Internal Waves 



569 



(*h' + c'*h = I 1 " -) Shh' (XVI. 49) 



is satisfied. 



It thus appears as if the frequently observed bulges in the sharp 

 discontinuity layer between the compensating superposed currents, flowing 

 in opposite directions in straits (for instance Bosporus, Dardanelles, Strait 

 of Gibraltar and others), can be explained in this way as a result of the bot- 

 tom configuration (see Defant, 1929, p. 51). 



In the case of currents in a water-mass where the density increase is con- 

 tinuous, stationary wave-like displacements of the isopycnals are also possible. 

 The stream lines will follow then the isopycnals. If a system of progressive 

 internal cellular waves (as discussed on p. 528) with a velocity of progress c 

 (equation (XVI. 29)) is superposed by a current with a velocity U = —c, the 

 waves become stationary. This is the case of a stratified water-mass flowing 

 with wavy stream lines. For a certain ratio e/x = XJX Z the wave length A of the 

 stationary waves can be computed, if U is known. Since the velocities in 

 the ocean are small, one obtains with sufficient accuracy 



L = 



In 



2nU 



V 



// 



2nU 



A 



k('*i 



(XVI. 50) 



With e/x = 10 2 and g/r = 10 6 one obtains for U = 2 m/sec, a horizontal 

 wave length of 40 km, where the height of the stationary waves will be 400 m 

 (compare Fig. 238). These are conditions which may occur in nature. These 



Fig. 238. Stationary waves in a stratified water mass. 



waves are of course free waves, which may appear because of a single impulse 

 and then they vanish because of frictional influences. The case is different when 

 external conditions (pressure disturbances on the surface, irregularities of the 

 bottom configuration) cause internal waves of cellular type. The phenomenon 

 of the so-called lee-waves belongs to this type. They occur behind a long out- 

 stretched bottom irregularity over which a horizontal current is flowing in 

 a transverse direction. Lord Kelvin (Lamb, 1932, § 246, p. 409) developed 

 the theory of these stationary lee-waves for an incompressible and homo- 

 genous heavy fluid. The examination of the lee-wave disturbances in a 



