616 REPORT—1890. 
The other actions are the bottom resistances and the viscosity of the 
water, which causes a definite change! in the internal motion of the water as 
the velocity falls below a point which is inversely proportional to the dimen- 
sions of the channel. 
That this last source of divergence from the simple kinetic law must 
make itself felt at some stage appeared to be certain. But the critical 
velocity at which the motion of the water changes from the ‘sinwous’ or 
eddying to the direct is inversely proportional to the depth, and by 
the kinetic law the homologous velocities in these experiments are pro- 
portional to the square roots of the depths only ; hence this action would 
seem to place a limit, if it were a limit, to the least tide at which the 
kinetic law would hold independently of the period, and this is not 
the case. Observation of the action of the water above and below the 
critical periods, however, confirmed the view that the limit was in some 
way determined by this critical condition of the water. For when water 
is running in an open channel above the critical velocity the eddies of 
which it is full create distortions in the evenness of the surface which 
distort the reflections, creating what is called swirl in the appearance of 
the surface. Now it was noticed and confirmed by careful observation 
that in the cases where similarity failed the swirl was absent at the 
commencement of the experiment, while it was easily apparent, par- 
ticularly on the ebb in the other experiments. Subsequently it appeared 
that the velocity of the water, particularly during the latter part of the 
ebb, which has great effect in the early stages, might be much affected 
by the bottom resistances, and hence not follow exactly the kinetic law. 
6. Theoretical Criterion of Similar Action.—The velocities of the 
water running uniformly in an open channel, 7 being the slope of the 
surface and m the hydraulic mean depth, is given by 
v=AVJ/in, 
where A is constant. 
If, then, 7 is proportional to e (the exaggeration of scale) and m pro- 
portional to h, since at the critical velocity y is inversely proportional to 
h, at this velocity he has a constant value. 
The function h’e=C is thus a criterion of the conditions under which 
similarity in the rate and manner of action of the water on the sand ceases. 
7. The Critical Values of the Criterion for Rectangular Tanks—Taking 
h to represent the rise of tide in feet, and e to be the vertical exaggeration 
as compared with a 30-foot natural tide by the simple hydrokinetic law, 
the values of this criterion have been calculated for each of the experi- 
ments and are given in Table I. 
Experiments I. and II., B, First Report, C=0:046, showed marked slug- 
gishness and local action; IV., B, C=0:058 and VIII., B, C=0-064, 
showed less, but still a certain amount of sluggishness and local 
action,? while in III, B, C=0-083, the rate of action was good and 
the action similar to the experiments with values for C higher than 
0:087,2 whence it would seem that the critical value of the criterion is 
about 0:087, and it may provisionally be assumed that C=0-09 indicated 
the limits of the conditions of similar action.” 
1 Reynolds on the Two Manners of Motion of Water, Phil. Trans. 1883, pt. iii. 
2 In both these experiments, IV. and VIII., B, the mean level of the tide was 
above the initial level of the sand, which would naturally increase the value of the 
criterior, 
