forces could be acting in each of the three directions,
but the force in each direction could be acting on areas normal to all of the
x, y and z directions, giving nine possible combinations. Hence, τ is a 3 × 3
tensor. The three normal stresses are τxx , τyy and τzz , whereas the six shear
stresses are τxy , τxz , τyx , τyz , τzx and τzy . We will return to the shear stresses
in §1.2.2.4 when considering viscosity, but first, the normal stresses creating
pressure must be understood.
A bewildering variety of quantities are called ‘pressure’ in fluid dynam-
ics, engineering, physics and medicine. It is common for students to confuse
these and get the wrong result, so it is helpful to begin with a precise set of
definitions. The hydrostatic pressure is the pressure due to the weight of the
fluid above. If the fluid above is a liquid and thus almost incompressible, the
hydrostatic pressure Ph is given by
Ph = ρ0 gh , (1.1)
where ρ0 is the liquid density, g is the acceleration due to gravity, which

has the standard value of 9.80665 ms−2 (Bureau international des poids et
measures, Paris, 2006), and h is the depth of liquid above. For water at 20◦ C,
ρ0 = 998 kg m−3 and for air at 20◦ C, ρ0 = 1.2 kg m−3 (Streeter and Wylie,
1979).
However, if the fluid above is a gas, such as the gases of the Earth’s at-
mosphere, a more detailed relation than (1.1) is needed. Fortunately, many
practical applications do not involve movements between the Earth’s surface
and high altitude, so that atmospheric pressure, Patm , does not vary as much
as other sorts of pressure to be discussed shortly. In addition, it is possible that
the fluid could be mechanically pressurised by some static pressure Ps above
the hydrostatic and atmospheric pressure and that flows are being consid-
ered relative to the sum of these unchanging pressures. These three pressures
added together give what we will call the ambient pressure, P∅ , (often called
the static pressure), which is thus given by
P∅ = Patm + Ph + Ps . (1.2)
Thus, P∅ is the pressure experienced when there is no fluid motion. Finally,

the symbol P will be used for total pressure (sometimes called the absolute
pressure), which includes the ambient pressure P∅ plus any dynamic pressure
due to fluid motion, p (total minus atmospheric pressure is called gauge pres-
sure in some engineering applications). The dynamic pressure can be negative,
but the total pressure is always positive.
Because the dynamic pressure is due to fluid motion, it will be of most

relevance in fluid-wave calculations, but the ambient pressure will be needed
too. In many circumstances, the dynamic pressure, p, is small relative to the
ambient pressure, P∅ , and because P∅ does not vary significantly, the term
‘pressure’ is often used to refer only to the pressure due to fluid motion.
When we consider fluid waves, it is variations in p that will be created by
waves. Thus
P = P∅ + p . (1.3)
In a motionless fluid, p = 0, so the total and ambient pressures are equal and
are equal to the average of the three normal stresses (and the normal stresses
are negative since the total pressure is positive), so P = P∅ = − 31 (τxx + τyy +
τzz ). If the fluid is moving but is incompressible, only the total pressure P is
given by this average, so P = − 13 (τxx + τyy + τzz ). If the fluid is moving and
is compressible, some part of the normal stresses is due to viscous resistance
to the rate of change of volume with time, which will be outlined in §1.2.2.4
below.
1.2.2.2 Strain
Linear strain is the ratio of one of the lengths of the substance when nor-
mal stress is applied along that length, to its length when not under stress.
Engineers typically study linear strain when learning how metal stretches
under load, for example, but for fluid flows, variations in the volume are
paramount.
Volumetric strain is the ratio of the volume, V , of the substance when
under some additional pressure to its volume when not under such pressure,
V∅ . If the contributions to normal stresses from fluid motion are all negative (so
that the dynamic pressure is positive), the substance is said to be compressed
by the motion. If these contributions are all positive (so that the dynamic
pressure is negative), the motion is said to put the substance under rarefaction
or sometimes ‘expansion’. Note that the ‘pressure’ we have just referred to is
the dynamic pressure p that is due to fluid motion, and thus the compression or
rarefaction is relative to the inherent compression due to the ambient pressure.
If a fluid is compressed, its density, ρ, will be higher than its ambient density
ρ0 , and if a fluid is rarefied, ρ will be lower than ρ0 . For example, in the
propagation of sound waves, ρ varies cyclically about ρ0 , being higher than
ρ0 during the compression half of the oscillation and less than ρ0 during the
rarefaction half of the oscillation.
Shear strain is the angle by which a substance is distorted owing to shear
stress.
10 Fundamentals
1.2.2.3 Relation between pressure and volumetric strain

