Figure 22: Twisting of two buttresses of a church in Valparaiso

DARWIN also mentioned such twistings of parts of buildings in Concepcion during the same earthquake and compared them to those in Valparaiso, also referring to the report by ARAGO in "l'Institut", 1839, who had observed similar phenomena in ancient Greek temples.

MALLET directly continues DARWIN's criticism of the explanation of such turning movements by a type of vorticose earth movement. He agrees with DARWIN that such a vorticose movement cannot be proven. But DARWIN suggested that the cause is a tendency in each stone with such weight to arrange itself in some particular position, with respect to the lines of vibration, in a manner somewhat similar to pins on a sheet of paper when shaken or like sand on Chladnis' acoustic plates. But this explanation is even more unlikely. In the latter case each stone would have a separate vortex and axis of rotation, in addition to the inconceivable angular velocity of motion at the extremity of the radius of the vortical movement.

MALLET's own explanation is based on the assumption that the base of a body has a "centre of adherence", as he calls it, like every body has a centre of gravity. Depending on the position of both centres to each other, there are the following possibilities during earthquakes:

"The centre of gravity of the body may be at such height above the base that it shall upset by its own inertia. This is the case with houses, towers, and walls' when they collapse during earthquakes.

The centre of adherence may be in point of the base plumb under the centre of gravity of the body, or in a vertical plane passing through it s centre of gravity, and in the direction of motion of the base', then the stone seems to move in the opposite direction. In reality the base under the body has slipped from under the body. The centre of adherence may be in points outside the line of its intersection by the plane, in such cases the ‚the effect of the rectilinear motion in the plane of the base will be to twist the body round upon its bed".[1]

For demonstrating this effect and this twisting motion MALLET had built a model of the two Calabrian pedestals where an observer could carry out a rectilinear movement in order to produce the twisting effect:

Figure 23: MALLET's model of the two Calabrian pedestals

After MALLET had proven that it was not necessary to presuppose a vorticose movement for the turning of bodies, as they were observed in the mentioned earthquakes, but that just a simple rectilinear movement is necessary, he asks the decisive question leading him to his theory of the nature of motions of earthquakes:

"If the earth shake both back and forwards - how is it that these and other displaced bodies are not replaced by the reverse motion - by the same sort of motion acting in the contrary direction?"[2]

It would have to be a horizontal movement that is either directed forwards or that is slower in one direction than in the other.

Since a pure forward movement would cause all bodies to fall into the same direction and would remain there: a situation which could not be observed, just as a reverse motion of bodies in their original position did not occur, a motion faster forwards than backwards must occur. But the only type of such a motion is the propagation of an elastic compression wave:

"The transit of a wave of elastic compression, or of a succession of these, in parallel or in intersecting lines, through the solid substance and surface of the distrubed country.[3]

With this formulation MALLET provided the basis and foundation for a theory of earthquakes, in contrast to previous theories, in which the first impulse is clearly separated from the way the wave motion propagates. The original impulse, that provokes the compressional wave, may have different causes. It can be caused by falling down of heavy masses or by breaking up of the earth strata due to the pressure of elastic and fluid matter from below, or as MICHELL had assumed, due to the pressure of hot vapour in caves due to cold water entering the cavity or due to the sudden explosion of a volcanoe where, as HUMBOLDT reported on the Cotopaxi, a mass of rocks of more than two hundred tons was ejected over a distance of nine miles. Independently of the cause, MALLET wanted, in his theory of wave motion in earthquakes, only to apply the laws of waves and vibrations in gaseous, fluid and solid bodies.

Although at his time it was still unknown how in an exactly analogous way particles in solid bodies move within the limits of elasticity just like those in fluids, MALLET thought that it was highy probable that the particles pass through similar circular or ellipsoid curves, as the WEBER brothers explained in their wave theory.

Figure 24: Diagram illustrating internal motions of a wave in a fluid

The strongly marked curve is the envelope of the wave as its moves towards the position shown by the similar dotted line. Wihin one wave length every fluid particle in the wave has described one complete revolution, as indicated by the arrows and circles. The arrows turne more and more depending on the depth of observation in the liquid.

