EARTHQUAKE MOTION DISCUSSED THEORETICALLY.
Ideas of the ancients (the views of Travagini, Hooke, Woodward,
Stukeley, Mitchell, Young, Mallet)—Nature of elastic waves
and vibrations—Possible causes of disturbance in the Earth’s
crust—The time of vibration of an earth particle—Velocity and
acceleration of a particle—Propagation of a disturbance as
determined by experiments upon the elastic moduli of rocks—The
intensity of an earthquake—Area of greatest overturning
moment—Earthquake waves—Reflexion, refraction, and interference
of waves—Radiation of a disturbance.
_Ideas of Early Writers._—One of the first accounts of the varieties of
motion which may be experienced at the time of an earthquake is to be
found in the classification of earthquakes given by Aristotle.[8] It is
as follows:—
1. Epiclintæ, or earthquakes which move the ground obliquely.
2. Brastæ, with an upward vertical motion like boiling water.
3. Chasmatiæ, which cause the ground to sink and form hollows.
4. Rhectæ, which raise the ground and make fissures.
5. Ostæ, which overthrow with one thrust.
6. Palmatiæ, which shake from side to side with a sort of tremor.
From the sixth group in this classification we see that this early
writer did not regard earthquakes as necessarily isolated events, but
that some of them consisted of a succession of backward and forward
vibratory motions. He also distinguishes between the total duration of
an earthquake and the length of, and intervals between, a series of
shocks. Aristotle had, in fact, some idea of what modern writers upon
ordinary earthquakes would term ‘modality.’
The earliest writer who had the idea that an earthquake was a
pulse-like motion propagated through solid ground appears to have been
Francisci Travagini, who, in 1679, wrote upon an earthquake which
in 1667 had overthrown Ragusa. The method in which the pulses were
propagated he illustrated by experiments.
Hooke, who, in 1690, delivered discourses on earthquakes before the
Royal Society, divides these phenomena according to the geological
effects they have produced; thus there is a _genus_ producing
elevations, a _genus_ producing sinkings, a _genus_ producing
conversions and transportations, and a _genus_ which produces what, in
modern language, we should term metamorphic action.
Woodward, in his ‘Natural History,’ written in 1695, speaks of
earthquakes as being agitations and concussions produced by water in
the interior of the earth coming in contact with internal fires.
Stukeley observed that an earthquake was ‘a tremor of the earth,’ to
be explained as a vibration in a solid. The Rev. John Mitchell, writing
in 1760, says that the motion of the earth in earthquakes is partly
tremulous and partly propagated by waves.
From these few examples, to which might be added many more, it will
be seen that an earthquake disturbance has usually been regarded as
a concussion, vibration, trembling, or undulatory movement. Further,
it can be seen in narratives of earthquakes that it had been often
observed that these tremblings and shakings continued over a certain
period of time. Although it had been noticed that large areas were
almost simultaneously affected by these disturbances, no definite idea
appears to have existed as to how earthquake motion was propagated.
Usually it was assumed that the disturbance spread through subterranean
channels.
The first true conception of earthquake motion and the manner of its
propagation is due to Dr. Thomas Young, who suggested that earthquake
motion was vibratory, and it might be ‘propagated through the earth
nearly in the same manner as a noise is conveyed through the air.’ The
same idea was moulded into a more definite form by Gay Lussac.
The first accurate definition of an earthquake is due to Mr. Robert
Mallet, who, after collecting and examining many facts connected with
earthquake phenomena, and reasoning on these, with the help of known
laws connected with the production and propagation of waves of various
descriptions, formulated his views as follows:—
An earthquake is ‘_the transit of a wave or waves of elastic
compression in any direction from vertically upwards to horizontally,
in any azimuth, through the crust and surface of the earth, from any
centre of impulse or from more than one, and which may be attended
with sound and tidal waves, dependent upon the impulse and upon
circumstances of position as to sea and land_.’
In brief, so far as motion in the earth is concerned. Mallet defined
an earthquake as being a motion due to the transit of waves of elastic
compression. In many cases it is possible that this is strictly true,
but in succeeding pages it will be shown that earthquake motion may
also be due to the transit of waves of elastic distortion.
To obtain a true idea of earthquake motion is a matter of cardinal
importance, as it forms the key-stone of many investigations.
