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,

From this formula, t=2pv^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=2pv^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 p × 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 maximum acceleration equals ^v2/_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 motion, or as ^v2/_a where v indicates velocity and a amplitude.

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 k₁, and the energy in a particle of the second shell k₂, 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.

In symbols, ^k₂/_k1 = ^r₁2/_r2² (1)

Assuming that energy is dissipated,

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, c₁, c₂, and at the angles ?₁, ?₂, &c.

Assuming that the displacement of an earth particle at c equals c b, and at c₁ equals c₁ b₁, and at c₂ equals c₂ b₂, &c., and let these displacements c b, c₁ b₁, c₂ b₂, &c., for the sake of argument, vary inversely as r, r₁, r₂, &c.

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.

Also r = ^h/_{sin ?}, and therefore the normal component c₁ b₁ at c₁ is equal to c ^{sin ?}/_h.

Also c₁ a₁ = c₁ b₁, cos ?.

and sin 2? is greatest when 2? = 90° or ? = 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.

---