196 M. K. Zoppritz on some Hydrodynamic Problems 



The velocity in the surface becomes 



ph 



and, when h is very great, changes into wi itself. Accordingly 

 to can also be written 



li — x 



This simple equation solves the important question how the 

 velocity is distributed in a very extensive sheet of water of 

 uniform depth, which is in stationary motion under the influ- 

 ence of winds acting constantly on its surface. It shows that, 

 if the bottom stratum is at rest, motions in the direction of the 

 wind take place in the rest of the strata, with velocities that 

 increase proportionally to the elevation Qi—x) above the bot- 

 tom, from to the value w in the surface. The question put 

 by various authors, mostly geographers, To what depth does 

 the influence of the trade-wind reach? is therefore to be 

 answered thus : — So far as the motion of the ocean under that 

 influence may be regarded as stationary, it extends to the bot- 

 tom of the sea, and even the deeper strata obey it according 

 to the measure given by the above law, provided that no other 

 causes (e. g. displacement-currents) put these strata into other 

 motion. In the case of the deeper strata being kept by an 

 extraneous cause in motion in exactly the opposite direction 

 to that of the upper strata, there must exist between them a 

 plane where the velocity is = 0. If this be considered as the 

 lower boundary plane of the upper bed, and its distance from 

 the upper surface be denoted by h ly for the motion in the upper 

 mass the following equation holds good :— 



Ju — x 



w = w — = 



ri\ 



The distribution of the velocity in the upper mass is there- 

 fore the same as if the lower w^ere a solid mass. 



The velocity found for stationary motion is dependent on 

 the friction-coefficient k only through w (a dependence which 

 vanishes when h is veiy great) ; it will consequently diminish 

 with the depth according to the same law in a, very viscous 

 liquid as in a very mobile one. In stationary motion the in- 

 fluence of friction appears solely in the participation of all the 

 strata in the motion which is communicated from without to 

 the upper surface only. 



Dependence on the coefficient of friction first comes in w T ith 

 the consideration of periodically variable motions, and gives 

 (to use a favourite expression with many writers) a measure 



