LECTURE 7 NOTES - SEISMIC WAVES (updated 10/15/97)
Instructor: Professor Barbara Romanowicz
Director of Seismological Laboratory
Office Hours: Thursday 2-4 pm , upon appointment only
475 Mc Cone Hall
SEISMIC WAVES
Introduction
Picture of vibrations leaving the seismic source in all directions.
Accounts of witnesses experiencing large earthquakes, consistently.
Distinguish 2 types of shaking.
1) - John Mitchell's report on the Lisbon earthquake of 1755:
"tremulous vibration" followed by a "wavelike undulation".
2)- Dec 16, 1857, Naples: Robert Mallet's 3 month visit; he wrote :
"The first principles of observational seismology"
- earthquake waves travel at different speeds through different materials
- seismic waves resemble soundwaves travelling through the air
3) - Alaska, April 1964
Report of John Williams, geologist in Anchorage.
Sitting in couch in living room, he wrote:
"At first we noticed a rattling of the building. The initial shaking lasted
perhaps 5 to 10 sec. The first shaking was followed without any noticeable
quiet period by a strong rolling motion which appeared to move from east to
west. After a few seconds of the strong rolling motion, .... (took his son and
ran to a parked car).. then the building was swaying in an east west direction.
Blocks were toppling, ground heaving, tres and poles were swaying
strongly..."
-Descriptions of the motion in an earthquake:
-duration and amplitude of waves
-pattern of wave arrival
-direction of ground movements ---> theory of seismic waves provides an
understanding of such descriptions.
P and S waves
Wave motion familiar to us from observations of waves on water. Throw a
stone into a pool, the surface is disturbed where the stone strikes and ripples
move outward from place of disturbance: these ripples form a "wave train"
which is produced by movements of the particles of water in the vicinity of
the ripples. The water does not FLOW: a cork on the surface will bob up and
down but not move away from its original position. The disturbance is passed
on further and further by the brief back and forth movements of water
particles which impart their motion to the next particles. In this way, water
waves carry energy from the broken surface where the stone lands to the edge
of the pool where they break.
Earthquake motions are analogous: the shaking we feel is the vibration of the
rocks and structure due to energy in seismic waves as they propagate.
Elastic body waves
Two kinds of "elastic waves" are produced and travel away from the source:
1) First, same in its properties as a sound wave; transmitted by alternate
compressions (pushes) and dilatations (pulls) on the air. Liquids and solid rocks
can be compressed, and the same type of wave travels through bodies of water such
as oceans/lakes, and through the solid earth.
-Direction of propagation
-Wavefront
Particles of rock move forward and backward in the direction of propagation
of the waves, which travel at equal speed in all directions: the particles
move perpendicular to the wavefront.
-Wave amplitude: amount of displacement forward and backward.
-These are P waves: from "primary" - arrive first, travel the fastest.
2) Unlike air, which can be compressed but not sheared, elastic materials
allow a second type of wave to propagate that shears and twists the material.
From an earthquake: "S wave" - secondary wave.
Behavior of the particles of rock in the passage of S wave is quite different:
S waves move the particles of rock transverse to the direction of propagation;
rock motions may be in a vertical or horizontal plane and are similar to the
transverse motions in the light waves (electromagnetic waves).
The presence of both P and S waves gives earthquakes an interesting
combination of effects that are absent in either the physical behavior of light
or the physical behavior of sound.
S waves do not travel through liquids or gases (shearing motion is not possible).
This sharp contrast between the properties of P and S waves can be used to detect
the presence of liquid zones deep in the Earth.
Polarization
E.g. polarization of light: cf. polarized eyeglasses to cut down on scattered
light. Only those light waves that are vibrating transversely to certain planes
(up and dowm, horizontally...) can pass through a polarized lens.
The transmitted light waves are said to be plane-polarized.
As S waves travel through the earth, they encounter structural boundaries
that refract or reflect them and polarize their vibrations.
Two types of S wave:
1) S wave polarized in a horizontal plane : SH
2) S wave polarized in a vertical plane : SV
Elastic behavior of rock
Elastic behavior is when the displacement due to the application of a force
follows a linear curve; displacement proportional to the applied force (linear
behavior): obeys Hooke's law (a contemporary of Newton). Illustrate by spring
and weight. When force ceases to be applied, displacement goes back to zero,
and elastic body recovers its original shape.
Similarly, in an earthquake, rock behaves like a spring and will experience
proportionally greater displacement in response to a larger force. In most
cases, will return to its originial position at the end of shaking.
Some important exceptions: soft soils, do not go back to original positions.
Shaking intensities become more difficult to predict: non-linear effects.
Motion of a spring
Illustrates the energy exchange between rocks and seismic waves.
