313 



purposes, the waves in the chaimel may be characterized essentially by the 

 properties associated with the significant waves. 



The Growth of Waves in the Channel 



The frequency spectra calculated at different fetches for the same air 

 velocity indicate how the waves grow as they move downstream along the 

 channel. A typical set of spectra for increasing fetch is shown in Figure 15. 

 These curves have been corrected by subtracting out the noise level, and 

 have been smoothed by a method similar to that discussed by Hidy and Plate 

 (1965). Near the leading edge of the water (small fetch), the observed 

 spectrum contains little total energy and is rather sharp. As the waves 

 travel downstream, the magnitude of the spectral density function increases, 

 the primary peaks broaden at first, then tend to sharpen up while the values 

 of fjn decrease. 



The growth of waves in the range of higher frequencies tends to be 

 limited as indicated in Figure 15- The complete mechanism for restraining 

 the growth of the high frequency components is not known. However, it can 

 be seen that the limitation in growth, in part, can be the result of attain- 

 ing a balance between gains in energy input from the air and losses by dis- 

 sipation. The dissipation of energy in small gravity-capillary waves is 

 probably related to the action of viscosity and surface tension. The loss 

 by viscous forces in waves is proportional to (ak) (Lamb (1932)) where a 

 is the amplitude of a wave. As proposed by Longuet-Higgins ( 1962a), the 

 loss resulting from surface tension can be related to the drain of energy 

 from larger waves when capillary ripples are formed near the crests of the 

 larger components. This particular mechanism indicates that the energy loss 

 is proportional to (a^^kS^,) . The subscript c refers to the capillary ripple 

 on the crest of a larger wave . If the interaction between components in the 

 wave train is a second or higher order effect (e.g., Phillips (1963)), the 

 action of dissipative processes should balance the input of energy from the 

 air motion in such a way that the net energy at equilibrium is smaller the 

 higher the frequency range . This seems to be suggested in the behavior of 

 the spectra shown in Figure 15- 



It is interesting to note that the growth of components in the lower 

 frequency range, say f < 3.5 cps in Figure 15, is approximately exponential. 

 Qualitatively, this type of growth has been predicted in the recent shearing 

 flow theories of Miles (see, for example. Miles (i960)). 



Similarity Shape of the Spectrum 



An important feature which was also exhibited by many of the spectra for 

 the channel waves was the tendency for growth in such a way that a similarity 

 shape in the spectral density function is maintained. The frequency spectrum 

 can be expressed, with Eq. (ll) in normalized form, as: 



(12) 



