Great Neapolitan Earthquake of 1857/Part I. Ch. XV

by Robert Mallet

CHAPTER XV.

TRANSIT VELOCITY OF THE WAVE FORM.

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Before passing to the subject of velocity of the wave, a few remarks should be made as to the methods of ascertaining its transit velocity. The transit velocity —that with which the form of the wave is transferred from point to point of the shaken surface— is so great that it can only be ascertained with the desirable precision, by means of a proper seismometer, of the self-registering class described in the author's fourth Report on the Facts of Earthquakes, ('Trans. British Association,' 1858), to be established prior to the shock; and the only known method of determination is based upon observation of the time of arrival of the wave, at each of three or more distant stations, within the shaken area.

In the facts which we can usually collect in the field, after the shock, we are limited to the casual observation by the ordinary time measurers (clocks and watches) of the moment of observed shock at several places.

Such observations are liable to multiplied sources of error; from errors in the indicated local time as shown by the time-pieces themselves, and errors of observation of the moment of true shock, as said to have been recorded by them, as well as ambiguous or doubtful statements; so that out of a considerable number of such time-facts, obtained in a seismic-shaken country, probably not above two or three can be really relied upon. Were we in possession of a large number of time records of considerable accuracy, such even as rated chronometers at the distant stations but still more, self-registering instruments, would afford, it would then be important to apply the method of least squares to their discussion, in the way that has been done by Dr. Julins Schmidt, astronomer at Bonn, to the Rhenish earthquake of 1846 ('Das Erdbeben vom 29 July 1846, im Reingebiet,' &c., von Dr. Jacob Nöggerath. Bonn, 1847. Pamphlet, 4to).

If we denote by $\alpha \beta \gamma .....\nu$ the surface distances in geographical miles from the seismic vertical = $\mathrm {Z}$,

$\mu \mu '\mu ''.....\mu ^{n}$ their respective differences of longitude in time from $\mathrm {Z}$,

$tt't''.....t^{n}$ the observed times at $\mathrm {A} \mathrm {B} \mathrm {C} ....\mathrm {N} \mathrm {T}$, the moment that the wave reaches the surface at the seismic vertical $\mathrm {Z}$,

$\tau \tau '\tau ''.....\tau ^{n}$ the transit periods or times of running over the distances $\alpha \beta \gamma .....\nu$, it is obvious that

$\tau =t\pm \mu -\mathrm {T}$

and $\tau ^{\prime }=t^{\prime }\pm \mu ^{\prime }-\mathrm {T}$

. .

and

. .

. .

$\tau ^{n}=t^{n}\pm \mu ^{n}-\mathrm {T}$

(I.)

and that

For places situated to the west of the seismic vertical . . . . . $\tau =t+\mu -\mathrm {T}$

(II.)

And for those to the east of the same $\tau =t-\mu -\mathrm {T}$

(III.)

While for those to the north or south of the seismic vertical or in the same meridian with it… $\tau =t-\mathrm {T}$

(IV.)

Add the respective velocities therefore

${\frac {\alpha }{\tau }};{\frac {\beta }{\tau ^{\prime }}};{\frac {\gamma }{\tau ^{\prime \prime }}}$ . . . . . . . . . . ${\frac {\nu }{\tau ^{n}}}$

(V.)

Obtaining an average transit time per second for three adjacent places, situated in one radius of surface, from the seismic vertical, Dr. Schmidt then applies the method of least squares to the discussion of all the remainder, and with a result undoubtedly important, where, as in his example, he has had thirty distinct observations of time, at as many different stations; but which, when the number of stations is very limited in which any real confidence can be reposed, possesses no advantage over a simple choice from the whole of the most trustworthy, and the reduction from these of the mean.^[1]

The question of transit velocity, however, although of great physical interest, and destined, no doubt, ultimately to connect itself in an important way with that of the velocity of the wave particle (or wave itself), is, as respects seismometry viewed as a branch of physical geology, of subordinate importance at present.

1. Or, as suggested to me by Dr. Robinson, giving each observation weight proportional to the length of its own wave-path, which gives the formula—

   $\mathrm {V} ={\frac {\alpha ^{2}+\beta ^{2}+\gamma ^{2}\mathrm {\&c.} }{\alpha \tau +\beta \tau '+\gamma \tau ''\mathrm {\&c.} }}$

   or, what is the same in result, though perhaps more convenient—

   $\mathrm {V} ={\frac {\alpha \tau +\beta \tau ^{\prime }+\mathrm {\&c.} }{\tau +\tau ^{\prime }+\mathrm {\&c.} }}$