194 Seismology and Earth Structure
5 The rainbow results from physical dispersion for light waves passing through
water drops in the atmosphere or a prism. Different frequencies (colors) of light travel
at different speeds through the water or prism, and thus refract at different angles,
separating initially white light into different colors.
ing frequency ω is from the oscillator’s natural or resonant frequency, ω 0 . The resonance curve shows how the damped harmonic oscillator responds to the frequency-dependent driving
force. The closer the driving frequency ω is to the oscillator’s
natural frequency ω 0 , the more the oscillator responds.
The resonance curve can be viewed in terms of the frequency
at which the peak occurs
ω p = (ω 2
0 − γ 2 /2) 1/2 = ω 0 (1 − 1/2Q 2 ) 1/2
(44)
and the amplitude of the peak
A(ω p ) = Q/(ω
2
0 (1 − 1/4Q
2 )
1/2
).
(45)
If the oscillator is undamped (γ = 0, Q = ∞) the peak occurs at
its natural frequency and shows an infinite response. Adding
damping lowers the amplitude of the peak and shifts it.
However, the shift is very small unless the system is much more
damped (Q < 2) than occurs for seismic wave attenuation. The
damping also spreads out the peak in frequency, so the more
the damping, the broader and lower the peak. To see why,
recall that the more the damping, the faster the oscillation
decays as a function of time (Fig. 3.7-11). As we will see in
Chapter 6, the spectrum of an undamped sinusoid is a sharp
line, or delta function, so additional frequencies, and thus a
broader peak, correspond to the decaying sinusoid. The phase
response also has significance, as we will see when we discuss
seismometers (Section 6.6).
The resonance curve concept appears in a wide variety of applications, because many physical systems can be viewed as
damped harmonic oscillators. Three commonly considered in
seismology are the attenuation of the earth’s normal modes, the
behavior of a seismometer, and the response of a building to
ground motion. An earthquake puts energy at various frequencies into the earth, exciting its normal modes (Section 2.9).
These modes form a set of damped harmonic oscillators, so the
amplitude spectrum of a long-period seismogram contains
peaks that correspond to the net resonance curve for each mode
multiplet. The width of a peak depends on the frequencies and
amplitudes of the mode’s singlets and the mode’s damping.
Seismometers can also be viewed as damped harmonic oscillators, whose natural frequency and damping control their
response to ground motion. In addition, as mentioned in
Section 1.2.2, buildings can be considered damped harmonic
oscillators. This concept is important in designing earthquakeresistant structures, because buildings are most vulnerable to
ground motion with frequencies close to their natural frequencies, so damping is added to reduce the resulting motion.
3.7.8 Physical dispersion due to anelasticity
An important consequence of seismic wave attenuation is
physical dispersion, in which waves at different frequencies
travel at different velocities. This differs from the geometrical
dispersion discussed in Sections 2.7 and 2.8, in which surface
waves of different frequencies have different apparent velocities at the surface because they sample different depths and
hence encounter material of different velocities. Thus, although
the intrinsic velocity of the rock at any depth is treated as
frequency-independent, dispersion occurs because of the depthvariable velocity of the material. By contrast, with physical
dispersion the intrinsic velocity of waves in the medium varies
with frequency. 5
To see how physical dispersion results from attenuation,
consider how a seismic wave changes shape. Assume that a
delta function wave, a pulse of infinite height and unit area
(Fig. 3.7-14), propagates through a homogeneous elastic
medium with intrinsic velocity c:
Frequency
Frequency
100
80
60
40
20
0
0
ω /2
0
ω 0
ω /2
0
3
ω 0
2
0
ω /2
0
ω 0
ω /2
0
3
ω 0
2
Q = 100
Q = 15
Q = 5
Amplitude
0
Q = 5
Q = 15
Q = 100
-π
-π /2
Phase,
φ
Fig. 3.7-13 Amplitude (left) and phase
(right) response of a forced, damped
harmonic oscillator with natural frequency
ω 0 . For greater damping (lower Q) the peak
decreases and both it and the phase shift are
broadened from the sharp values they have
with little damping.
