24 



Theory of Short and Long Waves 



merely approximate, the extreme form would have cusps at the crests, as 

 in the case of Gerstner waves (Fig. 15). This, however, is not the case. Stokes 

 and Mitchell (1893, p. 430) have shown that the extreme form has angles 

 of 120°, similar to a roofline. The ratio between the height and the length 

 of this steepest wave form was 0142 or approximately 1 :7, and its velocity 

 of propagation was 1-2 times greater than for waves with infinitely small 

 heights. 



Thorade (1931, p. 31) explains this striking result as follows. Figure 12 shows a small part 

 of a wave in the vicinity of the crest, where the water surface can be adequately represented by two 

 planes AB and A'B'; He adds the negative velocity — c according to the method of Rayleigh, and 

 he obtains a steady current towards the left, which is bounded by the crest acting as a solid wall; 



Fig. 12 Streamlines and equipotential lines in the crest of the highest possible Stokes's wave. 



The stream lines then follow the deathered lines CC; the system of lines CB, C'B' perpendicular 

 to CC represent equipotential curves, whose successive distance is inversely proportional to the 

 velocity. If one knows the angle 2a formed by two rigid walls, and provided that in greater depth 

 the stream lines are horizontal, then both systems of lines are completely determined, as shown 

 in Lamb (1932, § 63, p. 68). On account of the continuity of the current, the velocity in A must 

 be zero; it decreases in approaching A and increases again afterwards. Below A the horizontal 

 component increases, until it reaches —c when the stream lines are horizontal, but in the crest of 

 each stream line the vertical component is always zero. The condition of continuity and the condi- 

 tion of irrotational motion determine the entire field of streamlines. In this case an additional condi- 

 tion is that the surface BAB' is a free surface, and the fixed walls can therefore be removed without 

 changing the motion. This condition fixes the angle 2a. Calculation gives for the value of a = 60°. 

 The equation of continuity in the present case is dujdx+dwjdz = 0, the condition of irrotational 

 motion dwjdx—dujdz = 0. We introduce instead of rectangular co-ordinates (x, z) polar co-ordinates 

 (r,&) with their centre in A (Fig. 13). Then x = rsinft and z = a—rcos&, and the transformation 

 of co-ordinates gives for both conditions the equations: 



du dw du dw 



— sin# cos?? 1 -\ cos# H sin# = , 



dr dr rdd rd& 



tu dw du dw 



— cos# -j sin# sin# H cos# = . 



dr dr rd& rd& 



