Earthquakes and other earth movements

by John Milne

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 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.