5 The rainbow results from physical dispersion for light waves passing through
water drops in the atmosphere or a prism. Different frequencies (colors) of light travel
at different speeds through the water or prism, and thus refract at different angles,
separating initially white light into different colors.
ing frequency ω is from the oscillator’s natural or resonant frequency, ω 0 . The resonance curve shows how the damped harmonic oscillator responds to the frequency-dependent driving
force. The closer the driving frequency ω is to the oscillator’s
natural frequency ω 0 , the more the oscillator responds.
The resonance curve can be viewed in terms of the frequency
at which the peak occurs
ω p = (ω 2
0 − γ 2 /2) 1/2 = ω 0 (1 − 1/2Q 2 ) 1/2
(44)
and the amplitude of the peak
A(ω p ) = Q/(ω
2
0 (1 − 1/4Q
2 )
1/2
).
(45)
If the oscillator is undamped (γ = 0, Q = ∞) the peak occurs at
its natural frequency and shows an infinite response. Adding
damping lowers the amplitude of the peak and shifts it.
However, the shift is very small unless the system is much more
damped (Q < 2) than occurs for seismic wave attenuation. The
damping also spreads out the peak in frequency, so the more
the damping, the broader and lower the peak. To see why,
recall that the more the damping, the faster the oscillation
decays as a function of time (Fig. 3.7-11). As we will see in
Chapter 6, the spectrum of an undamped sinusoid is a sharp
line, or delta function, so additional frequencies, and thus a
broader peak, correspond to the decaying sinusoid. The phase
response also has significance, as we will see when we discuss
seismometers (Section 6.6).
The resonance curve concept appears in a wide variety of applications, because many physical systems can be viewed as
damped harmonic oscillators. Three commonly considered in
seismology are the attenuation of the earth’s normal modes, the
behavior of a seismometer, and the response of a building to
ground motion. An earthquake puts energy at various frequencies into the earth, exciting its normal modes (Section 2.9).
These modes form a set of damped harmonic oscillators, so the
amplitude spectrum of a long-period seismogram contains
peaks that correspond to the net resonance curve for each mode
multiplet. The width of a peak depends on the frequencies and
amplitudes of the mode’s singlets and the mode’s damping.
Seismometers can also be viewed as damped harmonic oscillators, whose natural frequency and damping control their
response to ground motion. In addition, as mentioned in
Section 1.2.2, buildings can be considered damped harmonic
oscillators. This concept is important in designing earthquakeresistant structures, because buildings are most vulnerable to
ground motion with frequencies close to their natural frequencies, so damping is added to reduce the resulting motion.
3.7.8 Physical dispersion due to anelasticity
An important consequence of seismic wave attenuation is
physical dispersion, in which waves at different frequencies
travel at different velocities. This differs from the geometrical
dispersion discussed in Sections 2.7 and 2.8, in which surface
waves of different frequencies have different apparent velocities at the surface because they sample different depths and
hence encounter material of different velocities. Thus, although
the intrinsic velocity of the rock at any depth is treated as
frequency-independent, dispersion occurs because of the depthvariable velocity of the material. By contrast, with physical
dispersion the intrinsic velocity of waves in the medium varies
with frequency. 5
To see how physical dispersion results from attenuation,
consider how a seismic wave changes shape. Assume that a
delta function wave, a pulse of infinite height and unit area
(Fig. 3.7-14), propagates through a homogeneous elastic
medium with intrinsic velocity c:
Frequency
Frequency
100
80
60
40
20
0
0
ω /2
0
ω 0
ω /2
0
3
ω 0
2
0
ω /2
0
ω 0
ω /2
0
3
ω 0
2
Q = 100
Q = 15
Q = 5
Amplitude
0
Q = 5
Q = 15
Q = 100
-π
-π /2
Phase,
φ
Fig. 3.7-13 Amplitude (left) and phase
(right) response of a forced, damped
harmonic oscillator with natural frequency
ω 0 . For greater damping (lower Q) the peak
decreases and both it and the phase shift are
broadened from the sharp values they have
with little damping.