CALCULATION OF MOTION OF GROUND 85 
Here #? is the restoring force per unit mass exerted by the spring when its 
length is increased by unity and 22 is the damping force or viscous resistance, 
per unit mass, to the motion of the mass m when moving relative to the stand 
with unit velocity. Hence we have 
y+ 2ky + pry = — Z. 
We see from this equation that if initially x=0, =0 and the mass is at 
rest in its equilibrium position, then when the ground begins to move we shall 
have y = —x because at the start y and y are zero so that j= —#. This merely 
means that at the start the mass m remains at rest relative to the fixed axes. 
But y will only be equal to —x during an interval small compared with the 
natural period of vibtation of the mass m. 
The above equation may be regarded as giving y when x is known or as 
giving x when y is known. In previous discussions of the theory of seismo- 
graphs, which the writer has seen, the motion of the ground or x has been as- 
sumed known and the resulting values of y have been calculated. 
The solution of the equation j+2kj+p2y = —Z= F(t) where F(#) denotes 
a function of the time ¢ and when y and y are zero at ¢=0 is 
1 t 
1S — I e~FC@-® sin u(t — 9) F(n)dn 
BK v0 
where 
(p? ass 2) eles 
By means of this equation it is easy to calculate the seismograph detiec- 
tions y as a function of ¢ for any simple values of F{(é). 
In homogeneous ground an explosion of dynamite would be expected to 
‘produce single waves, or short trains of waves, so that it is important to con- 
sider the motion of the seismograph due to single waves. To represent a single 
wave we may suppose that x =A(1—coswt) from t=0 to t=2z/w and that 
x =0 when ¢<0 and when t>27/w. This gives the wave shown in Fig. 1. 
x | | 
(0) T fs) 
Fig. 1 ’ Fig. 2. 
iu 
Before considering this case it will be convenient to consider the case when 
x=A(1+coswt) when ¢>0 and x=0 when ¢<0 which gives the series of 
waves shown in Fig. 2. 
The equation x =A(1—coswt) gives ¢= Aw? cos wi so that the expression 
for y becomes 
Ao? t 
V2 === e— (9) sin w(t — n) cos wndn. 
BK 0 
We shall consider two cases. One when the seismograph is undamped so that 
k=O and one when the seismograph is critically damped so that »=0. 
229 