Total energy = elastic + kinetic = constant (perfectly elastic)
(spring compression) + (velocity of spring parts)
Maximum wave amplitude : energy all elastic (v = 0, max displacement)
Equilibrium position : energy is all kinetic (v = maximum, no energy stored
in displacement)
Asssuming that there are no frictional or dissipative forces present:
movement will continue indefinetely with same amplitude.
In reality, friction between moving rocks gradually dissipates some of the
wave energy as heat; comes to rest eventually, unless some extraneous source
of energy is added.
Another effect contributes to gradual weakening of motion: as waves spread
from source (water waves, sound waves...) we observe a gradual weakening of
their amplitude with distance: initial energy is spread over a wider and wider
area, producing an attenuation called geometrical spreading.
The farther from an earthquake you are, the smaller the intensity of shaking
as a combination of these two effects.
PROPERTIES OF WAVES
Wave frequency
Pure music tone of a tuning fork has a pure pitch or frequency.
Frequency is the number of times that the sound waves compress or dilate in
a second; or water waves rise and fall in a second (in any time units).
Seismic waves: move back and forth.
Unit: hertz (Hz)
1 Hz = 1 cycle of motion per second.
Time between crests is the wave period T (reciprocal of frequency),
measured in seconds.
Human ears can detect sounds between 20 and 10,000 Hz. Sometimes a P
wave can refract out of the rock surface into the atmosphere and produce
audible sounds: heard as a rumble. Most earthquake waves have frequencies
lower than 20Hz (they are felt rather than heard).
Down to fraction of a millihertz: the longest vibration is 54 min
The simplest case: harmonic motion.
Sine wave with a single amplitude at a single frequency:
described by a few parameters:
elongation = distance from state of equilibrium
maximum elongation ==> amplitude A
wavelength l = distance between crests
period T = time to travel one wavelength
wave velocity v = wavelength/period: v = l /T
frequency of the wave f = the number of complete waves
that pass every second: f = 1 /T
Actual position of a wave depends on its position relative to the origin time
and distance: described by phase.
Waveforms that occur in earthquakes are more complicated; actual recording
is a superposition of short wavelengths and long wavelength waves. In fact,
they can be represented as a sum of harmonic waves: sum of sinusoidal
components.
The actual recorded ground motion can be analyzed using the
methods of Fourier by examining the individual component harmonic
waves separately.
Individual components can be shifted so that their peaks and troughs do not
coincide: phase shift. When they are summed, pattern looks different. Phase
is an important parameter.
Surface waves
When P and S waves arrive at the free surface, they generate other types of
waves.
Most important are Rayleigh and Love waves: propagate along the surface of
the Earth (motion of the particles of rocks decrease to zero with depth - energy
trapped near the surface- e.g. sound waves in whispering galleries).
In contrast, P and S waves are called "body waves".
Love waves can be some of the most destructive waves. Motion like SH waves.
Large amplitudes, horizontal shearing.
Rayleigh waves most closely resemble water waves: elliptical motion
forward, up, backward, down, in a vertical plane containing the direction of
propagation. Speeds of L (G) and R waves are always lower than speed of P waves
and less or equal than speed of S waves.
Dispersion property
waves with short wavelengths sample shallower layers than waves with long
wavelengths ==> they travel slowly. Surface waves disperse into long trains.
(contrast to water waves where short wavelengths travel faster).
Speeds of P and S waves and elastic moduli
P waves always arrive first because they travel faster than S waves. This
property (time interval between P and S waves) can be used to calculate the
distance from the source. At regional distances here in California, roughly
multiply time in seconds by 8 to obtain distance from the source in km:
10 sec ---> 80 km etc.
Actual velocities of P and S waves depend on the densities and inherent
elastic properties of the rocks. For linear elastic behavior, wave speed
depends on the measures of only two elastic properties, the elastic moduli:
1) incompressibility (K)
When uniform pressure is applied to the surface of a rock, its volume is
reduced: change of volume per unit volume = incompressibility. This type of
deformation occurs when P waves propagate in the Earth's interior. Applies
as well to liquids as to solids. The greater the incompressibility,
the greater the P velocity.
2) rigidity (m)
Apply equal but opposite tangential pressures to opposite faces of a cube of
rock. Cube will deform by shearing out of its rectangular shape , but without
change of volume. (usually measured in lab using cylindrical shapes). The
greater the resistance to shearing, the greater its rigidity. The greater the
rigidity, the greater the S-wave velocity.
Formulas for P and S waves velocities:
alpha = (K + 4m/ 3)/ rho (velocity of P waves)
beta = m/ rho (velocity of S waves)
where rho is Earth's density.
