222 Long Waves in Canals and Standing Waves in Closed Basins 



and time, as well as the initial state of the surface. The process consists in 

 oscillations around a state of equilibrium, and it is only in taking into con- 

 sideration these periodical changes in water level that many phenomena related 

 to these disturbances of the normal position of the surface can be correctly 

 understood. 



Of the theoretical investigations dealing with the oscillations produced 

 by variations in atmospheric pressure and wind in finite watermasses, the 

 following should be particularly mentioned. 



Chrystal (1909, p. 455) has given considerable attention to the influence 

 of different atmospheric pressure disturbances in developing seiches in a lake 

 with a parabolical normal curve, neglecting friction and the rotation of the 

 earth. Proudman and Doodson (1924, p. 140) have, on the contrary, 

 examined the oscillations produced by atmospheric pressure and wind in 

 a basin of constant rectangular cross-section, taking friction into account but 

 neglecting the rotation of the earth. Later on, Proudman (1929) dealt with 

 special cases, neglecting the friction, in which the influence of the earth's 

 rotation is particularly apparent. We are indebted to Stenij (1932) for an 

 extensive review, in which the problems are treated on a strictly mathematical 

 basis. We will only mention the essential features of these papers, especially 

 stressing the fundamental viewpoints and limiting ourselves to basins of 

 constant rectangular cross-section. 



The problem in its most general form is : the general equations of motion 

 and continuity have, with depth of the water h constant, and using the usual 

 symbols, the form 



8u r 8 , 8 2 u 



dv 8 8 2 v 



e7 +> = -gfy<n-v)+ v a?> (vi. 130) 



in which /= 2cos'm(p, v is the kinematic viscosity coefficient assumed to be 

 constant and —gQf} the variable part of the atmospheric pressure. If / = the 

 length of the basin and the origin of the co-ordinate system is taken in the 

 undisturbed water surface, we have to add as boundary conditions u = 

 for x = and x = I and for z = — h ; further an assumption for 8u/8z, for z = 

 to determine the influence of wind on the surface. Solutions are sought 

 for a given disturbance in the atmospheric pressure and for a given wind. 

 If the distribution of the atmospheric pressure is stationary the system of 

 equations is satisfied by the "equilibrium solution" 



rj = rj + constant , u = v = . (VI. 131) 