If we know the nature of the motion produced by an earthquake, we are
aided in tracking it to its origin, and in reasoning as to how it was
produced. If our knowledge of the nature of the motion of an earthquake
is incorrect, it will be impossible for us intelligently to construct
buildings to withstand the effects of these disturbances. We have thus
to consider, in this portion of seismology, a point of great scientific
importance, and shall deal with it at some length.
_Nature of Elastic Waves and Vibrations._—When it is stated that an
earthquake consists of elastic waves of compression and distortion, the
student of physics has a clear idea of what is meant and a knowledge
of the mechanical laws which govern such disturbances. The ordinary
reader, however, and the majority of the inhabitants of earthquake
countries, who of all people have the greatest interest in this matter,
may not have so clear a conception, and it will, therefore, not be out
of place to give some general explanation on this point.
The ordinary idea of a wave is that it is a disturbance similar to
that which we often see in water. Waves like these must not, however,
be confounded with elastic waves. A disturbance produced in water,
say, for instance, by dropping a stone into a pond, is propagated
outwards by the action of gravity. First, a ridge of water is raised
up by the stone passing beneath the surface. As this ridge falls
towards its normal position in virtue of its weight, it raises a second
ridge. This second ridge raises a third ridge, and so on. The water
moves vertically up and down, whilst the wave itself is propagated
horizontally.
To understand what is meant by elastic waves, it is first necessary
to understand what is meant by the term elastic. In popular language
the term elastic is confined to substances like india-rubber, and but
seldom to rock-like materials, through which earthquake waves are
propagated. India-rubber is called elastic because after we remove a
compressive force it has a tendency to spring back to its original
shape. The elastic force of the india-rubber is in this case the force
which causes it to resist a change of form. Now, a piece of rock may,
up to a certain point, like the india-rubber, be compressed, and when
the compressing force is removed it will also tend to resume its
original form. However, as the rock offers more resistance to the
compressing force than the india-rubber offers, we say that it is the
more elastic. It may be here observed that a substance like granite
offers great resistance, not only to compression or a change of volume,
but also to a change of form or shape; whereas a substance like air,
which is also elastic, only offers resistance to compression, but not
to a change of shape.
With these ideas before us we will now proceed to consider how, after
a body has been suddenly compressed or distorted, this disturbance
is propagated through the mass. For the elastic body let us take a
long spiral spring hung from the ceiling of a room and kept slightly
stretched by a weight. If we give this weight an upward tap from below,
say with a hammer, we shall observe a pulse-like wave which runs up the
spring until it reaches the ceiling of the room. Here it will, so to
speak, rebound, like a billiard ball from the end of a table, and run
towards the weight from which it started. Whilst this is going on we
may also observe that the weight is moving up and down.
Here, then, we have two distinct things to observe—one being the
transmission of motion up to the ceiling, which we may liken to the
transmission of an earthquake wave between two distant localities on
the earth’s surface, and the other being the up and down motion of our
weight, which we may compare to the backward and forward swinging which
we experience at the time of an earthquake.
These two motions—namely, the pulse-like wave produced by the
transmission of motion, and the backward and forward oscillation of the
weight or of any point on the spring—must be carefully distinguished
from each other.
First, we will consider the backward and forward motion of the weight.
The distance through which the weight moves depends upon the force
of the blow. The number of up and down oscillations it makes, say
in a second, depends upon the stiffness of the spring. The weight,
supposing it to be always the same, will move more quickly at the end
of a stiff spring than at the end of a flaccid one; that is to say, its
velocity is quicker. As in any given spring the number of up and down
oscillations are always the same in a given interval of time, if these
oscillations are of great extent, the weight must move more quickly
with large than with small oscillations.
At the time of an earthquake the manner in which we are moved backward
and forward is very similar to the manner in which the weight is moved.
If we stand on a hard rock-like granite, we are to a great extent
placed as if we were attached to a stiff quickly-vibrating spring.
If, however, we are on a soft rock, it is more like being on a loose
flaccid spring.
All that has thus far been considered has been a backward and forward
kind of motion, where there is a rectilinear _compression_ and
_extension_ amongst the particles on which we stand.
We might, however, imagine our rock, which for the moment we will
consider to be a square column, to be twisted, and thus have its
_shape_ altered. When the twisting force is taken off it seems evident
that the column would endeavour to untwist itself or regain its
original form. Now the force which a body offers against a change of
_volume_ may be very different from that which it offers against a
change of _form_.