Based on this observations MALLET was sure, that the surface of the solid earth, as an effect of an elastic wave or a strong earthquake stroke, undulates, i.e. carries out a wave movement. This propagation of a wave along the surface was observed, according to MALLET already in 1692 during a strong earthquake in Jamaica. Although it is difficult to reconcile our idea of solid bodies with such internal movements that pass through them, cause compressions or movements between the particles within the limits of their elasticity, MALLET is able, as GAY-LUSSAC before him did, to present several plausible examples: playing the piano in a reception hall may rock the walls of a house, a passing train may put the floor in vibration, houses and towers rock with the wind and high chimneys of factories sway back and forth. In all these cases there is no break in the cohesion of the particles of the vibrating bodies while ever they oscillate within its specific elastic limits.[4] MALLET refers to events similar to earthquakes, as they also provoke vibrations of the floor, such as collapses of coal mines in Northern Britain or the toppling of one or several large vertical stones in Stonhenge, that provoked such a tremor that all people living in the vicinity thought that an earthquake had occurred, which actually had happened according to MALLET's definition, but was not caused by a natural event. The decisive characteristic of MALLET's theory for being accepted was its superior explanatory power. With this theory he is able to explain all phenomena that were described in earthquake literature and that partly were left unexplained, with the uniform basis of wave motions, such as the collapse of walls in opposite directions by reflected seismic waves that throw down all those walls that remained intact during the first passage of the direct seismic wave. In the theoretical explanations of large earthquakes by MICHELL (Lisbon 1755), DARWIN (Concepcion 1833) and LYELL (Lisbon and Concepcion) there were in particular the water movements at the sea shores and in the interior lakes that could not be explained or that led to contradictory hypotheses. The recession of the sea directly after the first stroke, as observed in the Lisbon and Concepcion earthquakes - and the occurrence of a sea wave destroying everything half an hour later, was explained by MICHELL with the subsidence of a part of the bottom of the sea. But for LYELL the recession of the sea at the shore is due to the elevation of part of the bottom of the sea, during which the water will be attracted even at the sea shores that are far away. But none of these authors were able to explain the subsequent occurrence of the huge sea wave at the shores. DARWIN also just reports the destructions by the huge sea wave in Concepcion, but without any theoretical explanation of this phenomenon.

But MALLET was able to explain these phenomena with a system of wave motions that originate in a centre of disturbance but propagate at different velocities through the solid earth, water and air as sound waves. If the centre of disturbance is below the bottom of the sea, as in the case of Lisbon and Concepcion, the following order of wave motions, according to MALLET:

1. The elastic compressional wave due to the first stroke propagates with enormous speed of about 7.000 – 10.000 feet per second from the point vertically above the centre of the stroke into all directions produces real undulations along the surface of the bottom of the sea.

2. The "forced sea wave" emerges in the same moment when the undulation of the bottom of the sea meets shallow water at the coast producing a banking of water above itself and which in turn produces the strange receeding of the water at the shore and the changes at the shores of interior lakes.

   If a sound wave is created in the point of origin during the first stroke by the tearing apart rocks or earth strata or the explosion of a fire hearth, the following waves emerge:

3. A sound wave passing through the earth,

4. a sound wave passing through water that will arrive later due to its lower velocity than the one through the earth, but earlier as the

5. the sound wave passing through air.

Figure 25: The great sea wave (MALLET 1846)

According to MALLET's theory of the dynamics of earthquakes we have to distinguish these primary phenomena of various wave motions from the secondary phenomena that are created by the passing through of such waves. Both types have to be distinguished very precisely from those other simultaneous manifestations of force such as volcanic eruptions, permanent elevations and subsidence of land, etc., that, in contrast to previous opinions of HOOKE, LYELL, DARWIN, and HUMBOLDT, are not an integral part of an earthquake.

