PAPER BY PROF. HELMHOLTZ. 125 



then equation {5a) becomes 



S2 r<i w 



— a\ 



?, rx + s-i r-i 



Si Ti tC 



= a2. 



Si n 4- 6'2 r-2 



Siuce now Vi and r-z nave valnes that differ but little for the ordinary 

 ■waves (as the subsequent computations will show), and since for air and 

 water 



S2 773.4' • 



therefore this condition gives the rate of propagation of the wave 

 against the water as approximately 



For waves of low altitude equation I, Section vii of my paper of the 

 previous year,* neglecting the small quantities z and r, becomes 



> , o g-^ {s-i — si) 



Sitti'--^ .S2 a-i- = \, 



If we put to=10 metres which corresponds to a rather strong wind, 

 then for low waves of a constant moment of motion, we have 



ai = 9".98709 

 a2 = 0™.0]291 



A = 0°i.082782 



These waves of only 8 centimeters in length evidently can corre- 

 spond only to the first crumpling of the surface, such as a strong wind 

 striking upon it immediately excites. Only when the same wind blows 

 for a long time over these initial waves, and gives them a part of the 

 moment of motion of a long stretch of air, can waves be thereby pro- 

 duced with greater velocities of propagation. 



Hence in accordance with experience it follows that wind of a uni- 

 form strength striking a quiet surface of water can only produce more 

 rapidly running waves, namely, those that are longer and higher, when 

 it has acted for a long time on the waves that first arose, and has 

 accompanied these for a long distance over the surface of the water. 



At the same time it also becomes clear that for a uniform wind the 

 waves can only increase in size when the wind advances faster in the 

 same direction than the waves themselves. 



Energy of progressive waves on quiet water. — As in the case of the 

 moment of motion, so also with the storage of energy in the wave. 

 Our previous comparisons of the energy of different waves among 

 themselves has reference to the energy of relative motion of the fluid 

 with reference to the stationary wave. 



* [See page 107 of this collection of Translations.] 



