258 EXERCISES. 



35. From the general properties of equipotential and stream-curves, 

 prove that in a regular series of waves moving in deep water without 

 molecular rotation there is necessarily in the neighbourhood of the 

 surface a transference of fluid in the direction of the wave s propagation, 

 whether that surface satisfy the condition of a free surface or not. 



[Lord Rayleigh, Math. Tripos, 1876.] 



36. Prove that a wave of sudden rarefaction in a gas is an unstable 

 state of motion. [Thomson.] 



37. Prove that the dissipation-function of Art. 179 cannot be zero 

 at every point of an incompressible fluid unless the fluid move as a solid 

 body. [Stokes.] 



38. Assuming the formula 



&amp;lt;f) = ae- k v sin 7; (x - ct) 



as representing approximately the motion of a long train of waves on 

 deep water, prove by means of the dissipation-function of Art. 179 that 

 the diminution of a due to viscosity is given by the formula 



a=a e~* Ic W. [Stokes.] 



39. Explain in general terms, without calculation, why short 

 waves are more rapidly destroyed by viscosity than long ones. 



40. A long circular cylinder rotates- about its axis with uniform 

 velocity in an infinite mass of viscous fluid ; find the motion of the fluid 

 when it has become steady. [Stokes.] 



41. Work out the same problem for the case of a rotating sphere. 



[Kirchhoff.] 



42. Investigate the small oscillations of a spherical shell filled with 

 viscous fluid, and oscillating by the torsion of a suspending wire. 



[Helrnholtz.] 



43. A cylinder moves with uniform velocity in a direction per 

 pendicular to its length, in an infinite mass of viscous fluid ; prove that 

 the motion cannot be steady. Describe in general terms the nature of 

 the actual motion. [Stokes.] 



CAMBRIDGE : PRINTED BY C. J. CLAY, M.A., AT THE UNIVERSITY PRESS. 