The general relation between pressure and volumetric strain is given by the
bulk modulus of the fluid, Ev , usually expressed as
∂P
Ev = ρ , (1.4)
∂ρ
in which the volumetric strain can be identified once we realise that ρ/∆ρ =
∆V /V . It can be thought of as the ‘stiffness’ of the fluid since the higher
the bulk modulus, the more pressure is needed to cause a given volumetric
strain. In general, the bulk modulus is a function of temperature and fluid
composition. For liquids, the relation is often extracted from empirical data.
The bulk modulus will be used in §3.3.2 in the derivation of the speed of
sound, and in practice, the bulk modulus is actually determined from speed-
of-sound measurements (Fine and Millero, 1973). For an incompressible liquid,
which of course exists only as a theoretical approximation (albeit a very useful
approximation that we will employ many times), the bulk modulus is infinite.
For water Ev ' 2.2 × 109 Pa.
For gases, however, a much more convenient relation is available, and,
moreover, it can be derived from the fundamental theory of the mechanics of
gas molecules, called the kinetic theory of gases. The Ideal Gas Law relates
any value of total pressure, P , and the specific volume or volume per unit
mass, V, to any other ‘initial’ values of the total pressure and volume per unit
mass of the same gas, P0 and V0 . Here, care has been taken to introduce a
symbol for the initial state of the pressure, P0 , that is different to that of the
ambient pressure, P∅ , since some unchanging or ‘steady’ fluid flow might be
occurring (steady flows are defined in §1.2.4 below), causing some background
dynamic pressure; but if there is no such background flow, P0 = P∅ . Note that
V = 1/ρ and V0 = 1/ρ0 . The Ideal Gas Law is given by
P V κp = P0 V0 κp , (1.5)
where κp is the polytropic index whose value depends on the nature of the
heat transfer occurring during compression or expansion of the gas. If the
compression or expansion is isothermal, meaning that the temperature of the
gas does not alter during the compression or expansion, κp = 1. An isothermal
compression or expansion requires that heat flows out of or into the gas across
the boundaries of whatever is containing the gas without any restriction, so
that the temperature can stay constant. The bulk modulus for a gas is given
by Ev = κp P0 and, as just-noted, it will be used in §3.3.2 in the derivation
of the speed of sound. For isothermal air at atmospheric pressure and 20◦ C,
Ev ' 105 Pa.
If the compression or expansion is adiabatic or isentropic, meaning that
there is zero heat flow across the boundaries, κp = γ, where γ is the adiabatic
index. Adiabatic conditions need not imply the gas is in some container, like
Basic fluid dynamics 11
FIGURE 1.1
A set of oceanographic probes and sample bottles is winched into the ocean
from a research ship to a depth of over 1000 m. At such depths, water cannot
be considered incompressible, so the bulk modulus (1.4) is needed. For seawa-
ter, the bulk-modulus relation between stress and volumetric strain requires
an empirical relation, determined in practice by speed-of-sound measurements;
see Fine and Millero (1973) or Gill (1982). Photograph by Richard Manasseh.
the cylinder of a bicycle pump. If the expansions and contractions are very
rapid and are very small in magnitude, which is typical of most sound waves,
there is insufficient time for significant heat to flow one way on the crest of the
wave before it has to flow the opposite way on the trough, so in practice, the
heat flow is extremely small, if not exactly zero. The two cases of isothermal
and adiabatic behaviour represent the two idealised limiting cases for heat
transfer, so that 1 < κp < γ.
Now, the adiabatic index is related to the molecular structure of the gas
and is given by the formula
2
γ =1+ , (1.6)
nf
12 Fundamentals
∆u∆t
Infinitesimally
small element
∆θ
∆y under shear stress,
at time t + ∆t
FIGURE 1.2
The definition of the rate of shear strain
where nf is the number of degrees of freedom of the gas molecule. This is the
number of ways the atoms in the molecule can move and also move relative to
each other. In moving, the molecule possesses kinetic energy and thus heat.
In principle, the number of degrees of freedom of the molecule is purely a
geometric property of the molecule, but in practice, some degrees of freedom
only occur at high temperatures. For a gas such as argon (Ar) consisting of just
one atom, nf = 3 since the atom can move in the three dimensions of space.