Examples: velocity = 5.5 km/s in granite for P waves
= 3.0 km/s in -II- for S waves
= 1.5 km/s in water for P waves
= 0.0 km/s in -II- for S waves
Note: liquids have zero rigidity, the velocity of shear waves in water is zero.
Also, since K and m are positive, P waves travel faster than S waves.
Density ro increases with depth because of the increasing pressure. It would
appear that we would expect velocities to decrease with depth. It turns out
that the K and m increase more quickly with depth than the density (except
for m in molten areas). Generally : P and S wave speeds increase with depth in
the Earth (in expressions for P and S velocities, numerator grows faster than
denominator).
Anisotropy
Elastic rock type ---> constant elastic moduli at given p and T. However,
under some geological circumstances, elastic moduli may be different in
different directions through the rock ==> "anisotropy". Provides information
about geological conditions in the Earth. We will not worry about this here.
Reflection/refraction/conversion
Reflection: water waves arrive at a boundary such as a steep shoreline;
reflected back from this boundary: second wavetrain develops which interferes
with the incoming wavetrain.
Refraction: when waves travel obliquely to a boundary, they will change
direction of propagation at the boundary.
In case of seismic wave propagation, same phenomena plus an additional one:
Conversion: generation of a wave of different type (P to S or S to P).
Happens when incident wave is P or SV. No conversions for SH waves.
Conversion happens only when incidence is oblique.
Many striking earthquake effects can be explained by reflection and refraction
of waves.
S wave moving from deep source upward to surface. Will double its
amplitude and quadruple its energy at the surface because incident and
reflected wave add together. This is in accord with experience of miners, who
in many cases have not been aware of a strong earthquake occurrence.
Tangshan, China, 1976: coal miners underground knew there was a problem
when they felt moderate shaking and electricity failed. When they came to
the surface, city was devastated (250,000 dead).
Wave amplification in soft deep soils: sediments in alluvial valleys.
Soft rocks amplify the shaking by a factor of four or more depending on the
wave frequency and the thickness of the layer of alluvium (Cf Marina district
in Loma Prieta or SF 1906 eq).
Cf. two springs attached together: weaker one will have larger motion.
Earthquake waves in succession
Sequence of arrivals reflects different speeds.
1) P waves emerge generally at steep angle: vertical motion of ground,
which is more easily withstood than horizontal shaking (buildings etc built
to withstand vertical gravity). Not the most damaging waves.
2) S waves sometime later (~ half speed). SH and SV in two planes: horizontal
and vertical. Last longer than P wave trains. Buildings shaken sideways.
3) Love waves just after S waves or together. Ground shakes at right angles to
the direction of travel.
4) Rayleigh waves: shaking in longitudinal and vertical directions. Many
cycles: "rolling motion".
Surface waves attenuate at lower rate with distance than P or S aves: felt or
recorded at great distances from source, with long duration.
Dying end of earthqauke - coda: mixture of P, S, Love Rayleigh waves refracted
and reflected along the path. May trigger the collapse of structures already
weakened by earlier-arriving, more energetic S waves.
Wave diffraction
Trains of wave encounter an obstacle; most reflected, but some will run
around the obstacle into its shadow: dim illumination of the regions behind
obstacles and barries. Easier for longer waves. A striking example is the
diffraction of waves along the surface of the liquid core....
Resonance
Reflection and refraction may trap seismic energy in a geological structure
such as an alluvial valley (such traping of energy explains some heavy
damage, e.g. Mexico city in 1985). Cf. trapping of sound waves in a room where
they echo back and forth from the walls.
Waves begin to travel backward and forward and interfere, causing changes
in amplitude of waves: positive (constructive) interference at some frequencies,
destructive at some others. Fortunately, growth mitigated by friction
and geometrical spreading. In an enclosed structure, interfering seismic waves
create standing waves: ground surface moves purely up and down, eg. string held
at both ends, like in musical instruments. Standing waves of many frequencies
can be generated, specific frequencies: overtones or higher modes.
Large earthquakes cause the whole Earth to ring like a bell.
Love in 1911 predicted that a steel sphere as large as the Earth would have a
period of fundamental vibration of about 1 hour: confirmed with the Chilean
earthquake of May 1960 ---> 54 min (not too different from Love's prediction).
Analysis of ground records gave first unequivocal evidence that the
theoretically predicted free vibrations of the Earth had been generated.
Eigenfrequencies of resonant vibrations depend only on the properties of the
elastic globe (eg. a bell, or musical instrument): length, density and tautness of
string. From measuring the properties of eigenvibrations of the earth
(periods, amplitudes) we can learn about the distribution of density, elastic
moduli throughout the interior).
Two types of vibrations:
1) torsional vibrations (T): only horizontal displacements of rocks.
2) spheroidal vibrations (S): also radial vibrations.