CHAPTER VIII. 



TIDAL WAVES. 



165. ONE of the most interesting and successful applications 

 of hydrodynamical theory is to the small oscillations, under gravity, 

 of a liquid having a free surface. In certain cases, which are 

 somewhat special as regards the theory, but very important from 

 a practical point of view, these oscillations may combine to form 

 progressive waves travelling with (to a first approximation) no 

 change of form over the surface. 



The term c tidal,' as applied to waves, has been used in various 

 senses, but it seems most natural to confine it to gravitational 

 oscillations possessing the characteristic feature of the oceanic 

 tides produced by the action of the sun and moon. We have 

 therefore ventured to place it at the head of this Chapter, as 

 descriptive of waves in which the motion of the fluid is mainly 

 horizontal, and therefore (as will appear) sensibly the same for all 

 particles in a vertical line. This latter circumstance greatly 

 simplifies the theory. 



It will be convenient to recapitulate, in the first place, some 

 points in the general theory of small oscillations which will 

 receive constant exemplification in the investigations which 

 follow*. 



Let q lt q- z ,...q n be n generalized coordinates serving to specify 

 the configuration of a dynamical system, and let them be so chosen 

 as to vanish in the configuration of equilibrium. The kinetic 



* For a fuller account of the general theory see Thomson and Tait, Natural 

 Philosophy, Arts. 337, ..., Lord Eayleigh, Theory of Sound, c. iv.,Eouth, Elementary 

 Rigid Dynamics (5th ed.), London, 1891, c. ix. 



