continuity, and the condition of constant surface pressure. In addi- 

 tion, if the water is assumed to be frictionless, which it very nearly 

 's for purposes of wave theory, the flow must be irrotational. The 

 problem is thus reduced to the determination of an analytical expres- 

 sion, or the construction of a flow net, to satisfy these steady condi- 

 tions. Continuity requires that the flow system in each small element 

 of the wave be so constructed as to indicate no destruction or creation 

 of material. The condition of constant surface pressure is very 

 nearly satisfied for waves moving without wind action, and surface 

 pressure differences due to wind are probably not great as compared 

 with the hydrostatic pressure differences. Waves lose energy so 

 slowly that the assumption of constant energy along each streamline 

 seems reasonable. Irrotational motion occurs in fluids devoid of 

 viscosity and this condition may not be entirely satisfied in natural 

 waves; if the motion is not irrotational, the energy may be different 

 along different streamlines. For a complete discussion of rotational 

 and irrotational motion the reader is referred to Lamb, "Hydro- 

 dynamics," or Prandtl-Tietjens, "Fundamentals of Hydro- and 

 Aero-Mechanics." 



The aim of the classical analysis of wave motion has been to find 

 the surface form and wave velocity which satisfy the conditions out- 

 lined. 



Section 3. WAVES OF SMALL AMPLITUDE 



The observed permanence of oscillatory waves is utilized for the 

 mathematical analysis of the oscillatory wave problem. The solution 

 then requires the determination of the dynamic conditions required 

 for such permanence of wave form. This concept is fundamental to 

 the analysis of wave motion of the type under consideration audits 

 importance accounts for the preceding discussion of whether or not 

 waves actually do maintain their identity. Quantitatively, perman- 

 nence of form requires that every element of the wave form advance 

 with the same velocity. Consequently, a wave may be imagined as 

 being brought to rest relative to a fixed reference system by superim- 

 posing a mass velocity in the water equal but opposite to the wave 

 velocity. This does not modify the internal flow problem but does 

 simplify the analysis to one of steady flow. 



If water of constant depth is disturbed by small periodic impulses, 

 the wave length and velocity are related to the depth by the equation 

 (3). 



C=^||tanh ?^ (2) 



So long as the surface curvature is everywhere large enough to elimi- 

 nate the influence of surface tension and the wave height is small as 

 compared with d and L, this equation applies to both deep-water and 



