Internal Waves 561 



pagation c = a/x is determined by the force. In the equations for the am- 

 plitudes of the surface and of the boundary layer, the expression 



c(c- DiX(c- Ujfaat+lc- U0*qM-[(c- Vtfw, + (c- U^w 2Qi /q 2 



+ c(c-U x )<» x \Q]-g/*+4Qg 2 /x 2 = A (XVI.43a) 



always appears as denominator, wherein co x = cothxh x and co 2 = cothx/i 2 . 

 But this expression, when set equal to zero, is the equation which determines 

 the velocity of the free waves of the system. For U x = U 2 = this equation 

 becomes the equation of the velocity of free waves with the wave number x: 



c\co x co, + qJqJ - c 2 ((o x + co 2 )g/x +Agg 2 /x 2 = D 2 = . (XVI. 436) 

 The two solutions of this equation for long waves are 



Ci~- [g(h x + hj\ and j |^ ' h x h 2 l{h x + h 2 ) 



i/f* 



(XVI. 43c) 



The velocity e x is valid for waves at the surface, which are associated with 

 displacements of the boundary layer of smaller amplitudes than those at the 

 surface. c 2 is the velocity of free internal waves. In the case of forced waves 

 the case of resonance corresponds to D 2 = 0. Theoretically in this case the 

 amplitude of forced waves will be infinitely large. Note that the internal waves 

 will be infinitely large if the condition c = c 2 is exactly fulfilled. But this never 

 occurs, since the magnitude of e for tidal waves in the oceans is about 

 200 m/sec and that of c 2 only 2 m/sec. 



Taking into account the] Coriolis force, one obtains in the place of the 

 former (XVI. 436) denominator the expression: 



e 4 - (/?! + h,)ge 2 +zJ gg%/h 2 = D z (XVI. 43d) 



where: a 2 —f 2 /x 2 = e 2 . 



It has the same form as the relation (XVI. 436), transformed for tidal 

 waves, for the velocity of forced waves c, the influence of the earth's rotation 

 is entered through the quantity e . Whereas c has the magnitude of 200 m/sec 

 and the quantity D 2 is a large number, under certain conditions e can be 

 small or nearly equal to the velocity of free internal waves. With these values, 

 however, D 3 will be zero. If T t is the period of internal waves (equal to 12 

 pendulum hours), then 



£ 2 = c 2 [\- (T/Ti) 2 ] . (XVI. 43?) 



For the diurnal tide wave, T = 24 h and £ will be zero at 30° latitude, but 

 for the semi-diurnal lunar wave, T = 12 43 h and e will be zero at about 

 74° latitude. For small e equations (XVI. 43d) has, in the first approximation , 

 the form: 



D A =A g g 2 h x ho-g(h t + h 2 )e 2 . 

 Di will be zero for 



e 2 = {A Q lQ)h x h 2 l(h x + h 2 ) . (XVI. 43/) 



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