COMMISSION GEOLOGIOUE - Arkisto.gsf.fi

106 Bulletin de la Commission geologique de Finlande N : 0 212.
itself) varies with time. Indeed, eonsidering v as a funetion of the velo city
gradient, the equation (9) will be rather complicated. To simplify the solution
we assume v to be constant. This assumption influences our results
somewhat, but the general features should not change appreciably.
For constant v the solution of (9) reads
v = ar exp{-Val(2v)z} cos[a t- Val (2v)z].
( 10)
This solution satisfies the boundary eonditions
1° x = r sin (at), when z = 0,
2° v approaches zero when z approaches + infinity.
The distance of a particle from equilibrium is found from (10)
x = r exp [Va I (2v")zJ sin at -
The aeceleration of a particle is
Va I (2;)z].
( 11)
a = -a 2 r exp [Val (2v)zJ sin[at -
Val (2v)z).
( 12)
Formulas (10), (11), and (12) indicate that the particles are in harmonie
motion. The amplitude
r exp [Val(2v)z ]
decreases upwards. The phase of the motion is retarded. The phase shift
is Val(2v)z. The energy in the sand will be transferred from lower layers
upward as a damped transversal wave motion.
In solving equation (9), we assume that v is a eonstant. The solution
for v varying with z ean be found, for example, in dividing the sand mass
into a eertain number of separate layers where v can be assumed to be eonstant
and combining the different results as an entity.
Friction eertainly has a marked effeet on our solution. At the very
moment ovloz beeomes zero, the adjaeent layers begin to move as a rigid
body until the forces opposing the friction inerease enough to eause relative
movements again.
HORIZONTAL SHAKING WITH PEBBLES IN SAND
For the sake of simplieity it is assumed that the pebbles are spherical
and that they are embedded in the sand so far apart that their mutual
influenees ean be neglected. Let us consider the motion of a single pebble.
It is mainly linear and rotational to-and-fro motion eaused by frictional
forees.