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```c2 < c1.
2.1.2.2 The critical angle and total reflection
If c2 > c1 and θinc = θcrit = sin−1(c1/c2), then pz for the transmitted ray is zero.
This is known as the critical angle. If θinc > θcrit, the z slowness component for the
transmitted ray becomes imaginary,
(ptrans)z = −
(
1
c22
− p2x
)1/2
= −
(
1
c22
− sin
2 θinc
c21
)1/2
= − i sgn(ω)
(
sin2 θinc
c21
− 1
c22
)1/2
.
(2.1.9)
Note that the sign of the imaginary root is taken positive imaginary, i.e. Im(ωpz) <
0, so the transmitted wave decays in the negative z direction (as it must do physi-
cally). Thus
eiωptrans·x = eiωpx x eiωpz z = eiωpx x e|ωpz |z → 0, (2.1.10)
as z → − ∞.
In the wavefront diagram (Figure 2.3), the wavefronts for the transmitted wave
are perpendicular to the interface (from the x dependence in expression (2.1.10)),
prefl
pinc
ptrans
interface
normal
preflpinc
ptrans
(a) (b)
Fig. 2.3. Plane waves incident and totally reflected at a plane interface between
homogeneous half-spaces. The transmitted wave is evanescent. Part (a) shows the
wavefronts and (b) the slowness vectors. The dashed lines are constant amplitude
lines for the evanescent transmitted waves.
2.1 Plane waves 11
and lines of constant amplitude are parallel to the interface (from the z dependence
in expression (2.1.10)). Such a wave is known as an evanescent wave (we avoid
the term inhomogeneous wave as the word inhomogeneous is overloaded in the
subject of wave propagation, being used to describe different features in waves,
media and differential equations). This is in contrast to a travelling wave, as con-
sidered so far, where the constant amplitude and phase surfaces coincide. In the
evanescent wave, the propagation is parallel to the interface and the amplitude de-
cays exponentially away from the interface. No energy is transmitted away from
the interface in the second medium, so we describe the reflected wave as being a
total reflection. This contrasts with the situation when θinc < θcrit or c2 < c1, when
the transmission propagates energy away from the interface and we describe the
reflected wave as a partial reflection. Note that because of the signs used in the
exponent of expression (2.1.3), we have been able to identify the positive real or
imaginary root for pz with propagation in the positive z direction (travelling or
evanescent), and negative with negative propagation. Sometimes in the literature,
different signs have been taken, making the identification confusingly mixed. We
should also mention that Figure 2.3 has been drawn assuming no phase shift be-
tween the incident and reflected or transmitted waves. In fact, when total reflection
occurs, there is normally a phase shift (e.g. Section 6.3.1, equation (6.3.11)), and
the wavefronts should be shifted accordingly.
2.1.3 Time-domain solutions
As we are only considering non-dispersive waves (the velocity, c, is independent
of frequency, ω), it is straightforward to write the solution in the time domain. All
frequencies can be combined to describe the propagation of an impulsive signal.
In a homogeneous medium, a solution is
φ = f
(
t − p · (x − x0)
)
, (2.1.11)
which satisfies the wave equation (2.1.1) for any function f . For
∇φ = −p f ′ and ∇2φ = |p|2 f ′′ (2.1.12)
∂φ
∂t
= f ′ and ∂
2φ
∂t2
= f ′′, (2.1.13)
so provided |p| = 1/c, equation (2.1.1) is satisfied. (The prime indicates ordinary
differentiation of the function with respect to its argument. The arbitrary constant
p · x0 in the argument of the function f (t) is introduced as it is convenient to
measure phase with respect to some origin, x0.)
12 Basic wave propagation
φ φ
x x + �x
�t
t
t t + �t
�x
x
(a) (b)
Fig. 2.4. An arbitrary propagating pulse in: (a) the time domain; and (b) the spa-
tial domain. The pulse is propagating in the x direction.
R
xS
x
p
Fig. 2.5. A spherical wavefront and three rays in a homogeneous medium.
In Figure 2.4, we illustrate a propagating impulse in the temporal and spatial
domains. Obviously the shape of the pulse is reversed between the two domains.
2.2 A point source
So far we have considered plane waves, necessarily of infinite extent, and there-
fore not physically realizable. Let us now consider the wavefronts from a point
source which might represent a physical source, e.g. an explosion. We only con-
sider the geometry of the wavefronts and postpone a more detailed analysis.
Obviously in a homogeneous medium, the wavefronts from a point source
are spherical (Figure 2.5). Denoting the position vector of the point source as xS,
2.2 A point source 13
we define the radial vector, etc.
r = x − xS (2.2.1)
R = |x − xS| = |r| (2.2.2)
rˆ = r/R = sgn(r). (2.2.3)
Then the slowness vector must be
p = rˆ
c
, (2.2.4)
and the solution (2.1.11) reduces to
φ = f
(
t − r · r
cR
)/
R = f
(
t − R
c
)/
R. (2.2.5)
In Figure 2.5, we illustrate a spherical wavefront in a homogeneous medium. The
wavefront is defined by
t = T (x) = constant, (2.2.6)
where T (x) is the travel-time function. Thus the solution is
φ = f
(
t − T (x)
)/
R, (2.2.7)
where for a point source in a homogeneous medium, the travel-time function is
T (x) = |x − xS|/c = R/c. (2.2.8)
We will describe later, in much greater detail, the definition and methods of solving
for the travel-time function. For the moment we note that
p = ∇T, (2.2.9)
which follows as the slowness vector is perpendicular to wavefronts defined by
equation (2.2.6), and the variable (2.2.9) has the magnitude of the slowness (time
divided by distance). By analogy with potential fields (cf. electric potential and
electric field, gravity potential and acceleration due to gravity), given the travel-
time function, T (x), we can construct continuous lines perpendicular to wavefronts
in the slowness direction, called ray paths (e.g. Figure 2.5). As the wavefronts are
propagating in the direction of the rays, we can consider the energy as propagat-
ing along rays. Again later we will describe in much greater detail the properties
of rays in more complicated media (inhomogeneous and/or anisotropic media –
Chapter 5). For now we introduce them as a useful, intuitive concept. Given the
14 Basic wave propagation
transmission
reflection
direct
z
zS
z1 x
(a) (b)
Fig. 2.6. Wavefronts at a plane interface with a velocity decrease. Part (a) shows
the wavefronts at two times, and part (b) the ray paths. The direct wave is the solid
line, the reflection is the dashed line, and the transmission is the dotted line.
definition (2.2.9) we can compute travel times as
T (x) =
∫ x
x0
p · dx, (2.2.10)
where usually the integration is performed along a ray, although this is not neces-
sary.
2.2.1 A point source with an interface
Again we consider two homogeneous half-spaces with a point source in the
first medium. The cases of a velocity increase and decrease are significantly
different.
In Figure 2.6, wavefronts from a point source in a model with a velocity de-
crease are illustrated. The first wavefront is at a time before it reaches the inter-
face. Only the direct, spherical wavefront (the solid line) exists. At a later time,
the wavefront has interacted with the interface. A reflected wavefront (the dashed
line), also spherical but centred on the image point of the source, exists. The trans-
mitted wavefront (the dotted line) spreads out slower and is not a simple spherical
shape. Where the three wavefronts intersect the interface, the wavefronts satisfy
Snell’s law (as in Figure 2.2 except for a velocity decrease).
In Figure 2.7, wavefronts from a point source in a model with a velocity in-
crease are illustrated. The direct and reflected wavefronts are similar to before
2.2 A point source 15
transmission
reflection```   