NO. IB.] ON WAVES BET WEEN FRESH AND SALT WATER. 45 



From equation (1) p. 387 in Lamb (I. c. p. 33), the velocity of a water- 

 particle in its orbital motion, may be calculated. It follows that it is pro- 

 portional to the wave-height, and is always less than the velocity of propag- 

 ation of the waves, i. e. the wave-velocity. The velocity of a fresh-water par- 

 ticle in a wave-crest would, in the case of very long waves, be equal to the 

 wave-velocity, only if the wave-height H (from the mean level of the boundary, 

 to its highest level) be equal to the mean depth d, of the surface-layer. As 

 this is impossible, it only shows that in the case of long waves, the velocity 

 of the water-particles is always less than the wave-velocity, as long as the 

 wave-height is moderate enough for the waves to be approximately conform- 

 able to linear dynamical equations. In the case of short waves (wave-length 

 less than two or three times the depth of the water-layers) the greatest hori- 

 zontal velocity of the water near the boundary, should be equal to the wave- 

 velocity, only if the wave-height H be Ve (or Vaw exactly) of the wave-length; 

 and the velocity of the water near the surface is then considerably smaller. 

 This wave-height — Ve of the wave-length — is about the extreme limit of the 

 height of waves, if they are not breaking. 



It is very easily proved that, when restricting us to two-dimensional waves of a per- 

 manent type, the orbital velocity of the water-particles can be as great as the wave-velocity, 

 only when the waves have their extreme height; or in other words, the orbital velocity can 

 never be greater than the wave-velocity, as long as the waves do not break. For when 

 in the contrary case, the wave-motion is reduced to steady motion by superposing a velo- 

 city equal and opposite to the wave-velocity, the fresh-water at the very wave-crests, would 

 have a velocity opposite to the general flow of water ; and this is inconsistent with the repre- 

 sentation of the steady motion by means of stream-lines. In the extreme case in which the 

 velocity of the fresh-water (in the imagined steady motion) vanishes at the crests, these 

 must be infinitely curved so as to form sharp angles. This extreme case, however, cannot 

 be attained by boundary-waves. For, as can be immediately seen by drawing the stream- 

 lines in the immediate proximity of a wave-crest; the velocity of the salt-water would 

 then be infinite there. The velocity of the water-particles in the boundary-waves, is there- 

 fore always slower than the wave-velocity. 



D. THE EFFECT OF A LIGHT SURFACE-LAYER, ON THE RESISTANCE 



OF A VESSEL. 



As long as we confine ourselves to a merely qualitative examination, it 

 is not difficult to see in what manner a lighter surface layer resting upon the 

 salt water, may influence the resistance of a ship. 



