Internal Waves 



529 



as if between fixed walls and the period is the same for all cells. This is the 

 most elementary case of standing cellular oscillations. 



The theory of such cellular waves was developed by Love (1891, p. 307, 

 see also Haurwitz, 1931; Brunside, 1889, p. 392; Lamb, 1932, p. 379; 

 or Prandtl, 1942, p. 331). We can find the equation for the frequency only 

 after introducing an assumption for the vertical stratification. The simplest 

 assumption is a water-mass of infinite thickness, in which at rest the density 

 decreases with z (when z is counted positive upwards starting from the 

 bottom) according to an e power, so that 



Qo = Q B e~^ =Q B e~ z!H , (XVI.22) 



where H is the height, where the density has decreased to the et\\ part of the 

 density at the bottom. 



Let us discuss a two-dimensional case where the horizontal and the vertical displacement of 

 the water particles are represented by I and >] respectively. Both are functions of x, z, and of the 

 time t. They must satisfy the equations of motion as well as the equation of continuity. The con- 

 dition of invariability of density has to be added: .' 



dq dg dg dg 



— = \-u \-w — =0, 



dt dt dx dz 



(XVI.23) 



where u and w signify the horizontal and vertical velocities of the particles. The density g and 

 the pressure p contain the components at rest g and p the components of the disturbance g t and p t . 

 The latter are like u and w quantities of second order, so that products and squares can be neglected. 

 (XVI.23) then becomes 



8qi 

 dt 



d Qo 



dz 



w or Oj = — rj 



dz 



(XVI.24) 



This equation signifies that a particle now at the time / at a location x, z was originally at the 

 level z—rj. With this density structure (XVI. 22) becomes 



Qi = —y • 



H 



The linearized equations of the problem are in this case 



S 2 | 8 Pl 



Qo — ~ H 



dt 2 dx 







Qo 



d 2 r] 8p x 



dt* 



dz 



+ -- +g(Q + Qi) = 0, \ 



(XVI.25) 



e>£ dri 



— + — = 0. 



dx dz 



Then, with the given density distribution, a possible wave solution of 

 cellular type will have the following form: 



Ae zl2H cosxx 



1 e . 



^-fT~ cosez sine: 



Ltix x 



cos at , 



rj = Ae zl2H sin xx cosez cos at 



(XVI. 26) 



34 



