538 VISCOSITY. [CHAP, xi 



in which the kinetic energy (T) and the dissipation-function (F) can be 

 expressed as quadratic functions of the generalized velocities, with constant 

 coefficients. 



If the extraneous forces have not a single-valued potential, or if instead of 

 given velocities we have given tractions over the boundary, the theorems 

 require a slight modification. The excess of the dissipation over double the 

 rate at which work is being done by the extraneous forces (including the 

 tractions on the boundary) tends to a unique minimum, which is only 

 attained when the motion is steady*. 



Periodic Motion. 



298. We next examine the influence of viscosity in various 

 problems of small oscillations. 



We begin with the case of ' laminar ' motion, as this will enable 

 us to illustrate some points of great importance, without elaborate 

 mathematics. If we assume that v = 0, w = 0, whilst u is a 

 function of y only, the equations (4) of Art. 286 require that 



p = const., and 



du <$u 



This has the same form as the equation of linear motion of 

 heat. In the case of simple-harmonic motion, assuming a time- 

 factor e i(<rt+f) , we have 



the solution of which is 



u = Ae* +i >to + Be-* +i >to ...................... (3), 



provided =(<r/2v)* ........................... (4). 



Let us first suppose that the fluid lies on the positive side of 

 the plane xz, and that the motion is due to a prescribed oscillation 



) 



of a rigid surface coincident with this plane. If the fluid extend 

 to infinity in the direction of y-positive, the first term in (3) is 

 excluded, and determining B \>y the boundary-condition (5), we 

 have 



Cf. Helmholtz, I. c. 



