PAPER BY PROF. HELMHOLTZ. 127^ 



Hence it results that when waves of considerable progressive veloc- 

 ity trench upon quiet deep water the generally very small differences 

 [E^—JUJ-z) lose their negative and assume a positive value. 



Here also the energy that is given to the previously quiet water in 

 the form of an elevation of its surface and the living force of its mo- 

 tion must be abstracted from the atmosphere. In order to obtain a 

 sufficient amount for the formation of large waves, it will on this ac- 

 count be necessary that long layers of air shall blow over and shall 

 give up a part of their living force. 



In the first momeut when a new gust strikes the surface of the water 

 stationary waves only can be formed for which 21=0 and Ei—Ei=0 

 and a-i has the value given in equation (5a). The last condition shows 

 that these waves will be near the point of s]iirting, as we in fact often 

 see in the case of small ripples suddenly excited on the surface of the 

 water. Moreover in these small ripples, as Sir William Thomson has 

 shown, the capillary tension of the liquid comes into consideration, 

 which somewhat increases the store of energy of the billowy surface. 



In general therefore, stationary waves are not formed immediately 

 at the beginning, since the waves of constant momentum would leave 

 behind an excess of energy. But when from the very beginning waves 

 that have partly a positive and partly a negative difference of momen- 

 tum and of energy are successively produced on the quiet water, then 

 the sum of these differences can become zero. These systems of waves, 

 having different wave-lengths and progressive velocities, cause mani- 

 fold interferences as they progress, and, according to the principle given 

 by me for combination-tones (which in its application to the tidal wave 

 has already received a very beautiful confirmation by Sir William 

 Thomson's analysis of the tidal observations collected by the British 

 Association), waves of greater wave-length can gradually be formed. 



So long as the wind outruns the waves it steadily increases the store 

 of energy and the momentum of the waves, and furthermore, so long- 

 as the energies computed for stationary waves diminish and can form 

 a still lower minimum, the inclination to attain the form of least en- 

 ergy under the cooperation of all the small perturbations which the 

 other concurrent waves bring about, in the case of nature, will develop 

 still further. This will finally lead to the value corresponding to the 

 i'ormation of a cusp and to the foaming of the upper ridge in case this 

 can be produced by the given wind velocity. 



In April of this year [1S90] I endeavored by observations that I in- 

 stituted at the Cape of Antibes [near Marseilles] to arrive at some con- 

 firmation of these consequences drawn from theor3^ With a small port- 

 able anemometer I lueasured the strength of the wind directly at the 

 edge of the steep cliff" of the narrow tongue of land which projects 

 rather far into the sea. However, the observations showed that many 

 times a stronger wind must have prevailed out on the sea than I had 

 been able to observe on shore. I also counted the number of approach- 

 ing billows. 



