41 



9. Divide the initial wave crest in deep water into equal incre- 

 ments and draw lines perpendicular to all intermediate wave 

 crests froDi these division points to the shore. 



10. From the loiown period and height in deep water, compute 

 the power per unit of crest length and the height of the same wave 

 at different points in the diagram. The power per unit of crest 

 length is mversely proportional to the spacing of the orthogonal 

 lines measured along the wave crest. 



The method outlined is applicable to a gradually sloping bottom. 

 Very steep or vertical slopes, as around the end of a breakwater built 

 in deep water, present problems, especially in estimating the heights, 

 which can only be solved by supplementing this method with ejfperi- 

 mental results. 



Section 8. DAMPING OF OSCILLATORY WAVES 



After the forces generating waves cease to operate, the waves run 

 freely, losing energy by friction as they proceed. Waves in nature 

 probably are reduced in height by an increase in wave length as they 

 move outward from the source. In deep water, internal friction con- 

 verts mechanical energy into heat, while in shallow water both in- 

 ternal and bottom friction are operative. Waves generated in tanks 

 also lose energy by friction on the walls. The problem has been 

 analyzed theoretically by a number of investigators. Only a brief 

 summary of the general results will be presented here. Damping of 

 waves in tanks by bottom and wall friction is to be considered in a 

 separate report. 



Wave motion requires a distortion of the fluid elements and there- 

 fore viscous damping will occur even in deep water and in the absence 

 of vertical walls. The problem has been treated by Lamb (26) as 

 follows: 



If the wave surface profile is represented by the equation 



y=^Bm^(x-Ct) (49) 



where h =full amplitude (trough to crest) 



the average kinetic energy per unit area of surface is 1/16 pmh^ C^ and 

 the total energy is twice that, or 1/8 pmh^C^. From analysis of the 

 rate of distortion of fluid elements, the average rate of dissipation per 

 unit area is found to be 



