534 



Internal Waves 



to the direction of progress of the wave. The other equations remain un- 

 changed. The solutions become more complicated, because there exists now 

 also a variation in the amplitude of the wave cross to the direction of progress. 

 Three different types of waves are to be considered: 



(a) In a narrow channel Kelvin waves are possible (see p. 206): 



r\ = ?](z)e- {2oixla)y cos (at— xx) , 



-H e -(,^xlo)y C os(at—xx), and v = , 



u = 



dz 



(XVI. 32) 



with the phase velocity c = ajx. This solution is the same as the one in the 

 case without rotation of the earth, but the factor e"®"" 1 ** is being added to 

 the amplitude in each expression. This factor has here a greater significance 

 than in the case of the regular tide waves, as the values of x become larger 

 for internal waves. With ajx = 200 and 2w = 1-3 x 10~ 4 the amplitude per- 

 pendicular to the direction of progress of the wave decreases at a distance 

 of 15-3 km to j/e part of its value. Internal waves of this type can gain im- 

 portance in comparatively narrow channels only. 



(b) If the lateral boundaries are neglected one obtains: 



rj = rj (z) cos (at— xx) , 



a drj , v 



u = — f- cos(at — xx) , 

 x dz 



2co dn . , „ v 



v = r- sm(at — xx) , 



x dz 



(XVI.33) 



where 



c = 



>/(l-4coV) 



when c represents the velocity in the case of a non-rotating earth. 



(c) In a wide channel a wave of the following form is possible : if we assume 



x 2 + 



m c ii 



2tt2 



e 6 = 



! -4w 2 



Then 



. mn 

 sm-r-y- 



amn mn 

 zcoxb b 



a . mn 1 mn mn 

 -sm-yy- -y 



7] (z) cos (at— xx) , 

 dr\ 



£ 2 2cob 



dz 



cos(at—xx) , 



4eoV + o a 



9 9 



-TT- 



V = 



2(ox I x 2 + 



b 2 . mn dr\ . , ^ . 

 sin —j- y -/ sin (at — xx) , 

 m*n*\ b dz 



(XVI. 34) 



