Shallow Water Wave Transformation through External Factors 141 



internal friction on water waves, the condition for the normal tension remains the same as in the 

 case of pure water; however, the condition that the tangential stress must vanish is now replaced 

 by the one indicated above. We then obtain for disturbances of the surface with a wave number x 

 and with a frequency a a modulus of decay of t^ = (2 ^/2)/(v 1/2 k«t 1/2 ) instead of the value l/(2m 2 ). 

 The ratio of this to the modulus obtained on the hypothesis of constant surface tension will be 



r, , lvx*Vl* 1 



which is by hypothesis small. That is the damping time is the greater, the smaller the wave length 

 and many times smaller than with water that has not been contaminated. 