For a diatomic molecule, for example, nitrogen (N2 ) or oxygen (O2 ), two added
degrees of freedom are active at room temperature, so nf = 3 + 2 = 5. Hence,
for both N2 and O2 , γ = 1 + 2/5 = 1.4, and since the Earth’s atmosphere
is about 78% N2 and 21% O2 , the air has a value of γ very close to 1.4.
Meanwhile, carbon dioxide (CO2 ) is a molecule with three atoms arranged in
a line. It has a value of γ of about 1.3 at room temperature. The adiabatic
index is also is equal to the ratio of specific heats of the gas, CP /CV , sometimes
called the ‘heat capacity ratio’.
1.2.2.4 Relation between shear stress and the rate of shear strain
If a fluid is flowing in the x-direction with speed u, and this speed u varies
in the y-direction (at right angles to the x-direction), the fluid is in a state of
shear strain. Imagine that u is increasing in the y-direction. The shear strain is
then constantly increasing with time as the faster fluid at higher y constantly
overtakes the slower fluid. The ability of a fluid to continually suffer an ever-
increasing shear strain is the attribute that defines a fluid; as noted in §1.2.1,
it deforms indefinitely in response to shear stress.
The definition of the rate of change of shear strain can be understood with
the aid of figure 1.2. The rate of change of shear strain is the rate at which
the ‘tilt angle’, ∆θ, changes with time, i.e. ∆θ/∆t, where for small ∆θ,
∆u∆t
∆θ ' tan(∆θ) =
∆y
∆θ ∆u
⇒ ' (1.7)
∆t ∆y
Therefore, when the usual limits are taken as ∆y and ∆u tend to zero, the
rate of change of shear strain becomes du/dy.
However, the fluid does not experience stress without resistance. There
are chemical and physio-chemical forces between the particles of the fluid; for
example, there are attractive forces between the molecules of H2 0 in flowing
water, which transfer momentum from the faster-flowing portions of fluid to
the slower portions, slowing down the faster flows. Similarly, as faster-flowing
portions of gas overtake the slower portions, the molecules may collide, like-
wise transferring momentum. Owing to the random orientation of molecules,
these microscopic interactions inevitably transfer energy from the kinetic en-
ergy of the bulk flow to tiny and random molecular motions, and since the
amount of vibration of atoms and molecules is related to the temperature,
the fluid is thus heated. If the fluid is to continue deforming indefinitely, there
must be a shear stress continually applied to it that balances the shear stress
due to these inter-particle interactions.
The bulk fluid property that controls this transfer of momentum across
fluid layers is called dynamic viscosity (or sometimes ‘molecular viscosity’),
and is denoted with the symbol µ. The relation between the rate of shear
strain, du/dy, and the shear stress due to viscosity, τ , was first understood
by Isaac Newton; it is
du
τ =µ , (1.8)
dy
which is called the Viscosity Equation and traditionally called Newton’s Law
of Viscosity. For water, µ = 1.0016 × 10−3 Pa s at 20◦ C, and for air, µ =
1.81 × 10−5 Pa s at 20◦ C (Streeter and Wylie, 1979).
Now, a fluid flowing only in the x-direction with speed u could have gra-
dients in u in the z-direction as well as the y-direction, and indeed this is
occurring as fluid flows in any conduit (pipe, duct or channel); whatever the
value of the speed u in the middle of the conduit, u must be zero for those
fluid molecules physio-chemically bound to the walls. Thus the τ in (1.8) is
more properly written τyx , the shear stress in the x-direction acting on a plane
normal to the y-direction. Including the other directions gives
∂u
τyx = µ ,
∂y
∂u
τzx =µ . (1.9)
∂z
14 Fundamentals
FIGURE 1.3
A thin film of water flows down a glass feature window, forming waves affected
by surface tension. Photograph by Richard Manasseh.
Furthermore, the speed u could also vary in the x-direction, and as the flow
tries to compress (or expand) the fluid in the x-direction, energy could also be
transferred to heat, giving a form of normal stress in addition to that due to
pressure, the volume-viscous stress, with the corresponding parameter being
called the dilational viscosity or sometimes the ‘second coefficient of viscosity’.
The fluid could also be flowing in any and all of the x-, y- and z-directions
with velocity given by the vector u, where u = (u, v, w). Once flows in the
other directions are possible, for small strains, considering the geometry of
the deformed fluid element, (1.9) becomes

