50 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
applied to the sand either at the base or at the sides of the box and must be transmitted 
thru the sand. What is the character of this transmission? Evidently it must depend 
upon the amount of water contained in the sand and also upon the frequency of the vibra- 
tions. If the sand is fairly dry and the frequency slow, the sand will act very much as an 
elastic solid body and we may assume that the successive horizontal layers shear slightly 
over each other as a solid would do, and that the forces brought into play are proportional 
to the shear. The movement under these conditions, when we neglect the influence of the 
sides of the box, would be somewhat like the movement of a flexible rod fastened to the 
bottom of the box. The rod, however, would be bent with compressions and expansions on 
opposite sides, whereas the sand is distorted simply by the elastic shear of successive hori- 
zontal layers over each other; but the character of the motion in the two cases is very simi- 
lar. Tounderstand themovements of the sand we must consider the forces acting between 
the successive layers. The equation of motion of such a system (provided the motion 
is not too large) is 
d*y_ nd'y 
date fae 
where 2 is measured vertically upwards, and y in the direction of motion; ¢ is the time, p 
the density of the material, and n its coefficient of rigidity or shear. The solution of 
the equation if the column of sand were slightly distorted, and then allowed to vibrate 
without further disturbance, is 
en aT Me Acre 
= A sin—2z-sin—t 2 
y X, T., (2) 
where Ngee 
T,? P 
This represents a standing wave, of wave-length Ay and period 7. The period may have 
a great number of values, namely : 
wae Fi 
°— Fm +1 Vy @) 
and the corresponding wave-lengths are 
4H 
where H is the thickness of the sand; and 2m + 1 is any positive, odd, whole number. 
Introducing these values in (2) we get 
y= Asin=Z ent) De. sinet@ Gut ed ae (9) 
P 
The longest period with which the system can vibrate is 
T,=4H4/P (6) 
nv 
d@x but in addition there may be superposed the odd harmonics. 
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ss For the simplest vibration an originally vertical straight line 
: would be changed into a quarter of a sine eurve, as shown 
in fig. 25. Equation (6) is the expression used on page 39 to 
determine the free period of vibration of the strained rocks 
0 y near the fault-plane at the time of the earthquake. 
Fra, 25. Suppose, instead of vibrating freely, the base of the sand is 
stl fs : 
made to vibrate according to the expression B sin (re 7.e., With an amplitude B 
and a period P. 
