Also c1 a1 = c1 b1, cos θ.
∴ c1 a = c sin θ cos θ/h = c/h ∙ sin 2θ/2,
and sin 2θ is greatest when 2θ = 90° or θ = 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.