PROLMEL OPM SO BAVTHOSUS SHORE, THRRACE, | it 
differential movement of adjacent water particles. The laws of 
fluids require that this differential movement be distributed 
throughout the series of particles reaching to the surface, though 
experienced to a less degree as the distance from the bottoin 
increases. It has been shown that increased differential move- 
ment implies decreased wave-length. This shortening, accom- 
panied by steepening, may or may not be sufficient to cause 
breaking. Since these effects are greater at the bottom than at 
the surface, the lines of like phase will incline forward. These 
effects —the increased differential movement, the shortening 
waves, and the forward inclination of lines of like phase —fol- 
low from friction on the bottom, but it is immaterial whether 
this friction be that of the forward-moving or that of the back- 
ward-moving water. The forward inclination of lines of like 
phase indicates nothing as to the movement of the water asa 
body. The inclination of these lines may be arrived at in 
another way. The retarded particles below may be thought of 
as having a decreased angular velocity, and hence a less advanced 
phase than the upper particles in the same vertical line. This 
would require that lines of like phase should connect them with 
upper particles in advance of them in the direction of wave 
movement. 
Comparison of friction in forward and backward movement.— 
Looked at in cross-section, the area of the backward-moving 
water above the line AS (Fig.g) is less than the area of for- 
ward-moving water. The areas are equal when bounded below 
by one of the trochoid curves. The area of backward-moving 
water is made smaller by the substitution of a rigid plane for the 
depressed part of the trochoid, and that of the forward-moving 
water is made larger by the substitution of a flat bottom for the 
curve bulging upward. ‘This constriction and consequent greater 
friction of the backward-moving water makes itself felt in the 
form of the wave and in the bodily movement of the water. 
_ Asymmetrical form.— The velocity of propagation of wave 
crests depends purely upon the behavior of particles in the 
upper halves of their orbits, while the propagation of troughs 
