28 



Theory of Short and Long Waves 



8r 



D = 1 + xr — = 1 - *V 2 = 1 - x 2 A 2 e- 2 *? 



dy 



(11.20) 



The two equations (11.18) further give 



dp 



ca 



dp 



dy 



= (a 2 — xg) r sin & and = (a 2 — xg) r cos# -\-g— a 2 xr 2 



The boundary condition of uniform pressure at the free surface requires 

 that a 2 — xg — 0. This is identical with the equation (II. 7) for water of infinite 

 depth, which means that the velocity of propagation of the Gerstner waves 

 is also 



c = Vigtylji) . 



The equations for pressure are then reduced to 



cp/ca = and dp/dy = g(l -x 2 r 2 ) . 



Consequently, when y = 0, i.e. for the free surface, the pressure p must 

 be equal to the atmospheric pressure: 



P =Po + g 



y-^A 2 {\-e 2 *v) 



(11.21) 



The constant A, which is the wave amplitude, can only increase to such 

 an extent that D in (11.20) does not vanish or becomes negative. As y varies 

 between and oo, this requires that 



A 2 < 



1 



JA 2 



2.T 



(11.22) 



As long as A < Xjln, the equation of the surface is at any time a trochoid; 

 when the amplitude becomes equal to l\2n, we have the extreme case with 

 acute angles at the crest (cycloid). If A > l\2n, we have a curve with loops, 

 which is impossible for steady motion. This is in agreement with (11.22). 

 Figure 15 shows in thin, solid circles the orbits of the water particles and 



Fig. 15. Profile of a Gerstner wave: trochoid. Thin circles: orbits of the particles; dotted 

 curves: position of line of particles during passage of a wave. 