The paradigm shift in earthquake theory that MALLET had provoked with his definition of an earthquake as a passage of wave motions, can be seen best in the re-interpretation of those phenomena that had previously been considered as characteristic components of an earthquake, by pre-supposing the transport of solid, fluid, hot or vaporous or gaseous masses, such as the opening and closing of fissures and the ejections of solid and fluid substances from such fissures or holes such as ash, sand or water fountains, or the eruptions of fire, smoke, gas and vapour as described by DOLOMIEU and others in particular during the earthquake in Calabria in 1783. For MALLET all these phenomena are only secondary effects of the passage of one or several, partially reflected and overcrossing elastic waves. Even the example of radial fissures emerging from a central point, as it was observed in 1783 in Jerocarne, which were used by LYELL as a proof of his theory of elevations and subsidence[5], can be explained by MALLET as an effect of several wave motions.[6] Thus he can state, with full justification, in a final observation that all interference with secondary earthquake phenomena are also based on the mechanical wave theory:

"To this many other such curious, minute, accidental earthquake consequences might be added and explained, but it would be a tedious and useless labour, as the explanation of all such will be apparent easily to those acquainted with physics, were the conditions have been properly observed."[7]

MALLET improved and extended his wave theory of earthquakes in two further treatises: first in the 4th report on earthquake phenomena in the British Association for the Advancement of Science in 1858 and secondly in a treatise "On Earthquake Phaenomena" (3rd edition in 1859) that was first published in 1851 in the Admirality Manual of the British Navy. There, he calls the elastic wave as "large earth wave" progressing after the true earthquake shock with enormous velocity from the point perpendicular over the centre of the stroke into all directions thus creating undulations on the surface, the height of which continuously decreases with increasing distance from the point of origin. In order to gain a clear picture of the basic conditions of the propagation of seismic waves, MALLET illustrates his theory with a diagram:

Figure 26: Propagation of a longitudinal compression wave

This diagram is a vertical section of a part of the earth in the plane of the widest circle passing through the surface at h'h and through the centre of the stroke at A; Ap is the vertical from that point, whose depth below the surface is BA.

The wave propagates from the point of origin with a normal vibration and two transversal vibrations (assuming the earth mass as homogeneous); when we first neglect the transversal vibrations, then we can imagine the propagation of the wave into all directions as concentric spherical layers with the same volume at similiar phases of the wave. When R is the radius, the space between two spherical layers decreases as r2, and the overthrowing energy of the shock in the direction of r decreases in relation of the square of the distance from the location of origin.

The shock reaches the surface at B, perpendicular above the point of origin; but for all point around B it arrives in angles that increase until reaching the horizontal plane with increasing of the distance from the point of origin. The area and the horizontal shape of each wave that emerge on the surface of the earth is more or less circular or in ellipses, depending on the location of origin or the centre of disturbance. This centre may consist of a point, a line or several points of impulses. In the further propagation of such circular or ellipsoid lines, that MALLET called in his first treatise in 1846 in analogy to tide waves "earthquake cotidal lines" and later "coseismal lines", they will change their form according to the properties of the surface layer, although they will remain closed lines (figure 27).

Figure 27: Earthquake "cotidal lines"

"The diagrams (1, 2, 3) show the variable forms which the closed curves of the earth wave cotidal lines may assume, in the same uniform formation, according to the nature and position of the original disturbance. Where, as in (1) this is at a single point, the curves will be nearly circular; where the disturbance is along an uniform right line, they will be ovals (2); and where the original impulse comes simultaneously from several distinct points, the curves will be irregular closed figures of contrary flexure (3). All these are subject to the changes already described in passing from one formation to another."[8]

This illustration also shows that MALLET also thought about the different forms of the centres of such phenomena and that he did not reduce the location of origin of an earthquake to a common point of a volcanic hearth, as others accused him of later.[9]

For MALLET, the depth of the location of origin of the earthquake is of particular importance, because different systems of elastic waves can be described:

"Where successive fractures at different, but great, depths, take place in this way, two distinct systems of elastic waves, one of them having a vertical, and the other an horizontal or largely inclined direction of transit, will traverse the mass of the earth's crust, and be felt upon its surface at once; but the amplitude of the waves of the former system will be very small, as compared with the waves of the latter, and hence an undulating and oscillatory motion will be experienced at the surface, accompanied by a sharp upward jar or vibration at the same time,- circumstances which have been occasionally recorded as having been observed during earthquakes."[10]

In his first treatise on the dynamics of earthquakes MALLET has added illustrative diagrams to his theoretical considerations on the nature and mechanisms of elastic compression waves demonstrating their creation by flexure or fracture of rock strata.

