204 M. K. Zoppritz on some Hydrodynamic Problems 



k contains as unit the square centimetre ; therefore x must also 

 be expressed in centimetres, and be put = 10000. We thus get 



5000 

 s/t ~ 0-48. 0-12 ' 



which gives the value of t in seconds, and we find 



£=7,537,000,000 seconds = 239 years; 



so that, if the particles of the surface of an ocean of very great 

 (properly infinite) depth, at rest, begins at a point of time 

 t = to move forward with a constant velocity, half the velo- 

 city of the surface will first prevail at 100 metres depth after 

 239 years. 



If we inquire, After how long a time has a tenth part of the 

 surface-velocity penetrated to 100 metres depth ? we find it is 

 41 years. According to the principle expressed in equation 

 (15), at 10 metres depth the same velocities will prevail at the 

 end of 2*39 years and 0*41 year respectively. 



These numbers are very appropriate for giving an idea of 

 the slowness with which changes of velocity are propagated 

 downward ; for what has just been calculated for the propa- 

 gation of a commencing surface-movement in a liquid at rest 

 is just as valid for every change of motion in a moving liquid, 

 since the velocity already present and that which has newly 

 entered are algebraically added together. A stationary cur- 

 rent diminishing in velocity linearly with the depth w T ill on 

 this account be only extremely little altered by passing changes 

 of velocity affecting its surface (as, for example, by head- 

 winds or storms), except in the strata nearest to the surface.. 

 There will much rather prevail at every deeper-situated point 

 a mean velocity but very slightly variable with the time, and 

 determined by the mean velocity at the surface. The latter 

 velocity has the direction of the prevailing winds, and falls and 

 rises with them according to a law which cannot be more pre- 

 cisely determined. This fact of the limitation of considerable 

 action of transitory causes to the strata in the vicinity of the 

 surface is an additional justification of the assumption, made 

 to facilitate the calculation, that the ocean is of infinite depth. 



If the velocity of the surface is a periodic function of the 

 time, as are all winds dependent on the seasons of the year 

 and hours of the day, it must be possible to express them by 

 a finite series of the form 



< ^(0 = / 3 o + PiCos ( xA +/5 2 COS I — X 2 ) + 



As has been already shown in the preceding more compli- 



