Great Neapolitan Earthquake of 1857/Part II. Ch. XIII

by Robert Mallet

Chapter XIII.

Deductions from Facts Presented at the Certosa, Continued—Interval of Time Between the First and Second Shocks Calculated.

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We can calculate, therefore, from these data, to a good approximation, the interval of time that elapsed, between the arrival of the first and second shock, and thence the difference in transit velocity, of the two waves of shock; the first through the limestone, the second through the deep clays and gravels, &c.

The chimney may be regarded as a parallelopiped vibrating as a compound pendulum, upon $b$, and $b_{2}$ (Fig. 237) as points of suspension, whose centre of oscillation is in the plane of shock passing through the centre of gravity, and distant from $b$ or $b_{2}$ by two-thirds of the diagonal of the parallelopiped, in the same plane.

This distance = 4.33 feet is the length of the corresponding simple pendulum, with some small allowances for the hollowness and irregularity of form. The greatest possible are of vibration, is limited by that which would bring the centre of gravity, $c$, vertically over $b$ and $b_{2}$, Fig. 237, beyond which the mass must have fallen. This I found to be 21°, either side of the vertical through the centre of gravity, when the chimney was in its original undisturbed state.

The arc actually described, must have been less than this, and from other observations yet to be referred to I conclude, that this arc was not more than 15° at either side the vertical, or with a horizontal chord at the centre of gravity of about 12 inches. We shall not make a sensible error, however, by assuming the arc described, as the largest possible.

Taking the time of oscillation from the equation

$T=\pi {\sqrt {\frac {l}{g}}}\times (1+{\frac {a}{8\,l}}+{\frac {9a^{2}}{256\times l^{2}}}+{\text{. . . . .}})$

where $a$ = ver sine of haft the arc of vibration; we have

$T=3.14{\sqrt {\frac {4.33}{32.2}}}\times (1+{\frac {.066}{8\times 4.33}}+{\frac {9\times 0.66^{2}}{256\times 4.33^{2}}}+{\text{. . . . .}})$

${\text{or }}=3.14\times .3668\times 1.0019{\text{ approximately,}}$

${\text{and }}T=1.5635{\text{ seconds;}}$

correction for latitude being needless.

From the moment of arrival of the first shock, up to the arrival of the second transverse to it, the chimney had made, one half and one complete, vibration, and possibly had just commenced another. At the moment that the chimney relapsed upon its base it lost vis vivâ, and therefore time, before it rose again, to complete the arc westward. This minute loss of time we can only estimate, because although we know the velocity at the moment the mass of the chimney struck its base, on resuming the perpendicular ($\mathrm {B}$, Fig. 237), we do not know its hardness, elasticity, &c., upon which at loss also depends.

Neither can we calculate precisely, how much of the commencing arc of the third oscillation had been performed (if any), before the second shock reached the chimney, because we do not know the precise point round which it rotated, &c.

But where the are of horizontal rotation is great, as it is in this instance, where the chimney was twisted round 30° to the westward, the latter must have been extremely small or nil.

Calling these two small corrections $d$ and $f$, the whole interval in time ${\text{I}}$, between the shocks is

$I=T+{\frac {T}{2}}+d+f.$

We therefore obtain

$I=1.536+{\frac {1.536}{2}}+d+f.$

or without the two latter,

$I={\text{1.7304 seconds,}}$

and estimating $d$, assuming $f$ as small $=0$, we may consider

$I={\text{1.75 seconds.}}$

This is the difference in time, between the arrival of the two shocks; both of which started at the same moment and from the same origin. The evidence for the position of the latter has yet to be adduced; anticipating the fact, however, the Certosa, is distant from it, 16 1/2 geographical miles, or 33,415 English yards in the right line, which was the path of the second shock, arriving directly 15° W. of north to south, through the deep clays and gravels, &c., of the Vallone; while that of the first shock, was through the east lateral range of solid limestone mountains, but whose length (as respects the wave-path) we may consider as one-fifth of a mile greater.

From facts (also yet to be adduced), I found that the general velocity of translation of the wave of shock, through the limestone country, was at the rate of 240 yards per second. This, therefore, may be taken as the velocity of translation of the first shock here (through the limestone); the total time of its transit from the origin (surface velocity) is therefore

${\frac {33415}{240}}=139.230$ seconds;

and through the clays and gravels the whole time is

$139.230+1.750=140.980$ seconds.

The velocity per second of surface translation in the clays and gravels was therefore

${\frac {33415+402}{141''}}=239.84$ yards per second.

The shock through the limestone reached the Certosa from the Colline of Padula, the nearest point of rock in the path, through an intervening stratum of clays and gravels, by which there must have been some loss of vis vivâ. If we throw off the decimals for this correction, which we can only estimate, we have finally, for the surface velocity of translation, 240 yards per second in the limestone, and 239 yards per second in the clays and gravels, measures approaching so nearly to equality, as to warrant one or both of the following conclusions.

Either, the main primary or direct wave arrived, like the other through the limestone formations, deep beneath the clays and gravels of the piano, and shook the latter resting upon them, at their own rate of translation nearly; or (here at least), the bedding-joints and other breaches of homogeneity in the limestone rock produce a retardation of the wave transit therein, such as reduces the velocity to nearly that in the dense clays and gravels. In this extremely dislocated and overturned country I deem the latter as most probably the fact.

The measures of velocity for this locality are relatively, however, as nearly reliable as the data will admit. As absolute measures of transit velocity, they must be taken as mere approximations, as all our data for this are based upon the observations made as to time at the moment of shock, and unfortunately are not only few in number, but by no means to be relied upon as to exactness, in this locality.