PROFILE OF THE SUBAQUEOUS SHORE TERRACE 21 
falling forward into the diminished trough in the act of break- 
ing; the trough is more than filled and the excess of water ini- 
tiates a wave of translation exactly as in Mr. Russell’s experi- 
ments. 
Volume of undertow.— It is not necessary to suppose that the 
loss of velocity of the undertow is as rapid as the increase of its 
cross-section. This would be the case if all the upper water 
moving shoreward should reach the shore before turning back. 
The volume of the undertow would then also be constant 
throughout its course and its velocity would be inversely as its 
cross-section. But evenif the loss of motion due to friction and 
interference of the bottom be ignored, not all the shoreward 
moving water reaches the shore. The on-shore motion causes 
elevation of level over a belt of considerable width. This broad 
elevation constitutes a “ead which is the cause of outward flow 
below. It may be shown that the average position at which 
incoming particles turn back and join the undertow, is at the 
center of mass of the head. This head is greatest at. the edge 
of the water, hence more water turns back at that point than at 
any other, but the undertow which has its beginning here is 
constantly being augmented by that which returns toward deeper 
water without reaching the shore. 
slope be gentle, the beach shallow and very extended (as it sometimes is for a mile 
inward from the breaking point, if the wave be large) the whole inner portion of the 
beach is covered with positive waves of the first order, from among which all waves 
of the second order have disappeared. This accounts for the phenomenon of breakers 
transporting shingle and wreck and other substances shoreward after a certain point ; 
at a great distance from shore or where the shores are steep and abrupt the wave is of 
the second order, and a body floating near the surface is alternately carried forward 
and: backward by the waves, neither is the water affected to a great depth; whereas, 
near the shore the whole action of the wave is inwards, and the force extends to the 
bottom of the water and stirs the shingle shoreward; hence the abruptness also of the 
shingle and sand near the margin of the shore where the breakers generally run. 
The residuary waves given off after breaking are wide asunder from 
each other, are wholly positive, and the spaces between them, several times greater 
than the amplitude of the waves, are perfectly flat and in this condition they 
extend over wide areas and travel to great distances. These residuary positive waves 
evidently prove the existence, and represent the amount, of the excess of the positive 
above the negative forces in the wind wave of the second order.—fefort on Waves, 
p. 292. 
