Ratio of the Length and Height of Sea Waves. 113 



spot without interfering with the logical sequence of the 

 argument, inasmuch as it thus represents the same wave, 

 filled by the same instead of by changed particles of the 

 liquid to which its embodiment has been transferred. 



Let, therefore (Fig. 4), a, h, c, d, e represent the profile of 

 a wave from trough to trough, the dotted line /, g being the 

 mean or smooth water level. So far as the subsidence is 

 concerned we may wholly disregard the actual movements 

 of the particles, and conceive an indefinitely thin layer of 

 the liquid to be instantaneously fixed or congealed in the 

 shape of the wave a, h, c, d, e. Here i, c is the height, and 

 a, i, e the length of the wave. 



It will be seen that the area 6, c, d, h is that portion of 

 the liquid which has been raised above the mean level of 

 the ocean ; while the areas d, g, e and h, a, f are that of the 

 water which has been thereby depressed below the mean 

 level; whence the area 6, c, <5, h above the mean level is 

 equal to the sum of the areas d, g, e, and h, /, a below the 

 mean level, since the filling of the lower areas by the upper 

 would render the surface flat. 



Conceive now that the rigidity is slackened, so that the 

 ideal lamina becomes semi- viscous. The onward velocity of 

 a wave keeps it from sinking suddenly, as does that of a 

 hoop or a top ; its decline, therefore, is not due to its onward 

 velocity, and the slow sinking of a semi- viscous fluid may 

 justly represent the process of its actual subsidence. 



Taking this view then to be correct, we may^ under such 

 an assumption, consider the wave as wholly divested, not 

 only of any onward motion, but also of any rotatory move- 

 ment of the particles. This is nothing more than conceiving 

 the form of the wave to be embodied of the same particles 

 instead of successive ones. 



If the sinking of the upper area merely fiUed up the lower 

 areas, the length of the wave would still remain the same, 

 viz., a, i, e; but observation shows that the length absolutely 

 increases. Let the height of the wave have subsided to 

 c\ then instead of the profile being the curve f, h", c' d", e"^ 

 which it would be if the length remained unchanged, it is 

 represented by the curve a', h', c', d' e', whose length (the 

 dotted line e', i, a') exceeds e, i, a ; i\ d now represents the 

 height of the declining form. 



Now, in order to simplify matters, we may — the two 

 halves of the wave being symmetrical—treat only of the 



k2 



