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```chapters on ray
theory to emphasize the similarities and reuse results. The chapter then develops
various solutions of these equations, in homogeneous and inhomogeneous lay-
ers, using the propagator and ray expansion formalisms. The WKBJ and Langer
asymptotic methods, and the WKBJ iterative solutions, are included. The second
chapter of this group, Chapter 8: Inverse transforms for stratified media, then de-
scribes the inverse transform methods that can be used with the solutions from the
previous chapter. These include the Cagniard method in two and three dimensions,
the WKBJ seismogram method and the numerical spectral method. These methods
are then used in the final chapter of the group, Chapter 9: Canonical signals, which
describes approximations to various signals that occur in many simple problems.
These range from direct and turning rays, through partial and total reflections,
to the introductory Chapter 2 where the various singularities, discontinuities and
degeneracies of ray results had been emphasized. Particular emphasis is placed on
describing the signals using simple, ‘standard’ special functions.
The final chapter, Chapter 10: Generalizations of ray theory, describes recent
extensions of ray theory which increase the range of application and validity and
Introduction 5
include some of the advantages of the transform methods. The methods are Maslov
asymptotic ray theory which extends ray theory to caustic regions; quasi-isotropic
ray theory which extends ray theory to the near degeneracies that exist in weakly
anisotropic media; Born scattering theory which models signals scattered by small
perturbations in the model and importantly allows signals due to errors in the ray
solution to be included; and the Kirchhoff surface integral method which allows
signals and diffractions from non-planar surfaces to be modelled at least approxi-
mately.
2
Basic wave propagation
This introductory chapter introduces the reader to the concepts of seismic
waves – plane waves, point sources, rays and travel times – without math-
ematical detail or analysis. Many different types of rays and seismic signals
are illustrated, in order to set the scene for the rest of this book. The objective
of the book is to provide the mathematical tools to model and understand the
signals described in this chapter.
In this chapter, we introduce the basic concepts of wave propagation in a simple,
stratified medium. None of the sophisticated mathematics needed to solve for the
complete wave response to an impulsive point source in an elastic medium is intro-
duced nor used. Ideas such as Snell’s law, reflection and transmission, wavefronts
and rays, travel-time curves and related properties, are introduced. The concepts
are all straightforward and should be intuitively obvious. Nevertheless many sim-
ple questions are left unanswered, e.g. how are the amplitudes of waves found, and
what happens when wavefronts are non-planar or singular. These questions will
be answered in the rest of this book. This chapter sets the scene and motivates the
more detailed investigations that follow.
2.1 Plane waves
2.1.1 Plane waves in a homogeneous medium
In this section, we introduce the notation and nomenclature that we are going to use
to describe waves. We assume that the reader has a basic knowledge of waves and
oscillations, and so understands a simple wave equation with the notation of com-
plex variables used for a solution.
The simplest form of wave equation describing waves in three dimensions is the
Helmholtz equation
∇2φ = 1
c2
∂2φ
∂t2
. (2.1.1)
6
2.1 Plane waves 7
Table 2.1. Symbols, names and units used to describe
plane waves.
Symbol Name Units
φ field variable
A (complex) amplitude
t time [T]
x position vector [L]
ω circular frequency [T−1]
ν = ω/2π , frequency (Hz) [T−1]
k wave vector [L−1]
k = |k|, wavenumber [L−1]
c = ω/k, wave (phase) velocity [LT−1]
T 2π/ω = 1/ν, period [T]
λ 2π/k = c/ν, wavelength [L]
p = k/ω, slowness vector [L−1T]
|p| = 1/c, slowness [L−1T]
k · x − ωt phase [0]
When the velocity, c, is independent of position, the plane-wave solution of this
equation is
φ = Aei(k·x−ωt) (2.1.2)
= Aeiω(p·x−t). (2.1.3)
In Table 2.1 we give the definitions, etc. of the variables used in or related to this
equation (see Figure 2.1).
Note the signs used in the exponent of the oscillating, travelling wave given by
equations (2.1.2) or (2.1.3). The same signs will be used later in Fourier transforms
(Chapter 3). Various sign conventions are used in the literature. We prefer this one,
a positive sign on the spatial term and a negative sign on the temporal term, as
it leads to the simple identification that waves travelling in the positive direction
have positive components of slowness. For most purposes, it is convenient to use
the slowness vector, p, rather than the wave vector, k, to indicate the wave direction
as it is independent of frequency. Surfaces on which k · x is constant, as illustrated
in Figure 2.1, are known as wavefronts.
2.1.2 Plane waves at an interface
If a plane wave is incident on an interface, we expect reflected and transmitted
waves to be generated. Some continuity condition must apply to the field variable,
8 Basic wave propagation
λ k
x
Fig. 2.1. A plane wave in a homogeneous medium. At a fixed time, t , the lines
represent wavefronts where k · x is constant. Thus if k · x = 2nπ , the wavefronts
are separated by the wavelength, λ.
φ, its derivative or combinations thereof. Even without specifying the details of the
boundary condition, it is clear that, for a linear wave equation, all waves must have
the same frequency ω, and the same wavelength along the interface. Without these
conditions, the waves could not match at all times and positions. Let us consider a
plane interface at z = 0, separating homogeneous half-spaces with velocities c1 for
z > 0, and c2 for z < 0 (Figure 2.2). For simplicity we rotate the coordinate system
so the incident slowness vector, pinc, lies in the x–z plane, i.e. py = 0. Then the
wavelength along the interface is 2π/kx = 2π/ωpx , so the slowness component
px must be the same for all waves.
2.1.2.1 Snell’s law
As |p| = 1/c, the incident wave’s slowness vector is
pinc =

 px0
pz

 =


px
0
−
(
c−21 − p2x
)1/2

 = 1
c1

 sin θinc0
− cos θinc

 . (2.1.4)
The square root in pz is taken positive, so the minus sign is included as the incident
wave is propagating in the negative z direction (Figure 2.2). The angle θinc is the
2.1 Plane waves 9
z
prefl
pinc
ptrans
interface
preflpinc
ptrans
normal
θinc θrefl
θtrans
(a) (b)
Fig. 2.2. Plane waves incident, reflected and transmitted at a plane interface be-
tween homogeneous half-spaces. The situation when c2 > c1 is illustrated. Part
(a) shows the wavefronts and (b) the slowness vectors.
angle between the slowness vector and the normal to the interface and is taken so
0 ≤ θinc ≤ π/2. The reflected and transmitted waves must have slowness vectors
prefl =


px
0
+
(
c−21 − p2x
)1/2

 = 1
c1

 sin θrefl0
+ cos θrefl

 (2.1.5)
ptrans =


px
0
−
(
c−22 − p2x
)1/2

 = 1
c2

 sin θtrans0
− cos θtrans

 , (2.1.6)
where the angles θrefl and θtrans are similarly defined.
Equality of the x slowness component for these vectors immediately leads to
the reflection law
θinc = θrefl, (2.1.7)
and the refraction or Snell’s law
sin θinc
c1
= sin θtrans
c2
. (2.1.8)
10 Basic wave propagation
The situation illustrated in Figure 2.2 is when c2 > c1 so θtrans > θinc and the slow-
ness vector is refracted away from the normal. The opposite occurs when```