110 Basic Seismological Theory
the overtone branches cross, because these modes include both
P and SV energy. Some spheroidal modes involve primarily
radial motion, and some involve primarily tangential motion,
with a full spectrum in between.
However, the basic patterns seen for torsional modes also
apply for spheroidal modes. For a given frequency, modes
to the left (low l) correspond to core phases, and those in the
center correspond to mantle body wave phases, with the core
diffraction (dashed lines) being the boundary between these
groups of modes. The modes furthest to the right correspond to
Rayleigh surface waves. The modes corresponding to P-wave
phases are further to the left than their SV counterparts because
P waves travel faster than S waves. These ideas can be visualized by considering the radial eigenfunctions (Fig. 2.9-9d).
The two curves for each spheroidal mode represent the two
radial eigenfunctions U(r) (radial) and V(r) (tangential). Modes
with low l and higher n have larger deep displacements, so
superposition of modes with very low l yields the core phases.
For the low-order overtones (n = 2 is shown), the displacements
are closer to the surface as l increases. Thus the n = 0 branch
with l greater than about 20 corresponds to fundamental mode
Rayleigh waves, and the higher branches (n = 1, 2, etc.) correspond to successively higher Rayleigh wave overtones.
The equivalence between normal modes and propagating
waves gives us a powerful tool. For example, in Section 3.5.5
we will see models of wave propagation in the mantle that were
computed using modes, and so include core reflections, diffractions, and many other phases. Similarly, in Section 4.3.4 the
radiation patterns showing how various faults radiate surface
wave energy in different directions are computed using modes.
Thus we use either mode or wave methods, depending on
which seems easiest for a particular calculation. It is often
useful to do both and compare the answers, using each method
to bring different insight.
2.9.7 Observing normal modes
As with many basic seismological concepts, the idea of the
planet’s modes developed long before instruments became
available to observe them. As the theory of elasticity was
developed in the mid-1800s, there were discussions of finding
the “pitch” of the earth. 7 In 1882, Lamb modeled the earth as a
homogeneous steel ball, and calculated a fundamental mode of
78 minutes. In 1911, Love took into account the effect of gravity on radial motions of the earth, and revised the predicted
fundamental period to 60 minutes, not far from the actual
54 minutes. However, because making seismometers that can
detect such long-period motions is difficult (Section 6.6), it was
only after the great 1952 Kamchatka earthquake that this
mode was actually observed on a strainmeter recording.
7 Earth’s gravest observed mode, 0 S 2 corresponds to a note of E, twenty octaves below middle E on a piano. Johannes Kepler, among others, wrote about the “music of
the spheres,” and thought that each planet’s revolution around the sun corresponded
to a musical note. The earth’s 365.25 day revolution would correspond to a note of
C#, 33 octaves below middle C#.
8 Actually, because the earth vibrates at many frequencies, rather than just one, and
is laterally heterogeneous, a better but less poetic analogy would be that the earth
rattles like a dented garbage can.
Table 2.9-1 Some torsional and spheroidal modes.
Mode
Period
Description or associated phase
0 T 2
2,639.4
fundamental torsional
0 T 3
1,707.6
fundamental torsional
1 T 1
808.4
radial overtone
1 T 2
757.5
radial overtone
9 T 2
104.4
radial overtone
0 T 30
259.5
fundamental Love
0 T 130
68.9
fundamental Love
2 T 30
151.3
second-overtone Love
4 T 67
71.3
SH
10 T 40
71.4
SH diff
13 T 7
71.6
ScS SH
0 S 0
1,228.1
fundamental radial
1 S 0
613.0
radial overtone
0 S 2
3,233.5
football
0 S 3
2,134.4
pear-shaped
0 S 30
262.1
fundamental Rayleigh
0 S 130
75.8
fundamental Rayleigh
1 S 30
160.9
second-overtone Rayleigh
10 S 6
203.5
inner core PKJKP
11 S 5
197.1
inner core PKIKP
14 S 3
184.9
mantle ScS SV
1 S 1
19,500
Slichter
Sources: Dziewonski and Anderson (1981); Wysession and Shore (1994);
Dahlen and Tromp (1998).
Advances in seismic instrumentation, together with the
occurrence of the great 1960 Chilean and 1964 Alaska earthquakes, made it possible to identify and study large numbers
of modes. Over 40 modes were identified from the 1960
Chilean earthquake. The number of modes that have been
observed is now several thousands, due to continued advances
in seismometry, which permit recording at very long periods
(Section 6.6), an increase in the number of stations, more
powerful analytical techniques, and many large earthquakes.
Although none of the earthquakes has come close in size to
the 1960 Chilean event, the advances in instrumentation and
processing largely compensate. Large earthquakes are needed
to excite the gravest modes, and long lengths of seismograms
are needed to resolve their properties. As discussed in Section 6.3.3, this requires a seismogram that has significant energy extending over a time much longer than a mode’s period.
Fortunately this is the case for the largest earthquakes, leading
to the analogy that the earth rings like a bell after they occur. 8
Seismograms extending for many days are analyzed after the
largest earthquakes.
Table 2.9-1 shows the periods of several modes, some of
which have been discussed earlier. Note that for the fundamental ( 0 S and 0 T) overtone branches, modes with angular
orders greater than about 20 correspond to the fundamental
mode Rayleigh and Love waves with those periods and are
the overtone branches cross, because these modes include both
P and SV energy. Some spheroidal modes involve primarily
radial motion, and some involve primarily tangential motion,
with a full spectrum in between.
