Fig. 2.4-6 Particle motion plots for two time
segments of the radial and transverse components
shown in Fig. 2.4-5. SKS and SKKS, which are
primarily SV waves, are strongest on the radial
component (left), whereas S diff is primarily an
SH wave, and so is strongest on the transverse
component (right).
5 The transverse waves we see at a beach are not seismic waves in the water, but
instead propagate at the water surface and involve a rolling motion in two dimensions
similar to Rayleigh waves (Section 2.7.2).
2.4 Seismic waves 59
Transverse
SKS + SKKS
S diff
Transverse
Radial
Radial
and S-wave velocity, termed β or v S ,
β = (µ /ρ)
1/2
,
(46)
show that the seismic velocities depend in different ways on the
elastic constants of the material. Because the rigidity µ and the
bulk modulus K (Eqn 2.3.74) are positive, P waves travel faster
than S waves. Thus the first wave arriving from an earthquake
is always a compressional wave. As a result, the nomenclature P originally denoted the first-arriving, “primary” wave,
whereas S denoted the “secondary” wave.
Although both velocities depend on the rigidity, the shear
velocity does not depend on the bulk modulus K, because these
waves involve no volume changes. Because the shear velocity
is proportional to the square root of the rigidity, shear waves
cannot propagate through an ideal (µ = 0) fluid. However,
compressional waves propagate in an ideal fluid with a velocity
proportional to K 1/2 . Thus only compressional waves can
travel through the earth’s outer core or the ocean.
5
To get a feel for these wave velocities, consider typical values
for various parameters. The earth’s crust is approximately a
Poisson solid, with elastic constants λ ≈ µ ≈ 3 × 10 11 dyn/cm 2 .
Thus, for a density of 3 g/cm 3 , the P-wave velocity is 5.5 ×
10 5 cm/s, or 5.5 km/s. Similarly, the S-wave velocity is 3.2 ×
10 5 cm/s, or 3.2 km/s. Hence a P wave propagating with a
velocity of 5.5 km/s and a period of 2 s has a wavelength
(Section 2.2) of (5.5 km/s × 2 s) or 11 km. The frequency
is 0.5 s −1 (the unit s −1 is called a Hertz, or Hz), and the
wavenumber is 2π/11 = 0.57 km
−1 . On the other hand, a wave
with a period of 10 s and the same velocity has a wavelength of
55 km, and a frequency of 0.1 Hz. The longer-period wave has
a longer wavelength and a lower frequency.
0.001
Reflection
seismology
Surface waves
Body waves
Normal modes
Crustal
deformation
Frequency (Hz)
Period (s)
0.01
0.1
1
10
100
1000
1000 100
10
1
0.1
0.01 0.001
10
−7
10
4
10
5
10
6
10
7
10
−6
10
−5
10
−4
Fig. 2.4-7 Seismic spectrum showing the frequencies at which various
analyses are conducted.
The “seismic spectrum,” showing seismic waves of various
frequencies and types, is shown in Fig. 2.4-7. Studies of earthquakes typically use the period range from approximately 0.1 s
to more than 3000 s, or frequencies from 10 Hz to 3 × 10 −4 Hz
(0.1 mHz). Higher-frequency waves of 20–80 Hz generated by
explosions or other artificial sources are used in reflection
seismology to explore the earth’s crust. Still higher frequencies,
3–12 × 10 3 Hz (3–12 kHz), propagating primarily in the ocean,
are used by marine geophysicists to map the sea floor. At the
other end of the spectrum, ground motions with periods longer
than 10 4 s are due to slow crustal motions (Section 4.5) rather
than propagating seismic waves.
Earthquake sources generate both P and S waves, with the
S waves generally significantly larger. Figure 2.4-8 shows
seismograms of the three components (vertical, or up–down,
north–south, and east–west) of ground motion from seismic
waves generated by an earthquake ~280 km beneath two
seismic stations in Japan. The seismic waves are coming up
vertically toward the surface. The first arrival, a P wave, has
displacement along the direction of propagation, and therefore
appears primarily on the vertical component. The large later
arrival, a shear wave, has displacement perpendicular to the
direction of propagation, and thus appears most on the horizontal components.
segments of the radial and transverse components
shown in Fig. 2.4-5. SKS and SKKS, which are
primarily SV waves, are strongest on the radial
component (left), whereas S diff is primarily an
SH wave, and so is strongest on the transverse
component (right).