1 ∂u ∂v
τyx = µ + ,
2 ∂y ∂x

1 ∂u ∂w
τzx = µ + , (1.10)
2 ∂z ∂x
and similarly for τxy , τzy , τxz and τyz . This gives a total of six shear stresses.
Plus, in addition to the variation of u in the x-direction, possible variations
of v in the y-direction and w in the z-direction add to the volume viscous
stress. Thus, as noted in §1.2.2.1, there are nine stresses in the stress tensor,
τ , comprising the parts of the three normal stresses due to pressure plus the
three normal stresses due to volume viscosity, and the six shear stresses due
to Newtonian viscosity.
1.2.2.5 Surface tension

Surface tension is important for many fluid dynamical and fluid-wave phe-
nomena, such as ripples (§2.3.5.4) and bubble-acoustic vibrations (§9.3.1). A
liquid was defined in §1.2.1 to be a fluid where a distinct surface bounds its
volume. That is because attractive forces hold the molecules or atoms of the
liquid together, whereas, for gas, the molecules or atoms move comparatively
freely relative to each other and constantly bounce off each other. At the liq-
uid surface, the attractive force is defined to have a value of σ Newton per
metre, so that an imaginary line of length L ‘drawn’ on the surface would have
a force of σL Newtons acting at right angles to it. For a water-air surface,
σ = 0.07275 ± 0.00036 N m−1 at 20◦ C (Vargaftik et al., 1983), or, slightly
more precisely, σ = 0.07268 ± 0.00018 N m−1 at 22◦ C (Berry et al., 2015),
and decreases with temperature (see Vargaftik et al., 1983, for values at var-
ious temperatures). If the surface is curved, it can be shown in a few lines
(Batchelor, 1973) that these forces develop a component normal to the sur-
face, causing an increase in pressure, pσ , on the concave side of the curved
surface given by

1 1
pσ = σ + , (1.11)
r1 r2
where r1 and r2 are the radii of curvature of the surface in the two planes at
right angles to the surface. Clearly, if length scales are small, so r1 , or r2 , or
both are small, surface tension becomes important; conversely, in flows with
large length scales, surface tension is negligible. Examples of such cases are in
§2.3.5.4 and §9.3.1.
1.2.3 Conservation laws

