12 N. M. FENNEMAN 
takes account of the lower halves only. It results from a 
greater orbital velocity in the upper halves that crests are propa- 
gated more rapidly than troughs.t The necessary accompani- 
ment of this is the asymmetrical form, steeper in front than 
behind. 
Resulting currents—The constriction of backward-moving 
water mentioned above may be compensated either by greater 
velocity or by broadening the area of backward flow. Upon 
either of these assumptions, or upon the assumption of no com- 
pensation, certain conclusions follow from a geometrical inspec- 
tion of the diagram, and these conclusions agree with observed 
phenomena. 
Assume first that the deficiency in backward movement is 
uncompensated. This assumption involves an excess of forward 
movement which would be observed as a current, a well-known 
phenomenon where waves enter shallow water. On this same 
supposition of no compensation the area of the bottom covered 
by the backward-moving water is greater than that covered by 
the forward-moving water, and the velocity of that moving back- 
ward on the bottom is greater than of that moving forward at 
the same depth, because the former, being under the trough, is 
nearer to the surface. A current of this type would therefore be 
distinctly a surface feature which would not wash the bottom in 
the direction of its flow. It would, in fact, involve a certain 
amount of counter-current at the bottom, independent of any of 
the conditions which give rise to undertow. 
Assume next that the deficiency of area of backward mov- 
ing water is compensated in one of the ways above mentioned, 
either by greater velocity or by broadening the area. In either 
of these cases the backward movement on the bottom will be in 
excess, and will suffer more interference by friction than the 
forward movement will. This greater interference with the 
backward movement will favor, with each oscillation, a residual 
advance of the water as a whole, causing a progression in the 
™Compare also C. S. LyMAn, “A New Form of Wave Apparatus,” Journal of the 
Franklin Institute, Vol. UXXXVI, p. 187. 
