HOPKINS' Foundations of a Mathematical Theory of the Wave Motions of Earthquakes
Although MALLET's report about the dynamic of earthquakes that is explicitly mentioned by HOPKINS was read in the Irish Academy of Sciences more than one year before the report by HOPKINS on the geological theories of elevations and earthquakes in the British Association for the Advancement of Science, HOPKINS is neither a follower nor a successor of MALLET. His mathematical foundations of a theory of seismic waves were elaborated rather independently of MALLET and were presented as a second part of his geological elevation theory that he alone had already published in a first version in three parts in the Transactions of the Royal Society in 1839, 1840, and 1842 and being strongly influenced by DARWIN, as we had shown earlier . His report from 1847 also focussed on geological theory that dealt with the permanent changes of the earth's surface such as elevations and depressions. Only a third of the almost 60 pages of the treatise are dedicated to earthquake phenomena.

Vibrations of the earth's crust that occurred as secondary phenomena in addition to the processes of elevations and depressions are of low interest for geology, as they do not produce any permanent changes in rock formations where they propagate through. But they acquire special importance when they are linked to earthquakes.

Both, MALLET in his first article and HOPKINS in this context, refer to and rely on YOUNG who was the first to recognize the propagation of earthquakes in analogy to sound waves in air.[1]

While MALLET, in his first treatise, only concludes from the destructive effects on buildings that earthquakes are characterized by waves, HOPKINS already provides an application of the general mathematical theory of wave motions developed by POISSON and CAUCHY to earthquakes.

The important point is that HOPKINS already had a very clear notion of the complexity of wave motions of earthquakes. For he assumes that from the original location of the disturbance, regardless of its type and cause, completely different wave motions propagate which only separate later during the propagation process due to their different velocities.

Thus HOPKINS establishes several simplified models in his theory of wave motions that may occur in earthquakes. First he presents a distinction, which was of fundamental importance for the further development of the theory of seismic waves: longitudinal waves and transversal waves that had already been conceptualized in theory as analogies to sound waves in air and to water waves on the surface. For explaining the difference between these two types of waves, HOPKINS uses two models: the propagation of sound wave in closed tube filled with air, and the propagation of a water wave in an open canal.

In the first case of the closed tube filled with air and of unspecified length (figure 16) the vibration will propagate very quickly from the location of origin of the disturbance in the middle of the pipe to the left and to the right. The sections p' q' and p, q, are parts vibrating. Each of these sections is called a wave and each section has the same properties and the same length. Particles beyond q' and p' have not yet begun to vibrate and all those between p' and q, have stopped to vibrate and do not move anymore.

Figure 16: Propagation of vibrations along a cylindrical tube

According to this idealized model of a simple uninterrupted wave motion that has not yet reached the two end points A and B, HOPKINS lists the following characteristics of such a wave motion.

"(1.) The length p'q' (=l) of the wave will be constant.

(2.) The velocity (V) with which the wave will pass from one point to another (the velocity of propagation) is constant, and depends on the elasticity of the air.

(3.) Each particle will vibrate in succession exactly in the same manner. The time during which it will continue to be in motion, or that required for the wave to pass over it, = l/V, and is the same for each particle in succession.

(4.) The extent through which each particle moves in its vibration (the amplitude of vibration) is by hypothesis extremely small compared with the length of the wave; it will depend on the original disturbance. The direction of vibration will, at a sufficient distance from the original place of disturbance, be parallel to the axis of the tube, or perpendicular to the anterior and posterior bounding surfaces of the waves, those bounding surfaces being transverse sections of the tube perpendicular to its axis."[2]

These characteristics show that during the propagation of the wave motion the particles are either compressed or expanded during a period. Thus HOPKINS calls this type of waves either "waves of condensation" or "waves of rarefaction".

Since all gaseous, fluid and solid substances have a specific degree of compressibility, they are all more or less perfectly suited for transferring waves of this type. If the tube AB would be filled with water, for instance, the same type of wave would propagate in the same way as in a tube filled with air. But since the compressibility of water is much lower than that of air, the amplitudes of the vibrations will be much smaller. HOPKINS estimates the velocity of propagation in water being about four times higher than that in air. It depends on the relation between the elastic forces of water to its density.

While in the transfer of a vibration in a tube completely filled with water no other motion of particles can occur than that of "condensation" or of "rarefaction", the transport of a wave in an open canal is of a very different nature. Explaining the type of wave HOPKINS uses another simplified model of an open canal of uniform width and depth (figure 17).

Figure 17: ropagation of waves along the surface of water in a uniform canal

Assuming that a part of the fluid in the section pq of the canal is disturbed by a sudden elevation of the ground of the canal in this area, the surface of the fluid above will be elevated in almost the same degree and will then, according to the law of gravitation, try to re-establish its horizontal position. In this way two waves p'q' and p,q, are provoked that propagate in opposite directions along the surface of the fluid.

