water waves when the depth to the bottom 

 is greater than one-half the wave length 

 (d > L/2) . However, for shallow- water waves 

 the depth must be less than one twenty-fifth of 

 the wave length (d < L/25) . At intermediate 

 depths the entire equation (1.2) must be used. 

 In a low deep-water wave the water particles 

 move in circles. At any depth z below the sur- 

 face the radius of the circular path followed by 

 a particle is 



r = 



_H-g-2,rz/I, 



In this circle the speed is 



V = 27rr/r = TT 



He -2'^^/^ 



(1.5) 



(1.6) 



as the particles complete one revolution in the 

 timer. (See fig. 1.2.) 



A water particle at the surface remains at 

 the surface throughout its orbit. A water par- 

 ticle at a given average depth below the sea 

 surface is farthest removed from the surface 

 when it moves in the direction of wave 

 progress. 



In a low shallow-water wave the vertical 

 motion of the particles is negligible and the 

 horizontal motion is independent of depth. The 

 particles move back and forth, following nearly 

 straight lines. 



In a deep-water wave only half the energy 

 advances with wave speed, whereas in a shal- 

 low-water wave all the energy advances with 

 wave speed. The reason for this difference is 



DIRECTION OF PROGRESS 



Figure 1.2. — Movement of water particles in a deep- 

 water wave of very small height. The circles show 

 the paths in which the water particles move. The 

 wave profiles and the positions of a series of water 

 particles are shown at two instants which are one- 

 fourth of a period apart. The solid, nearly vertical 

 lines indicate the relative positions of water par- 

 ticles which lie exactly on vertical lines when the 

 crest or trough of the wave passes and the dashed 

 lines show the relative positions of the same par- 

 ticles one-fourth of a period later. 



that in a deep-water wave only the potential 

 energy varies periodically and advances with 

 the wave form, but in a shallow-water wave 

 both potential and kinetic energy vary periodi- 

 cally and both advance with the wave form. 

 These laws can also be stated by saying that 

 the energy advances at a rate which, in a deep- 

 water wave, equals half the wave speed, where- 

 as in a shallow-water wave it equals the wave 

 speed. 



Deep-Water Waves of Moderate and Great 

 Height 



Waves of moderate and great height are 

 those for which the ratio of height to length 

 (H/L) is from 1/100 to 1/25 and from 1/25 to 

 1/7, respectively. The form of these waves 

 cannot be represented by a sine curve. For 

 waves of moderate height the form closely ap- 

 proaches the trochoid, that is, the curve which 

 is described by a point on a disc which rolls 

 below a flat surface (fig. 1.3). Waves of great 

 height deviate from the trochoid; the troughs 

 are wider and flatter and the crests narrower 

 and steeper. Theoretically, the wave form be- 

 comes unstable when the ratio H/L exceeds 1/7. 

 Observational evidence indicates that instability 

 occurs at a steepness as small as 1/10. 



The wave speed increases with increasing 

 steepness (increasing values of H/L), but the 

 increase of speed never exceeds 12 percent. 



LWe ALONG WHICH DISC ROLLS 



FiGUEE 1.3. — Profile of a trochoidal wave (solid 

 lines) and of a sine wave (dashed lines). 



The water particles move approximately in 

 circles, the radii of which decrease rapidly with 

 depth. The particle speed is not uniform but 

 is greatest when the particles are near the top 

 of their orbit (moving in the direction of wave 

 progress), with the result that the particles 

 upon completion of each nearly circular motion 

 have advanced a short distance in the direction 

 of progress of the wave (fig. 1.4). Conse- 

 quently, there is a mass transport in the direc- 

 tion of progress of the wave. The mass trans- 



