PAPER BY PROF. HELMHOLTZ. 109 



The above given formulae show that when eos £ diminishes from its 

 upper to its lower value, then both D and $ and C 2 must continually 

 increase. For waves whose lengths remain constant the increase of $ 

 and D means an increase of the two velocities 6, and b 2 as well as their 

 sum, I e., the wind velocity w=b x +b 2 . If the latter remains constant, 

 then the wave length must necessarily diminish with increasing cos e. 



It follows from this, that within certain limits the same wind can 

 excite this form of waves of greater and smaller wave lengths. The 

 longer waves will at the same time have a relatively greater altitude. 

 This relation depends upon the store of energy that is accumulated in 

 the wave. 



VIII. THE ENERGY OF THE WAVES. 



When we investigate the energy of the waves of water raised by the 

 influence of the wind, and compare it with that which would be ap- 

 propriate to the two fluids uniformly flowing with the same velocity 

 when the boundary surface is a plane, we find that a large number of 

 the possible forms of stationary wave motion demand a smaller storage 

 of energy thau the corresponding current with a plane boundary. 

 Hence the current with a plane boundary surface plays the part of a 

 condition of unstable equilibrium to the above-described wave motion. 

 Besides these, there are other forms of stationary wave motion where 

 the store of energy for both the masses that are in undulating motion 

 is the same, as in the case of currents of equal strength with plane 

 bounding surfaces; and finally, there are those in which the energy of 

 the wave is the greater. 



The reason for this is to be found in the following circumstances: 

 In the undulating masses of water two forms of energy occur, namely: 



First, potential energy, represented by the water raised from the wave 

 valley to the wave summit. This quantity of work increases with the 

 increasing height of the wave, and must always be positive; it is only 

 absent for perfectly smooth surfaces. 



Second, living force is common to the two forms of motion under com- 

 parison, and according to the original assumption there is an equal 

 quantity of it in the portions of the fluid masses distant from the 

 boundary surface. The difference of the two modes of motion is not 

 affected by the participation of the more distant strata of fluid, the 

 difference between the two motions depends only on the strata that lie 

 near the boundary surface. The wave surface which we again imagiue 

 to ourselves fixed in space affords to the two fluids streaming along it 

 an alternately broad and narrow channel ; where the bed is broader the 

 fluid moves more slowly, the upper fluid above the wave valley, the 

 lower fluid under the wave summit. Thereby the living force of the 

 portion flowing through a broadening of the channel will be alternately 

 smaller, while that flowing through a narrowing of the channel will be 

 greater thau the living force in the corresponding part of the uniform 



