The determination of this coefficient is neces- 

 sary for an exact evaluation of energy transfer 

 by push. 



The pulling force of the wind always acts in 

 the direction of the wind. It is the same at the 

 wave crest and the wave trough but the eflfect 

 differs. Energy is transferred from the air to 

 the water (the movement of the surface layer 

 is speeded up) if the surface water moves in 

 the direction of the wind, but energy is given 

 off from the water to the air (the movement 

 of the surface water is slowed down) if the 

 surface water moves against the wind. If wind 

 and waves move in the same direction the water 

 particles move in the direction of the wind drag 

 while at the crest, but against the drag when 

 in the trough. (See fig. 1.2.) In the absence 

 of a mass transport speed the particle speeds 

 at the crest and trough are equal but in oppo- 

 site directions, so that the energy added by the 

 pulling force of the wind at the wave crest is 

 removed at the wave trough. In the presence 

 of a mass transport speed, however, the for- 

 ward motion at the crest is greater than the 

 backward motion in the trough (fig. 1.4) and 

 a net amount of energy is transferred to the 

 water. No satisfactory explanation of the 

 growth of waves has been given without assum- 

 ing a transfer of energy due to the wind pulling 

 at the water particles. 



Since the pulling force of the wind over the 

 ocean is known, the energy transfer from the 

 air to the water by wind drag can be computed 

 with considerable accuracy from the theoretical 

 values for mass transport speed given in Equa- 

 tion (1.7). Even when the wave speed exceeds 

 the wind speed, the effect of the wind drag 

 remains nearly the same because it depends 

 upon the difference between wind speed and 

 particle speed in the water. In general the 

 water particles move much more slowly than 

 the wind even when the wave form moves much 

 faster. If it is assumed that the wind cannot 

 transfer energy to the water by pulling at the 

 water particles, no satisfactory explanation can 

 be given of the fact that waves frequently have 

 a higher speed than the wind that produces 

 them. 



Energy is dissipated by viscosity, but the 

 viscosity of water is so slight that this process 

 can be neglected. There is no evidence that 

 energy is dissipated by turbulent motion in the 

 wave. The chief processes that can alter the 



wave height or the wave speed in deep water 

 are, therefore, the push of the wind, which be- 

 comes an air resistance if the wave travels 

 faster than the wind, and the drag or pull of 

 the wind on the sea surface. 



If the rate of energy transfer from the wind 

 and the rate at which the wave energy ad- 

 vances (p. 2) are known, it is possible to es- 

 tablish a differential equation from which the 

 relationships between the waves and wind 

 speed, fetch, and duration are obtained as spe- 

 cial solutions. The equation contains three nu- 

 merical constants (including the sheltering co- 

 efficient) which have to be determined in such 

 a manner that all of the data and the empirical 

 relations are satisfied. These conditions can be 

 met, and at the same time discrepancies be- 

 tween existing empirical relationships can be 

 accounted for. 



The growth of waves as determined in this 

 manner is illustrated in figures 1.8 and 1.9 

 which are constructed on the assumption that 

 a wind of constant speed of 40 knots started to 

 blow over an undisturbed water surface ex- 

 tending for 800 or more nautical miles from a 

 coast line. Figure 1.8 shows the height and 

 period of the waves as functions of the distance 

 from the coast for various wind durations. 

 Solid lines show the height and dashed lines 

 show the periods. First, small waves are 

 formed, probably by eddies striking the sea sur- 

 face. At the coast the waves remain low, but 

 off the coast they travel with the wind and grow 

 as they receive energy by push and pull. When 

 the wind has blown for 5 hours one finds that 

 with increasing distance from the coast the 

 waves increase rapidly in height and period out 

 to a distance of 35 miles. There the waves are 

 14 feet high with a period of 5.2 seconds. Be- 

 yond 35 miles similar waves are present but 

 there exists a striking difference between con- 

 ditions inside and beyond the 35-mile point. 

 Inside of 35 miles a steady state has been 

 reached, that is, at any given point the waves 

 do not change no matter how long the wind 

 lasts ; beyond 35 miles the waves continue 

 to grow for a length of time which depends 

 upon the distance from the coast. After 

 10 hours a steady state has been established 

 to a distance of 75 miles, after 20 hours to 

 a distance of 205 miles, and so on. In fig- 

 ure 1.8 the solid and dashed curves show 

 the steady state. Parts of the curves and the 



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