297-298] LAMINAR OSCILLATIONS. 539 



or, taking the real part, 



corresponding to a prescribed motion 



u = a cos (<rt + e) (8) 



at the boundary*. 



The formula (7) represents a wave of transversal vibrations 

 propagated inwards from the boundary with the velocity er//3, but 

 with rapidly diminishing amplitude, the falling off within a wave- 

 length being in the ratio e~ 27r , or ^L. 



The linear magnitude 



27T//3 Or (47TZ/ . 2?r/<7)* 



is of great importance in all problems of oscillatory motion which 

 do not involve changes of density, as indicating the extent to 

 which the effects of viscosity penetrate into the fluid. In the 

 case of air (y '13) its value is 1'28P* centimetres, if P be the 

 period of oscillation in seconds. For water the corresponding 

 value is '47P*. We shall have further illustrations, presently, 

 of the fact that the influence of viscosity extends only to a short 

 distance from the surface of a body performing small oscillations 

 with sufficient frequency. 



The retarding force on the rigid plane is, per unit area, 



rj ni I 



= fjL/Ba {cos (at + e) sin (art + e)} 

 Uz/=o 



= pit a* a cos (at + e + J TT) (9). 



The force has its maxima at intervals of one-eighth of a period 

 before the oscillating plane passes through its mean position. 



On the forced oscillation above investigated we may superpose any of the 

 normal modes of free motion of which the system is capable. If we assume 

 that 



u a A cos my + Bsin my (i), 



and substitute in (1), we find 

 whence we obtain the solution 



='S(A cos my + B sin my} e 

 * Stokes, 1. c. ante p. 518. 