In case there is no obstacle along the sides of the canal, and assuming that the depth of the canal is much smaller than the length of the wave, a wave of this type will have, according to HOPKINS, the following properties:

"(1.) The length (l) of the wave (not necessarily equal to pq) will be constant.

(2.) The velocity of propagation will depend on the square root of the depth of the canal nearly, that depth being much greater than the height of the crest of the wave.

(3.) Particles of the fluid situated in the same vertical section perpendicular to the axis of the tube, will have the same motion at the same instant. Every such section of particles will be carried in the direction of propagation through a certain space, during the passage of the wave, and will then be left at rest. Consequently a wave of this kind will be attended by a current, the velocity of which will depend on the height of the crest of the wave and the depth of the canal.

(4.) The elevation of the bottom pq being sudden, as we have supposed, the front of the wave will be steep, the descent from the crest to the posterior boundary being a gradual slope."[3]

Using simplified models, HOPKINS was able to demonstrate the fundamental difference between the two types of waves:

"In the first case the waves depend entirely on the compressibility and elastic force of the fluid, the motions being independent of gravitation; while in the latter the motion depends on gravitation, and is independent of the compressibility and elasticity."[4]

In addition he expresses an important statement: When both types of waves originate at the same time, they will usually separate very soon, due to the huge difference of velocities of their propagation.

HOPKINS continues to discusses the wave propagation in the centre of a fluid which he calls "waves of compression" and "waves of dilatation". The assumption is that the original disturbance inside the fluid is limited in space and in time. The vibratory motion will propagate in this case very rapidly to neighbouring particles, while the particles which were disturbed originally will remain without any motion.

According to the definition by HOPKINS, "the space, within which the vibratory motion will exist at any instant (i.e. the wave itself,.will be comprised) between two concentric spheres whose common centre is the centre of disturbance". If the vibratory motion is far enough away from the origin of the disturbance, it will be independent of the form of an original disturbance and will have properties analogous to the wave motion of a fluid in a closed tube. This wave motion will propagate in a spherical mode in all directions. Another case which serves for HOPKINS as a first step for explaining the more complex case of a wave motion in a solid substance into all directions, is the propagation of vibrations along a bar. He distinguishes two types of vibrations. One type is a longitudinal vibration - similar to the one in an elastic fluid in completely filled tube - while the other type is a transversal vibration analogous to the chord of a musical instrument. In the case of the longitudinal vibration the elasticity depends on the force which compresses or expands any substance or changes its volume to the original state, while in the case of the transversal vibration it is a force re- establishing the original unconstrained form once this form had been distorted without changing the volume. For determining the velocities of the propagation of both types of waves/vibrations, HOPKINS refers to POISSON's indication of the relation between the coefficient of both types of elasticity that is always smaller in transversal vibration than in longitudinal vibration. Thus the velocity of propagation of the longitudinal vibration will also be higher than that of the transversal one. In case of a brief disturbance, such as the stroke of a hammer, both types of vibrations are provoked, that separate at a certain distance from their common origin due to their different velocities of propagation.

After having explained the complexity of different vibratory motions in different substances in several simplified models, he deals with the last case of the propagation of vibrations in solid mass into all directions. This last theoretical case is the model that is directly applicable to the theory of earthquakes. First, this is also an idealized model, since HOPKINS assumes that the location of the disturbance inside the solid mass is relatively small in comparison to the space where the vibrations diverge, and he assumes that the force that caused the disturbance is operating only for a short time similar to a quarry blast. He also assumes the simplest case of a homogeneous mass with constant density and elastic properties. In this case, the original disturbance, regardless of the type, will provoke a longitudinal and a transversal vibration which overlap first but later separate from each other due to their different velocities of propagation.

HOPKINS then deals with another case being much closer to the reality of earthquakes, i.e. the reflexion and refraction of a wave when propagating from one medium to another of different density and elasticity. With the following diagram (figure 18) he demonstrates the laws of reflection and refraction of light:

Figure 18: Reflection and Refraction of a wave [5]

After these purely theoretical considerations HOPKINS tries to apply them to earthquakes. First he assumes in simplified model a homogeneous crust of the earth, later he extends it to more complicated models of an inhomogeneous structure of strata. As he had assumed in all previous considerations, he also assumes in this case that the original disturbance or the impulse from where the vibrations propagate occurs in a relatively small space, whatever might be the cause of this disturbance: be it a deep volcanic explosion, the collapse of the top of a subterranean cave or the sudden break up of solid rock.