In disturbances which take place in the rocky crust of our earth,
it would seem possible that we may have vibrations set up which are
either compressions and extensions or twistings and distortions. These
may take place separately or simultaneously, or we may have resultant
motions due to their combination.
The following are examples of possible causes which might give rise to
these different orders of disturbance:—
1. Imagine a large area stretched by elevation until it reaches the
limit of its elasticity and cracks. After cracking, in consequence
of its elasticity, it will fly back over the whole area like a
broken spring, and each point in the area will oscillate round its
new position of equilibrium. In this case there will be no waves
of distortion excepting near the end of the crack, where waves are
transmitted in a direction parallel to the fissure.
2. The ground is broken and slips either up, down, or sideways, as
we see to have taken place in the production of faults. Here we get
distortion in the direction of the movement, and waves are produced
by the elastic force of the rock, causing it to spring back from its
distorted form. In a case like this the production of a fissure running
north and south might give rise to north and south vibrations, which
would be propagated end on towards the north and south, but broadside
on towards the east and west. With disturbances of this kind, on
account of the want of homogeneousness in the materials in which
they are produced, we should expect to find waves of compression and
extension.
3. A truly spherical cavity is suddenly formed by the explosion of
steam in the midst of an elastic medium. In this case all the waves
will be those of compression, each particle moving backward and forward
along a radius.
Should the cavity, instead of being truly spherical, be irregular, it
is evident that, in addition to the normal vibration of compression,
transverse waves of distortion will be more or less pronounced,
depending upon the nature of the cavity.
The combination of these two sets of vibrations may cause a point in
the earth to move in a circle, an ellipse, the form of a figure eight,
and in other curves similar to these, which are produced by apparatus
designed to show the combination of harmonic motion. From these
examples it will be seen that we have therefore to consider two kinds
of vibrations—one produced by compression or the alteration of volume,
and the other produced by an alteration in shape.
Now the resistance which a body offers, either to a change in its
volume or in its shape, is called its elasticity, and the law which
governs the backward and forward motion of a particle under the
influence of this elasticity may be expressed as follows:
If T be the time of vibration, or the time taken by a particle to make
one complete backward and forward swing, D the density of the material
of which this particle forms a part, and E the proper modulus of
elasticity of the material, then,
_____
T = 2π √ D/E
_____
From this formula, T = 2π √ D/E, we see that the time of
vibration of the earth during an earthquake, or the rate at which we
are shaken backwards and forwards, varies directly as the square root
of the density of the material on which we stand, and inversely as the
square root of a number proportional to its elasticity.
_Velocity and Acceleration of an Earth Particle._—Another important
point, which the practical seismologist has often brought to his
notice, is the question of the velocity with which an earth particle
_____
moves. According to the formula, T = 2π √ D/E, we should expect
that a particle would make each semi-vibration in an equal time, and
from a knowledge of the density and elastic moduli of a body this
time might be calculated. Although the time of a semi-oscillation may
be constant, we must bear in mind that, like the bob of a pendulum
during each of its swings, the particle starts from rest, increases in
velocity until it reaches the middle portion of its half swing, from
which it gradually decreases in speed until it reaches zero, when it
again commences a similar motion in the opposite direction.
These pendulum-like vibrations are sometimes spoken of as simple
harmonic motions. If we know the distance through which an earthquake
moves in making a single swing, and the time taken in making this
swing, on the assumption that the motion is simple harmonic we can
easily calculate the maximum velocity with which the particle moves.
Thus, if an earth particle takes one second to complete a
semi-oscillation, half of which, or the amplitude of the motion, equals
_a_, the maximum velocity equals π × _a_.
Again, assuming the earth vibrations to be simple harmonic, the maximum
acceleration or rate of change in velocity will come about at the ends
of each semi-oscillation; and if V be the maximum velocity of
the particle, and _a_ the amplitude or half semi-oscillation, then the
V^2
maximum acceleration equals ———.
_a_
Later on it will be shown, as the result of experiment, that certain of
the more important earth oscillations in an earthquake are not simple
harmonic motion. Nevertheless the above remarks will be of assistance
in showing how the velocity and other elements connected with the
motion of an earth particle, which are required by the practical
seismologist, may be calculated, irrespective of assumptions as to the
nature of the motion.
