NO. IB.] ON WAVES BETWEEN SALT AND FRESH WATER. 43 



In all cases actually concerning us, the difference Jq between the den- 

 sities of the two water-layers, is very small, the specific gravity of sea-water 

 being always less than 103. In what follows this should always be assumed 

 to be the case. Tlie surface-waves are then, according to equations (1) and 

 (2), very low, compared with the boundary-waves by which they are caused. 

 In Fig. 4, PI. VI, the vertical scale of the surface-waves is exaggerated about 

 30 times, if we assume the water-layers to be made up of fresh-water and 

 pure sea-water of spec, gravity 103. 



The most peculiar property of the boundary-waves, in contradistinction 

 to ordinary waves on the surface of homogenous water, is their very slow 

 velocity (Jq assumed to be small). With moderate wave-lengths it varies as 

 the square root of Jq and stands to the velocity of equally long waves in 

 homogenous water, as ]/jq : ^2q \ As the wave-length increases, the velo- 

 city approaches the value 



--1% 



q VcZ+VD' 



which is its maximum value ; or, if D be great compared to d, 



Vm = igd, X l/jq/q 1 (4) 



This formula shows an obvious analogy between boundary-waves and waves 

 in shallow homogenous water, the maximum velocity of the latter being 

 yfgd. There are indeed several such analogies, as will be shown below. 



The average energy per unit area of the wave-systems considered, is 

 also small, being proportional to the difference of density Jq, in cases in 

 which the heights of the waves are the same. It is exactly twice the potential 

 energy of the waves and consequently varies, as in the case of ordinary 

 surface-waves, as the square of the wave-amplitude. 



Just as in the case of waves in shallow water, the ratio of transmis- 

 sion of wave-energy (see p. 36) is Va for short waves and increases asymp- 

 totically to 1, when the wave-length increases. In Fig. 1, PI. VI, it is repre- 

 sented as a function of the ratio v/vm of the wave-velocity v and the maxi- 

 mum velocity Vm of long waves in the same water. The ratio of transmis- 



1 This formula, as well as (3) and (4), are exact under the supposition that Jq is infi- 

 nitely small; if Jqjq is small, they are approximately true. 