In the following, MALLET describes how complex the system of wave motions can be when the crust of the earth is breaking when consisting of horizontal strata of rock or masses of individual rocks:

"Where the elevated crust consists of horizontally, or nearly horizontally stratified rock, or of several parallel superimposed masses of independent rock, of whatever sort, those that lie deepest will be first and most bent, and, ceteris paribus, will be the first to give way by fracture; those above them will break in succession, and when the area and amount of elevation are very great, each layer or plate of the whole elevated crust may be fractured successively at several places, so that from a single locus of elevation a number of earth waves may be produced and propagated in succession, each constituting a true earthquake shock: nay, even after numerous fractures have taken place, the further upheaval and tilting over of vast masses of the now broken up plate will, where its thickness is considerable, produce renewed mutual pressures and violent constraint, in the directions of the diagonals of the several tabular blocks, which will afterwards give rise to minor shocks, as in the further progress of elevation, the several masses of the ruptured crust are raised and freed from each other, and the compression and constraint of their elastic particles thus successively removed."[11]

MALLET presents and describes the individual phases of elevation and subsidence of the crust due to volcanic forces and the subsequent compressions and expansions of rock layers that eventually lead to a fracture in the crust in the following diagrams:

Figure 28: Start of the process of elevation according to MALLET (1846)

"The mass of stratified formation is in act of being elevated by the protrusion of lava from beneath, carrying up the loose materials reposing upon it. The arrows above and below the neutral plane indicate the directions of the compressions and extensions, and hence of the transit of elastic waves, if the flexure be of sufficient extent and rapidity to produce such."[12]

"A similar stratified mass is supposed to be in progress of subsidation, and the arrows show that all the previous forces remain operative, but changed in direction."[13]

Figure 29: Start of the subsidation according to MALLET (1846)

"In d, the flexure has been attended with fracture, and earth waves have passed outwards in all directions, and the diagram shows the directions of compression produced by the wedging together of masses of rock in progress of being lifted or tilted over, by continued upheaval after previous fracture, by which minor earth waves, subsequent to the great shock, may be propagated."[14]

Figure 30: Break up of the crust according to MALLET (1846)

In summary, the theory on the real origin and nature of the earthquake stroke is formulated by MALLET as follows:

"Such, then, I conceive to be the true origin and nature of the earthquake shock. It is produced by any force which disturbs the equilibrium of elasticity of the materials constituting the crust of the globe, and it is propagated from the locus of its origination, in accordance with the laws of transit of elastic waves through such materials."[15]

But for MALLET these theoretical considerations lead to the practical consequence to determine the transit velocity of elastic waves, their amplitude for any type of rock or material the earth's crust. All determining elements for his theory depend on elasticity and cohesion of mineral substances,whether they are the loose and rocky parts of the crust. However, only few of such values were known, to engineers and architects. Since in Britain earthquakes occur very rarely, only few observations and analyses of the soil could be carried out to gather data on the transit velocity. MALLET proposed the artificial production of an earthquake by explosions at different locations of the earth.

"But another, and much more rapid, and perhaps even certain, method, remains to be noticed, for obtaining part of our data as to the specific period of wave transit, viz. by direct experiment, which in all matters of inductive science may be pronounced, whenever it is possible, better than mere observation.

I have already stated that it is quite immaterial to the truth of my theory of earthquake motion what view be adopted, or what mechanism be assigned to account for the original impulse; so, in the determination of the time of transit of the elastic wave through the earth's crust, if we can only produce a wave, it is wholly immaterial in what way, or by what method, the original impulse be given.

Now the recent improvements in the art of exploding, at a given instant, large masses of gunpowder, at great depths under water, give us the power of producing, in fact, an artificial earthquake at pleasure; we can command with facility a sufficient impulse to set in motion an earth wave that shall be rendered evident by suitable instruments at the distance, probably, of many miles, and there is no difficulty in arranging such experiments, so that the explosion shall be produced by the observer of the time of transit himself, though at the distance of twenty or thirty miles, or that the moment of explosion shall be fixed, and the wave period registered by chronometers, at both extremities of the line of transit.

For this alone very moderate charges of powder will answer, but if the explosion be made out at sea with sufficient energy, there will not only be produced the transit of the earth wave and the sound waves through the sea and air, but the accumulation and subsequent coming in of the great sea wave, so that all the phenomena of the natural earthquake are thus placed within our power of production, observation, and control."[16]

These explosion experiments for producing artificially an earthquake were permitted and were carried out by MALLET with utmost care and in different geological formations.