Conservation laws are at the heart of fluid dynamics and thus fluid wave
problems. The laws were determined independently by different scientists over
history, but they can all be expressed by a single relation, the Reynolds Trans-
port Theorem. There are three conservation laws of fluid mechanics and fluid
dynamics:
• Conservation of mass
• Conservation of momentum
• Conservation of energy
Of these, the conservation of mass and momentum are more frequently used in
fluid wave problems, and so will be detailed below. The Law of Conservation of
Energy is often used when fluid dynamics is applied in an engineering context,
and an example of this will occur in §8.5.2.
1.2.3.1 Conservation of mass

The statement that mass is conserved is one of the most fundamental state-
ments in classical physics, which can be expressed as ‘mass can neither be cre-
ated nor destroyed’. Those superficially familiar with nuclear reactions might
disagree since both nuclear fusion and fission reactions result in the conversion
of a small fraction of mass into energy, hence in principle forever ‘destroying’
16 Fundamentals
Infinitesimally
u − ∆u/2 small element, u + ∆u/2
Inflow - cross-sectional - Outflow
ρ − ∆ρ/2 area ρ + ∆ρ/2
A×
∆x -
FIGURE 1.4
Transport of mass in one dimension for an infinitesimally small element of
fluid.
that mass. However, the principle of mass and energy equivalence, elucidated
by Albert Einstein in 1905, and encapsulated by the relation E = mc2 where
E is energy, m is mass and c is the speed of light, effectively unites the Law
of Conservation of Mass with the Law of Conservation of Energy. Excluding
nuclear reactions, however, it is convenient to express the conservation of mass
and energy with separate laws.
The conservation of mass in a fluid-dynamics context is easily expressed
by first imagining a one-dimensional flow in which fluid with density ρ is
transported by a flow with speed u along a channel in the x-direction, as
shown in figure 1.4. The channel has a cross-sectional area A× , say. After
travelling an infinitesimally small distance along the channel, ∆x say, imagine
that the speed has changed from u − ∆u/2 at the entry to the infinitesimally
small element, to u + ∆u/2 at the exit, and the density has likewise changed
from ρ − ∆ρ/2 to ρ + ∆ρ/2. The volume of this short segment is A× ∆x.
Therefore, mass is exiting this short segment of the channel of length ∆x at
a rate (ρ + ∆ρ/2)(u + ∆u/2)A× while it is entering at a rate (ρ − ∆ρ/2)(u −
∆u/2)A× , and the net change of mass in transiting the short segment would
be [(ρ − ∆ρ/2)(u − ∆u/2) − (ρ + ∆ρ/2)(u + ∆u/2)]A× .
Since ∆ρ and ∆u are by definition small, their product is negligible, so
the net change of mass becomes −(∆ρu + ρ∆u)A× . In general, ∆ρ and ∆u
could be positive or negative. Meanwhile, it is possible that the density inside
the short segment is changing with time, so the rate of net increase in mass
inside the short segment is (∆ρ/∆t)A× ∆x. Since mass can neither be created
nor destroyed, it can be neither gained nor lost in its journey along the short
segment of the channel, so that
(∆ρ/∆t)A× ∆x = −(∆ρu + ρ∆u)A× . (1.12)

Dividing (1.12) by the volume A× ∆x of the small segment and taking the
limits as the small quantities tend to zero gives
∂ρ ∂ρ ∂u
=− u−ρ ,
∂t ∂x ∂x
∂ρ ∂
⇒ = − (ρu). (1.13)
∂t ∂x
This reasoning is an application of the Reynolds Transport Theorem to mass.
Now, imagine that rather than being a segment of a channel, the small
volume is in the middle of a flowing mass of fluid, and, adopting Cartesian
coordinates, imagine that mass is able to flow into and out of the volume in
the y and z directions as well, with velocity components u, v and w in the
x, y and z directions respectively. The analysis is the same: dividing by the
volume ∆x∆y∆z at the equivalent of the step that led to (1.13) gives
∂ρ ∂ ∂ ∂
= − (ρu) − (ρv) − (ρw), (1.14)
∂t ∂x ∂y ∂z
or, using standard vector-calculus notation, use the operator ∇ for the deriva-
tives in space, which has the advantage that the coordinate system does not
need to be specified. Then (1.14) has the compact form
∂ρ
= −∇ · (ρu). (1.15)
∂t
This is often called the Continuity Equation.
1.2.3.2 Conservation of momentum

