Elastic String vibrating in a Viscous Medium. 743 



string at any point x as a function of x. We are, I think, 

 justified in assuming that, for its fundamental vibration, 



. 7TX 



where q is its displacement at its middle point and I is its 

 length, since we know that this is true for an undamped 

 string, and observation does not show any sensible variation 

 from this form of displacement when the string is vibrating 

 in a viscous medium. 



The velocity, v x , of any point, will then be given by 



. irx 

 v x = v sin —r-, 



where v is the velocity at the middle point. The mean 



2 

 velocity will therefore be equal to - v, and the equivalent 



2 

 momentum of the whole string to -Mi;, where M is its 



mass. 



In the same way, the mean frictional force will be given 



2 

 bv — B.v, where R is the force which would be necessary to 



" 7T 



overcome the internal and external friction of the whole 

 string if every point of it were moving with unit velocitv. 

 The normal pressure on the string, at any point, tending to 



bring it back to its position of equilibrium, is equal to -, 



where t is the tension to which the string is subjected, 

 and p the radius of curvature of the string at that point. 

 For small curvature, 



1 CpQx 7T 2 . 7TX 



-=71 = — -J2- £ sin -p. 

 p dx z r I 



Therefore, for any point, x, on the string, the normal pressure 

 is given by 



T7T 2 . ttx 



--F? SU1 T- 



The total force on the string tending to restore it to its 

 position of equilibrium is equal to 



:0 



T7T 2 . TTX 7 2lTT 



The equation of motion of the string, considered as a 



