CALCULATION OF MOTION OF GROUND 89 
7 they only agree from ¢=0 to ¢=1. We see that to obtain the actual motion 
of the ground it is necessary to use a much longer period with a critically 
damped seismograph than with an undamped one. 
Fig. 8 shows the curves for x and —y when w=p= so that the periods of 
the waves and of the seismograph are equal. The equations for x and y in this 
case are 
x 
y 
The seismograph is less sensitive and x and —y do not agree as well as when 
the period of the seismograph is greater than that of the waves. 
1 — cos (30#)° 
— }sin (30#)° + (wt/12)e—**/6. 

loy 
Fig. 9. 
If the period of the seismograph is one third that of the waves and the 
damping critical then we may take 
x = 1 — cos (30#)° 
so that w=7/6 and k=7/2. The equation for y is then 
Szi + 8 
100 
—xrt/2 
1 
j= a cos (30¢ — 36-87)° + 
Fig. 9 shows the curves for x and —10y in this case. —10y is plotted instead 
of —y because the maximum value of y is only one twentieth that of the 
maximum value of x. 
Comparing Fig. 9 with Fig. 7 we see that the deflections of a critically 
damped seismograph with period three times that of the waves are somewhat 
similar in form but about ten times larger than those of a critically damped 
seismograph with period one third that of the waves. 
It appears that, while in all cases the initial deflections are equal and op- 
posite to the motion of the ground, the only case in which this remains true 
approximately for more than a fraction of a wave is that in which the period 
of the seismograph is large compared with that of the waves and the seis- 
mograph is undamped. 
The ratio of the maximum value of y to the maximum value of x is about 
w?/(w? — yu?) or if T is the period of the waves and T”’ that of the seismograph 
it is T’?/(T’2— T*). This is equal very nearly to unity when T’/T is large and 
is nearly equal to (T’/T)? when T’/T is small. When 7’ and T are nearly 
equal it becomes large but not for the first wave. 
Very short period seismographs are difficult to make sufficiently sensitive 
233 
