Internal Waves 565 



But there are also cases, where these conditions are not sufficient for the 

 explanation of the internal breakers. The circumstances under which internal 

 waves occur, permit the development of dynamic instability. The discussion 

 of the following simple case will be sufficient. Two layers of great thickness 

 are superposed. Let u be the velocity of the upper, lighter water-mass, 

 u the velocity of the lower, heavier one. The theory shows (see Lamb, 1932, 

 p. 373; V. Bjerknes, 1933. p. 381; Hoiland, 1943), that in this case the 

 velocity of internal waves at the discontinuity surface is given by the ex- 

 pression 



qu+q'u .. 

 c = - — rV ±1 



g p— P , lu —u 



QQ 



(XVI. 44) 

 * 6+ Q \Q+ 9 I . 



The first term on the right-hand side represents the convective velocity 

 and may be called the mean velocity of the two currents. Relatively to this 

 the velocity of waves of the length A = (2tz/x) is represented by the term 

 on the right-hand side under the square root. The first term under the root 

 represents the velocity of progress of the internal waves in the system at 

 rest (p. 518 (XVI. 7)), i.e. the velocity of progress of pure gravity waves. 

 The second term under the root with the negative sign is the velocity of pure 

 inertia waves. The term for the gravity waves is always positive, if q > q', 

 which is of course always the case. Pure gravity waves are always stable. 

 The inertia term, on the other hand, is always negative. Thus it has always 

 an unstable effect and therefore weakens the static stability of the gravity 

 waves. This weakening may become so effective as to produce dynamic 

 instability. This happens if 



(u'~ uf > £ tn^l , (XVI.45) 



X QQ 



i.e. if for a given wave length and discontinuity of density, the difference 

 in velocities on the discontinuity surfaces become great enough. 



In the cases under consideration the thickness of the two superposed water 

 layers are small. This causes a further decrease of the term at the right side 

 of the inequality (XVI.45). Let h and h' be the thickness of the two layers. 

 Then we obtain as inequalities 



(u'-uf >-{q-q) 



1 + ' 



(XVI. 46) 



x iQCOiYixh cothx/?' 



If we simplify the case by assuming h = h\ and if we consider, that xh is 

 small, we can write with sufficient accuracy 



1 



coth^/z 

 Then the inequality becomes 



tanhx/j = xh 



«>y(«* £ 



qq' 