5 The transverse waves we see at a beach are not seismic waves in the water, but
instead propagate at the water surface and involve a rolling motion in two dimensions
similar to Rayleigh waves (Section 2.7.2).
2.4 Seismic waves 59
Transverse
SKS + SKKS
S diff
Transverse
Radial
Radial
and S-wave velocity, termed β or v S ,
β = (µ /ρ)
1/2
,
(46)
show that the seismic velocities depend in different ways on the
elastic constants of the material. Because the rigidity µ and the
bulk modulus K (Eqn 2.3.74) are positive, P waves travel faster
than S waves. Thus the first wave arriving from an earthquake
is always a compressional wave. As a result, the nomenclature P originally denoted the first-arriving, “primary” wave,
whereas S denoted the “secondary” wave.
Although both velocities depend on the rigidity, the shear
velocity does not depend on the bulk modulus K, because these
waves involve no volume changes. Because the shear velocity
is proportional to the square root of the rigidity, shear waves
cannot propagate through an ideal (µ = 0) fluid. However,
compressional waves propagate in an ideal fluid with a velocity
proportional to K 1/2 . Thus only compressional waves can
travel through the earth’s outer core or the ocean.
5
To get a feel for these wave velocities, consider typical values
for various parameters. The earth’s crust is approximately a
Poisson solid, with elastic constants λ ≈ µ ≈ 3 × 10 11 dyn/cm 2 .
Thus, for a density of 3 g/cm 3 , the P-wave velocity is 5.5 ×
10 5 cm/s, or 5.5 km/s. Similarly, the S-wave velocity is 3.2 ×
10 5 cm/s, or 3.2 km/s. Hence a P wave propagating with a
velocity of 5.5 km/s and a period of 2 s has a wavelength
(Section 2.2) of (5.5 km/s × 2 s) or 11 km. The frequency
is 0.5 s −1 (the unit s −1 is called a Hertz, or Hz), and the
wavenumber is 2π/11 = 0.57 km
−1 . On the other hand, a wave
with a period of 10 s and the same velocity has a wavelength of
55 km, and a frequency of 0.1 Hz. The longer-period wave has
a longer wavelength and a lower frequency.
0.001
Reflection
seismology
Surface waves
Body waves
Normal modes
Crustal
deformation
Frequency (Hz)
Period (s)
0.01
0.1
1
10
100
1000
1000 100
10
1
0.1
0.01 0.001
10
−7
10
4
10
5
10
6
10
7
10
−6
10
−5
10
−4
Fig. 2.4-7 Seismic spectrum showing the frequencies at which various
analyses are conducted.
The “seismic spectrum,” showing seismic waves of various
frequencies and types, is shown in Fig. 2.4-7. Studies of earthquakes typically use the period range from approximately 0.1 s
to more than 3000 s, or frequencies from 10 Hz to 3 × 10 −4 Hz
(0.1 mHz). Higher-frequency waves of 20–80 Hz generated by
explosions or other artificial sources are used in reflection
seismology to explore the earth’s crust. Still higher frequencies,
3–12 × 10 3 Hz (3–12 kHz), propagating primarily in the ocean,
are used by marine geophysicists to map the sea floor. At the
other end of the spectrum, ground motions with periods longer
than 10 4 s are due to slow crustal motions (Section 4.5) rather
than propagating seismic waves.
Earthquake sources generate both P and S waves, with the
S waves generally significantly larger. Figure 2.4-8 shows
seismograms of the three components (vertical, or up–down,
north–south, and east–west) of ground motion from seismic
waves generated by an earthquake ~280 km beneath two
seismic stations in Japan. The seismic waves are coming up
vertically toward the surface. The first arrival, a P wave, has
displacement along the direction of propagation, and therefore
appears primarily on the vertical component. The large later
arrival, a shear wave, has displacement perpendicular to the
direction of propagation, and thus appears most on the horizontal components.