Chapter XVI 



Internal Waves 



1. Basic Facts and Theory of the Internal Waves 



In stratified water and in water in which the density varies with depth, waves 

 may occur which are of different type from those appearing on the free sur- 

 face and which have been discussed in chapters II to V. Because the waves 

 occur inside the water-masses they are called boundary or internal waves. 

 Their principal characteristic is that the largest vertical displacements of the 

 water particles are to be found at the boundary surface between different 

 strata or at some intermediate depth below the free surface. The amplitude 

 of these internal waves is usually considerably larger than that of the ordinary 

 waves at the free surface. The appearance of waves at the boundary surface 

 between two water layers has for a long time escaped the attention of ob- 

 servers, because even when the amplitude of the oscillation at the boundary 

 surface is large, the free surface of the upper layer is only slightly disturbed 

 and remains practically at rest. Only recently, thanks to more numerous 

 and accurate observations, has it become possible to study these internal 

 waves more thoroughly and to prove their importance in oceanography. 

 (a) Internal Waves at Boundary Surfaces 



Stokes (1847) was the first to develop the theory of internal waves at the 

 boundary surface in a fluid consisting of two layers of infinite thickness. 

 An excellent treatment of the problem can be found in Lamb (1932, p. 370). 

 If we put the origin of the co-ordinates in the undisturbed boundary surface, 

 we can write for the velocity potentials in the superposed layers the expressions 



<p = Ce* z cosxxe ia < , \ t 



y = C'e-* z cosxxe ia ' . J 



The accents relate to the upper fluid, and the z-axis is taken positive up- 

 wards. These expressions satisfy the equations of continuity (II. 1) for both 

 layers. The motions represented by them decrease rapidly with increasing 

 distance from the boundary surface and practically vanish at a great distance. 

 The equation of the boundary surface disturbed by the internal wave can 

 be written 



rj = acosxxe iat . (XVI. 2) 



