42 



Therefore 



|(ipmOT)=-lMm3(7^A^ 



(50) 



dh 



= -2vhw? 



dt- — (^1) 



A=A,g-2W'»' (52) 



The time, U necessary for the height to be reduced in the ratio e : 1 is 



1 U 



Assuming that the wave length-velocity-period relationship is un- 

 affected by viscosity 



L=5.12T' C=5.12T 



and the distance Lr of wave travel corresponding to the same ratio of 

 height reduction is 



Lr=Ctr=l.71 — 

 »■ " 



v = 1.0X10-'Jtysec. X,= 1.71X10ST* 



Wave travel required for a ratio of reduction in height of e: 1 



(54) 



The theory shows a tremendously slow rate of damping by internal 

 friction for even short period waves. 



The theory outlined starts from the assumption of a sinusoidal 

 wave form but the theory may be taken as indicating a very slow rate 

 of damping of all waves in deep water. 



The damping of waves in shallow water has been considered by 

 Hough (27) who also found that the modulus of decay, or time of travel 

 necessary for the wave to be reduced in amplitude in the ratio e : 1 is 



^ (55) 



tr = 



Sl'TT^ 



For fluids having small viscosities, such as water, the velocity of the 

 wave and the relationship between depth, length, and period is un- 

 altered by friction. Hough states that: 



This agrees with the formula given by Professor Lamb for the case of waves 

 in deep water. We see now that it holds for waves of any wave-length in water 

 of any depth, provided that the bottom is perfectly smooth and that the internal 

 viscosity is sufificiently small to allow of our approximations. 