_Propagation of a Disturbance._—We may next consider the manner in
which a disturbance, in which there are both vibrations of compression
and of distortion, is propagated. The first or normal set of vibrations
are propagated in a manner similar to that in which sound vibrations
are propagated. From a centre of disturbance these movements approach
an observer at a distant station, so to speak, end on. The other
vibrations have a direction of motion similar to that which we believe
to exist in a ray of light. These would approach the observer broadside
on.
If the disturbance passed through a formation like a series of
perfectly laminated slates, each of these two sets of vibration might
be subdivided, and we should then obtain what Mallet has termed
ordinary and extraordinary normal and transverse vibrations.
In consequence of the difference in the elastic forces on which the
propagation of these two kinds of vibration depends, the normal
vibrations are transmitted faster than the transversal ones—that is
to say, if an earthquake originated from a blow, the first thing that
would be felt at a point distant from the origin of the shock would be
a backward and forward motion in the direction to and from the origin,
and then, a short interval afterwards, a motion transversal, or at
right angles to this, would be experienced.
From the mathematical theory of vibratory motions it is possible to
calculate the velocity with which a disturbance is propagated. As the
result of these investigations it has been shown that normal vibrations
travel more quickly than transverse vibrations.
Deductions from experiments on small specimens are, however,
invalidated by the fact that the specimens used for experiments are,
of course, nearly homogeneous, whilst the earthquake passes through
a mass which is heterogeneous and more or less fissured. Mallet, by
experiments ‘on the compressibility of solid cubes of these rocks,
obtained the mean modulus of elasticity,’ with the result that ‘nearly
seven-eighths of the full velocity of wave-transit due to the material,
if solid and continuous, is lost by reason of the heterogeneity and
discontinuity of the rocky masses as they are found piled together in
nature.’ The full velocities of wave-transit, as calculated by Mallet
from a theorem given by Poisson, were—
For slate and quartz transverse to lamination, 9,691 feet per second.
„ „ in line of lamination, 5,415 „ „
This more rapid transmission in a direction transverse to the
lamination, Mr. Mallet observes, may be more than counterbalanced by
the discontinuity of the mass transverse to the same direction.
_The Intensity of an Earthquake._—The intensity of an earthquake
is best estimated by the intensity of the forces which are brought
to bear on bodies placed on the earth’s surface. These forces are
evidently proportional to the rate of change of velocity in the body,
and, as the destructive effect will be proportional to the maximum
forces, we may consistently indicate the intensity of an earthquake by
giving the maximum acceleration to which bodies were subject during
the disturbance. On the assumption that the motion of a point on
the earth’s surface is simple harmonic, the maximum acceleration is
directly as the maximum velocity and inversely as the amplitude of
_v_^2
motion, or as ————— where _v_ indicates velocity and _a_ amplitude.
_a_
The next question of importance is to determine the manner in which
earthquake energy becomes dissipated—that is, to compare together
the intensity of an earthquake as recorded at two or more points at
different distances from the origin. First let us imagine the origin
of our earthquake to be surrounded by concentric shells, each of
which is the breadth of the vibration of a particle. Going outwards
from the centre, each successive shell will contain a greater number
of particles, this number increasing directly as the square of the
distance from the origin. Let the blow have its origin at the centre,
and give a vibratory movement to the particles in one of the shells
near the centre.
This shell may be supposed to possess a certain amount of energy,
which will be measured by its mass and the square of the velocity of
its particles. In transferring this energy to the neighbouring shell
which surrounds it, because it has to set in motion a greater number
of particles than it contains itself, the energy in any one particle
of the second layer will be less than the energy in any one particle
in the first layer; the total energy in the second shell, however,
will be equal to the total energy in the first shell. Neglecting the
energy lost during the transfer, if the energy in a particle of the
first shell at any particular phase of the motion be K_{1},
and the energy in a particle of the second shell K_{2}, these
quantities are to each other inversely as the masses of the shells—that
is, inversely as the squares of the mean radii of the shells.
K_{2} _r__{1}^2
In symbols, ————— = ————————— (1)
K_{1} _r__{2}^2
Assuming that energy is dissipated,
K_{2} _r__{1}^2 _r__{1}^2
————— > ————————— = _f_ ————————— (2)
K_{1} _r__{2}^2 _r__{2}^2
where _f_ < 1 is the rate of dissipation of energy which is assumed to
be constant.