However, the basic patterns seen for torsional modes also
apply for spheroidal modes. For a given frequency, modes
to the left (low l) correspond to core phases, and those in the
center correspond to mantle body wave phases, with the core
diffraction (dashed lines) being the boundary between these
groups of modes. The modes furthest to the right correspond to
Rayleigh surface waves. The modes corresponding to P-wave
phases are further to the left than their SV counterparts because
P waves travel faster than S waves. These ideas can be visualized by considering the radial eigenfunctions (Fig. 2.9-9d).
The two curves for each spheroidal mode represent the two
radial eigenfunctions U(r) (radial) and V(r) (tangential). Modes
with low l and higher n have larger deep displacements, so
superposition of modes with very low l yields the core phases.
For the low-order overtones (n = 2 is shown), the displacements
are closer to the surface as l increases. Thus the n = 0 branch
with l greater than about 20 corresponds to fundamental mode
Rayleigh waves, and the higher branches (n = 1, 2, etc.) correspond to successively higher Rayleigh wave overtones.
The equivalence between normal modes and propagating
waves gives us a powerful tool. For example, in Section 3.5.5
we will see models of wave propagation in the mantle that were
computed using modes, and so include core reflections, diffractions, and many other phases. Similarly, in Section 4.3.4 the
radiation patterns showing how various faults radiate surface
wave energy in different directions are computed using modes.
Thus we use either mode or wave methods, depending on
which seems easiest for a particular calculation. It is often
useful to do both and compare the answers, using each method
to bring different insight.
2.9.7 Observing normal modes
As with many basic seismological concepts, the idea of the
planet’s modes developed long before instruments became
available to observe them. As the theory of elasticity was
developed in the mid-1800s, there were discussions of finding
the “pitch” of the earth. 7 In 1882, Lamb modeled the earth as a
homogeneous steel ball, and calculated a fundamental mode of
78 minutes. In 1911, Love took into account the effect of gravity on radial motions of the earth, and revised the predicted
fundamental period to 60 minutes, not far from the actual
54 minutes. However, because making seismometers that can
detect such long-period motions is difficult (Section 6.6), it was
only after the great 1952 Kamchatka earthquake that this
mode was actually observed on a strainmeter recording.
7 Earth’s gravest observed mode, 0 S 2 corresponds to a note of E, twenty octaves below middle E on a piano. Johannes Kepler, among others, wrote about the “music of
the spheres,” and thought that each planet’s revolution around the sun corresponded
to a musical note. The earth’s 365.25 day revolution would correspond to a note of
C#, 33 octaves below middle C#.
8 Actually, because the earth vibrates at many frequencies, rather than just one, and
is laterally heterogeneous, a better but less poetic analogy would be that the earth
rattles like a dented garbage can.
Table 2.9-1 Some torsional and spheroidal modes.
Mode
Period
Description or associated phase
0 T 2
2,639.4
fundamental torsional
0 T 3
1,707.6
fundamental torsional
1 T 1
808.4
radial overtone
1 T 2
757.5
radial overtone
9 T 2
104.4
radial overtone
0 T 30
259.5
fundamental Love
0 T 130
68.9
fundamental Love
2 T 30
151.3
second-overtone Love
4 T 67
71.3
SH
10 T 40
71.4
SH diff
13 T 7
71.6
ScS SH
0 S 0
1,228.1
fundamental radial
1 S 0
613.0
radial overtone
0 S 2
3,233.5
football
0 S 3
2,134.4
pear-shaped
0 S 30
262.1
fundamental Rayleigh
0 S 130
75.8
fundamental Rayleigh
1 S 30
160.9
second-overtone Rayleigh
10 S 6
203.5
inner core PKJKP
11 S 5
197.1
inner core PKIKP
14 S 3
184.9
mantle ScS SV
1 S 1
19,500
Slichter
Sources: Dziewonski and Anderson (1981); Wysession and Shore (1994);
Dahlen and Tromp (1998).
Advances in seismic instrumentation, together with the
occurrence of the great 1960 Chilean and 1964 Alaska earthquakes, made it possible to identify and study large numbers
of modes. Over 40 modes were identified from the 1960
Chilean earthquake. The number of modes that have been
observed is now several thousands, due to continued advances
in seismometry, which permit recording at very long periods
(Section 6.6), an increase in the number of stations, more
powerful analytical techniques, and many large earthquakes.
Although none of the earthquakes has come close in size to
the 1960 Chilean event, the advances in instrumentation and
processing largely compensate. Large earthquakes are needed
to excite the gravest modes, and long lengths of seismograms
are needed to resolve their properties. As discussed in Section 6.3.3, this requires a seismogram that has significant energy extending over a time much longer than a mode’s period.
Fortunately this is the case for the largest earthquakes, leading
to the analogy that the earth rings like a bell after they occur. 8
Seismograms extending for many days are analyzed after the
largest earthquakes.
Table 2.9-1 shows the periods of several modes, some of
which have been discussed earlier. Note that for the fundamental ( 0 S and 0 T) overtone branches, modes with angular
orders greater than about 20 correspond to the fundamental
mode Rayleigh and Love waves with those periods and are