Earthquakes and other earth movements

by John Milne

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,

^k₂/_k1 > ^r₁2/_r2² = f ^r₁2/_r2² (2)

where f < 1 is the rate of dissipation of energy which is assumed to be constant.