26 Theory of Short and Long Waves 



If we develop the ^-functions in series and substitute (11.16) for rj, we obtain 



(11.17) 



VI 



g^l. 



The velocity of progressive waves with a wave profile not changing its 

 type slightly increases with the relative wave height, i.e. with the steepness 

 of the waves; however, the discrepancy with the simple formula is not great, 

 as ajK appears only in the second order. The somewhat more exact value 

 derived by Levi-Civita also corresponds with the first terms of (11.17). 



The Stokes waves of the permanent type show the characteristic that 

 they possess relatively to the undisturbed surface a certain momentum in 

 the direction of wave propagation. The individual particles in these progressive 

 waves of permanent type do not have closed orbits, but they show a slow 

 but constant advance in the direction of the wave propagation. This is ex- 

 plained by the fact that, in the case of rather large wave heights, a particle 

 which moves in the wave crest at the surface to the rigth has a smaller velocity 

 in the wave trough in the opposite direction and that it does not come back 

 to its initial position below the former wave crest. The particle, in this way, 

 does not cover circles but loops. According to Stokes, the velocity of this 

 current in a depth z is 



u = y}a % ce-- xz = n 2 d 2 ce { - 4nz)l * 



when d is the ratio of the wave amplitude 2a = H to the wave length X (d is 

 called the wave steepness). The velocity of the current decreases rapidly with 

 increasing depth. Rayleigh (Thorade, 1931, p. 31) showed that, in an 

 irrotational wave motion there always must be a water transport, which is 

 also valid for other waves; Levi-Civita (1922, p. 85) has given the theoretical 

 proof of this statement. The mean velocity Q is given by the equation 

 Q = ^(a/h) 2 gh/c. If a = 10 cm, h = 500 cm and c = 50 cm/sec, we obtain 

 for Q = 2 cm/sec; compared with c, this is a small amount, which should 

 not be neglected generally. 



3. Gerstner's Rotational Waves 



In 1802, Gerstner developed a wave theory which gives an exact solution 

 of the hydrodynamical equations for waves over infinite depth. The exact 

 equations, according to Gerstner, express a possible form of wave motion. 

 Contrary to the Stokes waves, the motion in this fluid is rotational. The 

 vorticity is greatest at the surface and diminishes rapidly with depth. The 

 sense of the vorticity is opposite to that of the revolution of the individual 

 particles in their circular orbits and, therefore, unfavourable for the generation 

 of such waves, because it is impossible to originate these waves by the action 

 of ordinary forces on the surface at rest. 



The derivation of the equations is facilitated by using the Lagrangian 