According to HOPKINS the most important point is the determination of the position and the depth of a volcanic centre or another source of the disturbance where earthquakes originate. In this context he makes statement that acquired special importance for subsequent attempts at solving the difficult problem:

"We may observe that the roughest approxiamation to this position would constitute a very important geological element, and might sweep away much that is vague and fanciful in geological speculation."[6]

With his graphical presentation of the centre of the disturbance HOPKINS starts a series of experiments in order to determine by geometrical models from different velocities and directions of vibratory motions, both the depth of the centre and also the epicentre, as it was called later.

Figure 19: Position of the Centre of disturbance by HOPKINS (1847) [7]

For determining the distance of the centre of disturbance from the surface HOPKINS proposes two methods: the first method assumes that both the velocity of the wave (V) inside the solid crust as well as the velocity of the wave rolling from one point of the surface to another one are known. HOPKINS calls the velocity of the surface wave "apparent horizontal velocity" = v. Another determining element is the horizontal direction in any point that the surface waves passes. This direction is identical to a line between a point on the surface of the earth and that point that is on the surface, perpendicular to the centre of disturbance inside the earth i.e. the epicentre as it was called later.

In the determination of the velocity of the wave inside the earth HOPKINS first only refers to the longitudinal compression wave which reaches the surface first and which leads to a horizontal projection with a certain direction.

According to this first method the formula for determining the depth of the centre is:

where a is the distance of the point (P) at the surface where the velocity of the surface wave can be measured, to the epicentre (C)

While such a method, which has to suffice so many pre-conditions, is of little practical use, HOPKINS proposed another method - still valid today - for determining the depth of the centre. For he uses the difference in velocity between longitudinal compression waves and transversal waves, a difference that he had already discussed in detail in his theoretical considerations.

"It may be worth while also to indicate a formula for determining the direct distance OP, and depending on the retardation of the wave in which the vibrations are transversal, with reference to the wave in which they are normal. Let V1 be the velocity of the latter wave. V2 that of the former; V1 will be greater than V2. Also let t1 t2 be the respective times in which the waves pass from O to P, and T the observed interval between the arrivals of the two waves at P."[8]

From this the following formula results:

But HOPKINS is also aware of the fact that for determining the temporal difference T an instrument is needed which is accurate and sensible enough to detect the difference between the two waves that start to propagate simultaneously from point O. He also states, that the distance OP must be big enough in order to allow for the separation of the two waves.

These considerations by HOPKINS were always based on the assumption - which never applies to the reality of earthquakes - that the density of the media through which vibrations propagate is homogeneous. Therefore he modifies his theory of earthquakes accordingly by two of the most important cases:

  • first by the case that a wave propagates through a mass with different elasticities:

  • second, by the much more complicated case that the original point of disturbance O is situated in a fluid.
From this point the wave starts to propagate first through the fluid and then through the solid crust and finally through the sea above.

For the first case HOPKINS proposes a simplified geometrical model (figure 20) of a single horizontal stratum with density and elasticity that differ from the ones of the mass below where the original point of disturbance lies, and he describes it in the following way.

Figure 20: Propagation of a wave through masses of different elastic properties [9]

For the more complicated case of earthquakes and seaquakes where vibrations propagate through three different media, HOPKINS presents the following model:

Figure 21: Propagation fo a wave in a fluid mass [10]

Further considerations HOPKINS discusses only briefly, refer to the case that the location of the disturbance is not very small but of rather large dimensions, and to the case when the sea floor is not a uniform, as indicated in figure 21. In this case, the sea wave will not continue to maintain its circular form.

Maybe the most important proposal, with the most practical method for determining the epicentre, that has been used later many times[11], has also been formulated by HOPKINS:

"Suppose for instance two points were known, at which the vibratory wave arrived at the same instant, then, if we bisected the line joining these two points, and drew another line perpendicular to it through the point of bisection, the latter line would pass through C, assuming always the spherical form of the general vibratory wave. In some cases also, as in those of earthquakes in the neighbourhood of insulated volcanoes, circumstances may indicate, antecedently to any instrumental observations, the approximate position of the point in question."[12]

Given this large number of remarkable and influential foundations of a theory of vibratory waves and wave motions of earthquakes it is a pity that HOPKINS' interest turned away from seismological topics and turned to the mathematical treatment of purely geological problems, such as the movement of glaciers[13]. All his theoretical considerations were not linked to any concrete historical earthquake. This was also the reason for the exceptional importance of the contributions by Robert MALLET who himself highly estimated the theoretical models and considerations by HOPKINS but who overshadowed him with his own theories. For what MALLET added to his theoretical considerations that were first based only on his knowledge which he collected from the literature on historical earthquakes, was his famous investigation of the strongest earthquake at that time, the so-called "Neapolitan Earthquake". This study is considered with full justification the first attempt to determine all propertie of an earthquake according to wave theory and thus to be the basis for all further endeavours in this field[14].

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