ON THE ACTION OF WAVES AND CURRENTS. 515 
would present a general similarity amongst themselves, or even similarity 
under particular relations among the conditions, were still open questions. 
The experiments, as shown in Table I., Table IJ., made this year 
emphatically confirm the conclusions (1) as to the existence for each rise 
of tide of a critical period at which the rate and manner of action begin 
to change, being similar for all smaller periods; (2) these experiments 
also confirm the general similarity of the final states of equilibrium as 
regards slopes for periods smaller than the critical period, as shown in 
Table II. 
The experiments (Experiments IV. and VIII., B) this year also show 
that with tides 0:094 and 0:097 foot the periods 34°4 and 35°4 seconds are 
greater than the critical periods, although the results show a nearer 
approach to similarity as regards manner and rate of action than the 
results obtained last year in II., B, with the tide ‘088 foot and period 35:4 
seconds, while the final conditions of similarity were approximately reached. 
With tides 0:088 foot and periods 69°3 seconds the results in rate and 
manner of action are emphatically different from those with less than the 
critical period, and with tides of 0°042 foot and periods 50°5 seconds still 
greater differences are presented. 
On the other hand, it is found with (V., B) tides 0:042 foot and 
periods 50°5 seconds that if the sand be given a condition correspond- 
ing with the condition of final equilibrium, as if the period were above 
the critical period according to the simple hydrokinetic law, this is a 
state of equilibrium; and, further, that it is not a state of indifference is 
shown, since on diminishing the period the sand readily shifted so as to 
bring it nearer the theoretical slope for the new period. This shows 
that the state of equilibrium follows the simple hydrokinetic law for 
periods greater as well as less than the critical period, which is thus 
shown to be critical only as regards rate and manner of action in 
reducing the sand from the initial level state to the final condition. 
The experiments carefully considered suggest that there is some 
relation between the rise of tide and critical period. They do not, how- 
ever, cover sufficient range to indicate what this relation is with any 
exactness. The critical period diminishes with the rise of tide, but much 
faster than the simple ratio. 
5. Causes of the Change in Manner and Rate of Action.—The change 
in the action which sets in at the critical period is the result of some 
action, of which no account is taken in the simple hydrokinetic law. 
A list of five such sources of possible divergence from the hydrokinetic 
law is included in last year’s report (p. 339), and with a view to obtain 
an indication of some relation between the rise of tide and period (or 
vertical exaggeration, as compared with the standard tide of 30 feet, by 
the kinetic law), which relation would be a criterion of the limiting 
conditions under which the simple kinetic law may be taken as approxi- 
mately accurate, these five discarded actions were carefully considered. 
The fouling of the sand by the water, although it comes in as pre- 
venting further action, cannot take any part in imposing these limits, 
since it is at the immediate starting of the experiments that the actions 
observed to fail. For the same reason the limits cannot be in any way 
due to the drainage from the banks, as these banks have not appeared 
above water. 
Again the limit cannot be due to the size of the grains of sand because 
it would then occur at particular velocities, whereas this is not the case. 
LL2 