The Law of Conservation of Momentum is traditionally known as Newton’s
Second Law, and it was first expressed by Isaac Newton in 1687. It is the
statement that force is equal to mass multiplied by acceleration, i.e.
F = ma, (1.16)
where F is the force vector, m is mass, and a is the acceleration vector. The
right-hand side of (1.16) is the rate of change of momentum with time. To
prepare for fluid-dynamical calculations, first, make a trivial re-arrangement
of (1.16),
F
a= . (1.17)
m
The purpose of this re-arrangement is to highlight that in fluid dynamics, the
greatest challenge is often posed by the determination of the motion of the
fluid for a given applied force. The calculation of the velocity resulting from
the acceleration can be extremely difficult, as we will see shortly.
18 Fundamentals
In contrast, the determination of motion under Newton’s Second Law is

easy for a solid. For example, if a solid mass of 1 kg is acted upon by a
horizontal force of 10 N (recall that is 10 kg m s−2 ), Newton’s Second Law
(1.17) predicts that it will accelerate in the horizontal direction at 10 m s−2 .
After 0.5 s, it will be travelling horizontally at 5 m s−1 , and if the force
ceases to be applied to it after 0.5 s, it will continue to move horizontally at
5 m s−1 . The Law of Inertia (Newton’s First Law) states that it will continue
to move horizontally at 5 ms −1 forever unless another force is applied to
it. Furthermore, far more complicated systems of accelerating solid masses,
such as the parts of an engine or the motions of planets and spacecraft, are
also amenable to direct integration of (1.17). The latter example results in
predictions accurate centuries or even millennia in advance and our ability to
send machines and people to other worlds.
Now imagine the same force is exerted in the same way, but to 1 kg of
water in an open container. The outcome will be completely different: the
mass of water will immediately become greatly distorted, break into many
drops and be splattered widely.
This stark contrast illustrates a vital difference between solids and fluids.
A portion of fluid can accelerate, not just by changing its velocity with time
at a fixed point but also by moving into a region within the fluid mass where
the velocity is different. This ability caused the fluid mass to distort and thus
break up. A less-dramatic example is water flowing steadily in a wide pipe
that feeds all its flow into a narrow pipe; as the water moves into the narrow
section, its speed increases, even through at every point in the pipe, the speed
is not changing with time. It is mathematically expressed by recognising that
the velocity is a function of space as well as time and that the differentiation
of velocity with respect to time must necessarily include the use of the chain
rule of differentiation. For example, for motion of an incompressible fluid in
the x-direction only, the rate of change of the x component of velocity, u, is
given by
Du ∂u ∂u
= +u . (1.18)
Dt ∂t ∂x
The term ∂u/∂t is called the local acceleration and represents the change
in velocity with time only: exactly the sort of acceleration a solid mass would
experience. The operator D/Dt is called the material derivative or total deriva-
tive though many other terms are used for it.
The term u∂u/∂x has appeared because of the ability of a fluid to change
its velocity by moving into a region where the velocity is different. It is of im-
mense importance. It is a nonlinear term, usually called the advective acceler-
ation (or sometimes ‘convective’ although it is nothing to do with heat-driven
convection). This term, and its full three-dimensional equivalent, accounts for
most of the complexity and beauty of fluid flows. This nonlinearity causes
phenomena such as the vortex shedding that causes the flap of a flag and gen-
erates the fundamental notes of many musical instruments, and it creates the
chaos of turbulence. The unpredictability of the weather is a consequence of

