x
i 1
a
c
i 2
v 2
v 1
Medium 1
Medium 2
(b, − c)
b − x
(0, a)
(x, 0)
Hypocenter
10
174°W
0
TVO
x 5 500 at 1 Hz, 21 000 at 3 Hz
W
T wave
P
Tonga
trench
SOFAR channel
175°W
176°W
5
E
Depth (km)
Fig. 2.5-12 Top: A P wave generated by an
earthquake and reflected between the ocean
floor and surface is trapped in the SOFAR
channel and propagates as a T wave.
Bottom: T waves recorded in Tahiti from
an earthquake in Tonga. The amplitudes
exceeded the gain on the seismometers,
causing them to clip at the top and bottom.
The high-frequency ringing of the T waves
distinguishes them from body and surface
waves. (Talandier and Okal, 1979.
© Seismological Society of America.
All rights reserved.)
2.5 Snell’s law 71
with them. This approach, studying wave propagation using
ray paths, is called geometric ray theory. Although it does not
fully describe important aspects of wave propagation, it is
widely used because it often greatly simplifies the analysis and
gives the correct answer or a good approximation.
The most obvious application of rays is for computing travel
times. To find when a plane wave generated at one position will
arrive at another, we use the travel time, which is the length
of the ray path divided by the velocity. Thus, if waves follow
complicated paths, their travel time is the sum of the travel
times for each portion of the ray path. The travel time for a ray
that has traveled through several media, sometimes as a P wave
and sometimes as an S wave, is found using the appropriate
path length and velocity for each segment.
The concept underlying this approach is Fermat’s principle,
a famous result from optics, the study of light. Fermat’s principle states that the ray paths between two points are those
for which the travel time is an extremum, a minimum or maximum, with respect to the nearby possible paths. The simplest
case is two points in a homogeneous halfspace; the time needed
to traverse the straight line connecting the points is less than
for adjacent paths (Fig. 2.5-13). A second ray path for which
the time is a minimum compared to adjacent paths is that of the
reflected ray satisfying Snell’s law. The direct ray path corresponds to an absolute minimum of the travel time, whereas
the reflected ray corresponds to a local minimum.
Snell’s law can be derived from Fermat’s principle. Consider
the possible ray paths (Fig. 2.5-14) between the point (0, a) in
medium 1, with velocity v 1 , and the point (b, −c) in medium 2,
with velocity v 2 . The ray paths can be parametrized by the
point (x, 0) where they cross the interface. The travel time as
a function of x is
Fig. 2.5-14 Derivation of Snell’s law for refraction using Fermat’s
principle. The ray path between points on opposite sides of the interface
is that for which the travel time is a minimum.
Source
Receiver
Fig. 2.5-13 Two ray paths (solid lines), one for the direct ray and
one for the reflection obeying Snell’s law, connecting two points in a
homogeneous halfspace. The travel time for these paths is less than
for nearby paths (dashed), in accord with Fermat’s principle.
i 1
a
c
i 2
v 2
v 1
Medium 1
Medium 2
(b, − c)
b − x
(0, a)
(x, 0)
Hypocenter
10
174°W
0
TVO
x 5 500 at 1 Hz, 21 000 at 3 Hz
W
T wave
P
Tonga
trench
SOFAR channel
175°W
176°W
5
E
Depth (km)
Fig. 2.5-12 Top: A P wave generated by an
earthquake and reflected between the ocean
floor and surface is trapped in the SOFAR
channel and propagates as a T wave.
Bottom: T waves recorded in Tahiti from
an earthquake in Tonga. The amplitudes
exceeded the gain on the seismometers,
causing them to clip at the top and bottom.
The high-frequency ringing of the T waves
distinguishes them from body and surface
waves. (Talandier and Okal, 1979.
© Seismological Society of America.
All rights reserved.)
2.5 Snell’s law 71
with them. This approach, studying wave propagation using
ray paths, is called geometric ray theory. Although it does not
fully describe important aspects of wave propagation, it is
widely used because it often greatly simplifies the analysis and
gives the correct answer or a good approximation.
The most obvious application of rays is for computing travel
times. To find when a plane wave generated at one position will
arrive at another, we use the travel time, which is the length
of the ray path divided by the velocity. Thus, if waves follow
complicated paths, their travel time is the sum of the travel
times for each portion of the ray path. The travel time for a ray
that has traveled through several media, sometimes as a P wave
and sometimes as an S wave, is found using the appropriate
path length and velocity for each segment.
The concept underlying this approach is Fermat’s principle,
a famous result from optics, the study of light. Fermat’s principle states that the ray paths between two points are those
for which the travel time is an extremum, a minimum or maximum, with respect to the nearby possible paths. The simplest
case is two points in a homogeneous halfspace; the time needed
to traverse the straight line connecting the points is less than
for adjacent paths (Fig. 2.5-13). A second ray path for which
the time is a minimum compared to adjacent paths is that of the
reflected ray satisfying Snell’s law. The direct ray path corresponds to an absolute minimum of the travel time, whereas
the reflected ray corresponds to a local minimum.
Snell’s law can be derived from Fermat’s principle. Consider
the possible ray paths (Fig. 2.5-14) between the point (0, a) in
medium 1, with velocity v 1 , and the point (b, −c) in medium 2,
with velocity v 2 . The ray paths can be parametrized by the
point (x, 0) where they cross the interface. The travel time as
a function of x is
Fig. 2.5-14 Derivation of Snell’s law for refraction using Fermat’s
principle. The ray path between points on opposite sides of the interface
is that for which the travel time is a minimum.
Source
Receiver
Fig. 2.5-13 Two ray paths (solid lines), one for the direct ray and
one for the reflection obeying Snell’s law, connecting two points in a
homogeneous halfspace. The travel time for these paths is less than
for nearby paths (dashed), in accord with Fermat’s principle.