568 Internal Waves 



extended over several hundred metres. Similar variations but not as large 

 were found in some cross-sections of the "Meteor" Expedition. The oc- 

 currence of such stationary wave-like variations of oceanographic properties 

 in straits was demonstrated by several longitudinal sections through these 

 straits. It was pointed out earlier and it can be proven that they are caused by 

 topographical features of the bottom. 



Let us assume two superposed water-masses of different density q' , h' and 

 q, h (density and thickness of the upper and lower layer respectively), where 

 both water-masses have the same velocity c. It can easily be shown theor- 

 etically that a disturbance of the bottom causes stationary wave-like dis- 

 placements, both at the surface of the upper water-mass (the free surface) 

 and at the boundary surface between the two water-masses. Under certain 

 circumstances the displacement of the boundary surface may become very 

 great, many times greater than those of the free surface. If one assumes 

 a simple wave-like profile with the wave length I = Injx it becomes evident 

 that the stream lines in both water-masses follow the same contour as the 

 bottom configuration and the boundary surface also takes part in these 

 stationary displacements. In the denominator of the equation for the am- 

 plitude of these wavy stream lines appears a value TV of the form: 



N ^^Icothxhcothxh' ^^\-c 2 (ianhxh-\-coihxh')^ + (l - -)- 9 . (XVI. 48) 



Q I X \ Q J X 



This expression N = is identical with equation (XVI. 9) and gives the 

 velocity c of free waves of the wave length X = 2n\x at the surface or at the 

 boundary surface in a system at rest (p. 519). Equation (XVI. 48) means 

 that the stationary displacements, caused by the bottom configuration can 

 become very large, if the velocity of the currents of the two water-masses 

 equals the wave velocity of the free waves at the surface or of the waves at 

 the boundary surface. Small irregularities in the bottom configuration are 

 thus, under certain conditions, accompanied by great, wave-like, stationary 

 displacements at the boundary surface. In the case that the length of the 

 bottom irregularity becomes great, equation (XVI. 48), N — gives the two 

 velocities for which this occurs. These velocities are identical with equa- 

 tions (XVI. 17). The first value c ± gives always very great values and cannot 

 be accepted as the velocity of a basic current. The second current velocity c 2 

 on the other hand has relatively small values. Here the stationary displace- 

 ments are greatest at the boundary surface, while the free surface remains 

 practically undisturbed. Thus for q— q' = 1 xlO -3 and if h' = 50 m and 

 h = 100 m. c 2 becomes 58 cm/sec. This is a reasonable value for basic currents. 



The current velocities of the two superposed water-masses are generally 

 different (in the upper water-mass c', in the lower one c). In that case par- 

 ticularly large stationary wave-like displacements of the boundary surface 

 will occur if the condition 