nonlinearity in the conservation of momentum. Concomitantly, the existence
of this nonlinear term also makes the equations of fluid dynamics unsolvable
in general.
While waves on the surface of the water (chapter 2), sound waves (chap-
ter 3), internal waves (chapter 4), and rotating-fluid waves (chapter 5) are
all derived by neglecting this nonlinear term, in chapters 6 and 7, effects of
nonlinearity will be considered; and the effects of nonlinearity feature in many
practical applications in Part II of this book.
A more physical derivation of the acceleration, and also more general
since we will not assume incompressibility, can be achieved by considering
the transport of momentum, just as for the transport of mass, as shown in
figure 1.5. Indeed, this is simply the application of the Reynolds Transport
Theorem to momentum, and exactly the same reasoning is applied to mo-
mentum as to mass. Momentum is exiting the short segment of length ∆x in
Infinitesimally
u − 12 ∆u small element, u + 12 ∆u
Inflow - cross-sectional - Outflow
1 1
ρu − 2
∆(ρu) area ρu + 2
∆(ρu)
A×
∆x -
FIGURE 1.5
Transport of momentum in one dimension for an infinitesimally small element
of fluid.
figure 1.5 at a rate (ρu + 21 ∆(ρu))(u + 12 ∆u)A× while it is entering at a rate

(ρu − 12 ∆(ρu))(u − 12 ∆u)A× . Neglecting the product of the two small quan-
tities, the net change of momentum becomes −[∆(ρu)u + ρu∆u]A× . Just as
for the transport of mass that led to (1.13), take limits as the small quantities
tend to zero, giving a change of momentum equal to
D(ρu)
.
Dt
Considering the force in (1.16), the force is composed of two parts; the
forces acting on the body of the infinitesimal element of fluid, and the forces
acting on the surface of an infinitesimal element of fluid. Of the body forces,
the most significant is usually that due to gravity, with a value per unit volume
of ρg; this is the key to waves on the surface of the sea, for example. There may
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I Theory and classical applications 1

3.4 Building acoustics . . . . . . . . . . . . . . . . . . . . . . . . 113

4 Internal gravity waves 119

5 Waves in rotating fluids 133

6 Introduction to some nonlinear wave theories 147

7 Nonlinear wave interactions 165

II Further applications 179

9 Bubble acoustics 203

10 Surface-wave breaking in weather and climate 229

10.3.1 Energy transfer from whitecapping to microscale

11 Rotating-fluid waves in space and planetary systems 241

12 Nonlinear environmental waves 253

13 Streaming in medicine, industry and geophysics 265

• Used to denote any variable any any

f Frequency (in cycles per second) R s−1 Hz

U Speed constant, usually a velocity scale R m/s

∇ Vector differential operator, ‘del’ or ‘nabla’ R m−1

Aims and motivation

1. to list useful key results for selected fluid-wave problems;

2. to show full derivations of selected key results;

An immense variety of phenomena involving mechanical waves occur in

How to use this book

to test students’ comprehension and a few harder questions. The majority of

Approach and acknowledgements

I am very grateful to Filippo Nelli, Danica Tothova, Elissa Goodrich, Justin

Theory and classical

1.1 Summary of key points

where Patm is the atmospheric pressure, conventionally taken to be

where µ is the dynamic viscosity, and u is the speed of flow in the x-

where σ is the surface tension coefficient which takes a value of 0.07275 ±

where the kinematic viscosity is given by ν = µ/ρ0 .

the period of the waves, T , where T = 1/f , is given by (1.69) on page 36

• Dimensionless numbers relevant to fluid waves include those derived in

Traditional Abbreviation Function Ratio of forces

• The forced response of any linear oscillator is given by (1.83) on page

where a is the real amplitude of the response, Φ is the phase, F̂ is the

where, for a water-air surface, σ = 0.07275 ± 0.00036 N m−1 at 20◦ C

• Useful textbooks include Batchelor (1973) for an applied-mathematical

1.2 Basic fluid dynamics

1.2.2 Constitutive relations for fluid continua

where ρ0 is the liquid density, g is the acceleration due to gravity, which

Thus, P∅ is the pressure experienced when there is no fluid motion. Finally,

Because the dynamic pressure is due to fluid motion, it will be of most