THE INFLUENCE OF THE FOUNDATION ON THE APPARENT INTENSITY. 51 
The solution of equation (1) under these conditions is 
a B PNP a on goo 
YS waa igp Cote (w HT) sin ¢ (T) 
cos 2 7 — 
Xr 
where X, the length of a distortional wave of period P in the sand, supposed 
of indefinite extent, equals py. Equation (7) shows that a vertical 
straight line in the sand is distorted into a cosine curve with its maximum 
amplitude at the surface. Fig. 26 shows the form of this curve; S is the 
surface and only that part of the curve is followed which lies between S 
and the bottom at the distance H below it. At the surface x = H, since 
is measured from the bottom, and the amplitude becomes 
B 
cos 2 wit (8) 
xr 

Fig. 26. 
and this varies between B and infinity, according to the value of the ratio of = If H 
; : r : 
is any even number of times ry the denominator becomes 1 and the amplitude becomes 
B. If H is any odd number of times a the denominator becomes 0 and the amplitude 
infinite. If, instead of varying the depth, we suppose it constant and vary the period 
of the disturbance, we get similar results. Replace » in (8) by its value and the surface 
amplitude becomes ; 

B 
“eres (9) 
fete 
P 
The free periods of the system are given by equation (3), and if P has one of the values 
there given, the denominator of (9) becomes the cosine of an odd number of times ue 
which is 0, and the amplitude becomes infinite. Practically, of course, friction or a slip- 
ping of the sand particles would prevent the amplitude from becoming extraordinarily 
large. 
We see, therefore, from (7), (8), and (9) that the surface would vibrate with the same 
period as the base and that it would always be in the same or in the opposite phase; that 
its amplitude would never be less than that of the base and that it would in general be 
larger and might become indefinitely large when the depth of the sand is an odd number 
of times a quarter wave-length, a wave-length being determined by the density and 
rigidity of the mass and the period of vibration; or what amounts to the same thing, 
when the period of vibration becomes equal to one of the free periods of the system. If, 
however, the frequency of the vibration should be too great, the bond between the differ- 
ent grains of sand would be broken, and the conditions upon which the above conclusions 
are based would no longer hold; the sand grains would slip over each other, and the 
amplitude of the upper surface would be diminished. If we apply the above theory to 
Mr. Rogers’s experiments with dry sand, we find it in close agreement with his results. 
When the frequency is very low, the sand moves with the box; in terms of the theory, 
we are dealing with only the upper part of the curve in fig. 26, and since H is very small 
in comparison with the surface amplitude as given by (8) becomes B, the amplitude of 
