Page:Great Neapolitan Earthquake of 1857 Vol 2.djvu/412

CHAPTER XVI.

OF LOCAL DISTURBING CAUSES PRODUCING ABRUPT PERTURBATIONS OF THE WAVE OF SHOCK.

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In the general equation of wave motion

$V=a\cos \left({\frac {2\pi }{\lambda }}(x-at)\right)$

the velocity of transit $a$, is constant for a given uniform medium, and the vis vivâ of any particle $\Delta m$, whose density is given, is $={\frac {1}{2}}\Delta ma^{2}$* ($a^{2}$ being the intensity) for the whole undulation or complete phase, and is also constant.

If the mass of the medium in wave motion, at any moment of the transit, is always the same, the thickness of the successive spherical couches, between similar points of phase, must diminish as $r^{2}$ increases, $r$ being the distance from the origin; and the vis vivâ can only remain constant, with a constant velocity in the direction of $r$, by supposing movement in the ordinates transverse to $r$, whose range increases as that in $r$ diminishes, and the extinction of the wave in the direction of $r$, will continually tend towards $\lambda =0$. But the vis vivâ may also be constant, and the extinction of the wave be due to a continually-increasing amplitude, and decreasing wave velocity, as in the analogous case