_Area of greatest Overturning Moment._—Although the rate of dissipation
of the impulsive effects of an earthquake may follow a law like that
just enumerated, it must be remembered that if the depth of the origin
is comparable with the radius of the area which is shaken, the maximum
impulsive effect as exhibited by the actual destruction on the surface
may not be immediately above the origin where buildings have simply
been lifted vertically up and down, but at some distance from this
point, where the impulsive effort has been more oblique.
At the _epicentrum_ we have the maximum of the true intensity as
measured by the acceleration of a particle, or the height to which
a body might be projected, but it will be at some distance from
this where we shall have the maximum intensity as exhibited by an
overturning effort.
This will be rendered clear by the following diagram.
In the accompanying diagram let O be the origin of a shock, and O C the
seismic vertical equal to _r_. Let the direct or normal shock emerge at
C, C_{1}, C_{2}, and at the angles θ_{1}, θ_{2}, &c.
Assuming that the displacement of an earth particle at C equals C
B, and at C_{1} equals _c__{1} _b__{1}, and at C_{2} equals _c__{2}
_b__{2}, &c., and let these displacements C B, _c__{1} _b__{1}, _c__{2}
_b__{2}, &c., for the sake of argument, vary inversely as _r_, _r__{1},
_r__{2}, &c.
[Illustration: FIG. 9.]
The question is to determine where the horizontal component C A of
these normal motions is a maximum.
First observe that the triangle O C _c_ is similar to _a_, _b_, _c_.
_h_
Also _r_ = —————, and therefore the normal component _c__{1}
sin θ
sin θ
_b__{1} at C_{1} is equal to C —————.
_h_
Also _c__{1} _a__{1} = _c__{1} _b__{1}, cos θ.
sin θ cos θ _c_ sin 2θ
∴ C_{1} _a_ = C ———————————— = ———— ∙ ———————,
_h_ _h_ 2
and sin 2θ is greatest when 2θ = 90° or θ = 45°.
That is to say, the horizontal component reaches a maximum where the
angle of emergence equals 45°.
This question has been discussed on the assumption that the amplitude
of an earth particle varies inversely as its distance from the origin
of the shock. Should we, however, assume that this amplitude varies
inversely as the square of the distance from the origin, we are led to
the result that the area of greatest disturbance is nearer to the point
where the angle of emergence is 55° 44′ 9″. Both of these methods are
referred to by Mallet, but the first is considered as probably the more
correct.
_Earthquake Waves._—Hitherto we have chiefly considered earthquake
vibrations; now we will say a few words about earthquake waves. If
we strike a long iron rod at one end, we can imagine that, as in the
long spring, a pulse-like motion is transmitted. If the rod be struck
quickly, the pulses will rapidly succeed each other, and if struck
slowly the pulses will be at longer intervals. Each individual pulse,
however, will travel along the rod at the same rate, and hence the
distance between any two will remain constant; but that distance will
depend on the interval between the blows producing these pulses being
equal to the distance travelled by one pulse before the next blow is
struck.
From this we see that an irregular disturbance will produce an
irregular succession of motions; some will be like long undulations in
a wide deep ocean, whilst others will be like ripples in a shallow bay.
Again, consider the bar to be struck one blow only, and then left to
itself. The bar will propagate a series of pulses along its length, due
to the out and in vibration of its end. These will succeed each other
at regular intervals, and will be mixed up with the pulses we have
previously considered.
From this we see that in an earthquake, if it be produced by one blow,
the motion will be isochronous in its character; but if it be due to
a succession of blows at regular intervals, the motion will be the
resultant of a series of isochronous motions, and will be periodical.
If the impulses are irregular, you have a motion which is the resultant
of a number of isochronous motions due to each impulse, but these
compounded together in a different manner at each instant during the
earthquake, and giving as a result a motion which is in no sense
isochronous. This approaches more nearly to the actual motions we feel
as earthquakes.
If we can imagine the ground shaken by an earthquake, made of a
transparent material which transmitted less light when compressed,
and we could look down upon a long extent of this at the time of an
earthquake, we should see a series of dark bands indicating strips of
country which were compressed. The distances between these bands might
be irregular. Keeping our attention on one particular band, this would
be seen to travel forward in a direction from the source. If we kept
our eye on one particular point, it would appear to open and shut,
becoming light and dark alternately.
As to the existence of these elastic waves in actual earthquakes we
have no direct experimental evidence. The only kind of wave with which
we are familiar is a true surface undulation, which, although having
the appearance of a water-wave, may nevertheless represent a district
of compression.