A STUDY OF PROGRESSIVE OSCILLATORY WAVES IN WATER 



1. INTRODUCTION 



Present knowledge of the motion of oscillatory waves in water con- 

 sists almost wholly of theoretical studies made by mathematicians 

 seeking either a solution of the classical hydrodynamical equations 

 for an incompressible heavy liquid, which satisfies certain conditions 

 dictated in part by logic and in part by experience, or a solution by 

 geometrical methods based on observation of natural phenomena. 

 Certain well-determined rigorous solutions of the problem, as posed, 

 differing chiefly in their definition of the motion of the water particles 

 involved have been obtained. 



Gerstner's theory,^ which was the first solution obtained, is based 

 on geometrical considerations, and states that the wave surface is a 

 trochoid, the paths of the water particles being circles. The diam- 

 eters of these circles, or particle orbits, decrease with increasing depth 

 below the surface according to an exponential law. The motion of 

 the wave is rotational. The Gerstner solution rigorously satisfies the 

 equations of motion and limiting conditions of the problem regardless 

 of wave height. 



Stokes criticized Gerstner's solution for requiring a rotation, since 

 Lagrange had shown that the motions of liquids generated from rest 

 under the influence of impulsive forces are necessarily irrotational; 

 but did not otherwise question the exactness of the Gerstner solution. 



This criticism led Stokes and others to search for a solution to the 

 problem of surface waves, which would satisfy all the limiting condi- 

 tions and be such that the eddy vector, or rotation, would everywhere 

 be zero. Approximate solutions through a fifth approximation were 

 obtained by Stokes, who failed, however, to prove the convergency of 

 the series used in the approximations and consequently the exactness 

 of his solution. Levi-Civita, following Stokes' work, succeeded in 

 showing the series to be convergent and obtained a rigorous solution of 

 the wave problem for the condition of infinite depth identical to that 

 of Stokes. 



In these latter solutions it is found that the particle orbits are not 

 closed paths but open, indicating a current (defined as "mass trans- 

 port") in the direction of wave propagation. Also, the velocity of 

 propagation of the wave is dependent upon its height; whereas Gerst- 

 ner's solution indicates independence from the wave height. 



' References to all works cited will be found in appendix II, Bibliography. 



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