43 



In nature, the dissipation of energy is very gradual and correction 

 for energy dissipation may be unnecessary. If a correction must be 

 made in wave refraction studies, the computation proceeds by a 

 series of approximations using the heights computed on the assump- 

 tion of no energy loss as a first approximation. 



Equation 54, which applies to all waves in all depths of water, was 

 derived on the assumption that the bottom is perfectly smooth and 

 the flow viscous, Kemembering that the frictionless case gives large 

 velocities at the bottom in shallow water and that the bottom in 

 nature is always rough, it appears likely that turbulent friction at 

 the bottom is present in the usual case. Compared with the probable 

 thickness of the boundary layer, even sand grains in a smooth layer 

 present a considerable roughness. The bottom is generally marked 

 by ripples and observation shows that vortices do form around these 

 ripples. In very shallow water, there are often observed clouds of 

 sand which may result from hydraulic flow in the sand from points 

 under the crest to the trough, but which certainly indicate that the 

 motion near the bottom is not laminar whatever may be the origin 

 of the clouds. 



U. S. GOVERNMENT PRINTING OFFICE 



