GROWTH AND DECAY O 



ward again at the instant of passage of the top of the next crest. (See 

 fig. 2.) 



It is obvious that if any given water particle is close to the surface 

 of the water the vertical distance between the points that it occupies 

 in its orbit when it is at the top of a wave crest and when it is at the 

 bottom of the succeeding trough is equal to the vertical distance be- 

 tween the crest and the trough. In other words, the diameter of its 

 orbit is equal to the height of the wave from crest to trough and is 

 entirely independent of the length of the wave from crest to crest, or 

 of the speed of its advance. 



It is further clear that the velocity with which each water particle 

 circles its orbit is governed (a) by the distance the water particle 

 must cover during each circuit and (b) by the length of time during 

 which each circuit is completed. The length of the circuit of a water 

 particle at the surface is, of course, equal to the circumference of the 

 orbit, or (the diameter of the latter being equal to the height of the 

 wave) about 3.14 times as long as the vertical height of the wave, crest 

 above trough, at the time. The time occupied by it in each circuit 

 is equal to the time interval that intervenes between the passage of 

 every two successive crests past any given point, for each water particle 

 is at the top of its orbit when each successive crest passes by. This 

 time interval, in turn, depends on the velocity at which the wave forms 

 are advancing, and on the distance from one crest to the next, i.e., on 

 the so-called "period" of the wave, as explained on page 31. 



Since there can be no one fixed relationship between the heights 

 of waves and their periods, there is no one fixed relationship between 

 the velocities at which the wave forms advance and the velocities at 

 which the water particles circle their orbits. But the orbits traced 

 by the individual water particles are invariably much shorter than 

 are the distances from crest to crest, because wind waves at sea are 

 always many times longer than they are high. Hence (since the 

 period is the same for the wave as it is for the water particles of which 

 it is composed), the velocities of the water particles around these 

 orbits are always much lower than the velocity at which the wave 

 form is advancing. In the case of a wave, for example, 200 feet long 

 from crest to crest and 10 feet high from crest to trough, each water 

 particle at the surface would trace an orbit of 10 x 3.14 feet, or about 

 31 feet, i.e., a little more than one-seventh as long as the wave, so that 

 the velocity at which it circled its orbit would be a little more than one- 

 seventh as great as that at which the wave form was advancing. And 

 the longer a wave is, relative to its height, the greater will be the differ- 

 ence between the velocity at which it is advancing and the velocities at 

 which its water particles are circling their orbits. If, for example, 

 the 10-foot wave just discussed were 400 feet long instead of 200 feet. 



