THE CALCULATION OF THE MOTION OF THE 
GROUND FROM SEISMOGRAMS 
By H. A. WiLson 
Rice Institute, Houston, Texas 
(Received January 30, 1932) 
ABSTRACT 
The equation of motion of a mechanical seismograph is —£=y+2ky+p?y where 
x is the ground displacement and y the seismograph deflection. This equation may be 
solved for y when x is supposed known or for x when y has been observed as a function 
of the time. In this paper both of these ways of solving the equation are considered. 
The motion of the seismograph due to a train of waves starting at t=0 is considered 
and also the motion due to the arrival of a single wave. In each case seismographs with 
several periodic times and either undamped or critically damped are considered. 
Curves are given showing the motion of the ground and the calculated motion of the 
seismograph. The motion of the ground corresponding to several simple assumed seis- 
mograms is also worked out and shown by means of curves. The motion corresponding 
to a given seismogram depends greatly on the periodic time and damping of the seis- 
mograph. Finally the ground motion is deduced from two actual seismograms due to 
dynamite explosions. An integraph is described which enables the calculations to be 
done more quickly. 
N THE case of seismograms obtained with mechanical seismographs hav- 
ing a definite periodic time and damping coefficient the motion of the 
ground can be deduced from the seismograms without serious difficulty. This 
of course is well-known, but so far as the writer is aware few if any results of 
such calculations have been published, at any rate for seismograms obtained 
in geophysical prospecting.! 
The seismograph consists essentially of a heavy mass supported on 
springs, or in some other way, so that it is free to oscillate. The motion of the 
ground relative to this mass is magnified and recorded. 
Consider a mass m supported by a spiral spring from a stand resting on 
the ground. Let y denote the upward displacement of the mass m, relative 
to the stand, from its equilibrium position, so that y is equal to the decrease 
in the length of the spring due to the upward displacement of the mass relative 
to the stand. When y is zero the force exerted by the spring is equal and op- 
posite to the weight of the mass. 
Let x denote the upward displacement of the ground and stand due toa 
wave in the ground relative to axes supposed fixed. The equation of motion 
of the mass m is then 
mé+ jy) = — mp*y — mky. 
1H. Arnold, Zeits. f. physik. Erdkunde 10, 269-317, gives calculations of the ground mo- 
tion from several seismograms. 
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