Two fundamental problems in the oscillatory motion in a viscous incompressible fluid are considered. In Chapter II, the solution for a fixed point source of momentum fluctuating sinusoidally in strength in a given direction and producing no mass is derived. The potential flow due to an oscillating mass-dipole is found to give rise to a momentum source which is referred to as the irrotational oscillatory point source of momentum. This is also a solution of the linearised equation of motion, which is the important approximating equation for small values of M/pѵ2 , the only nondimensional parameter for the point source. Another solution of the linearised equation is the one with constant pressure due to a ring source of circulation (strength A cos ωt per unit time), in the limit as the radius, ω0 , of the ring tends to zero such that πω20 A = M/p (const.). A linear combination of these two solutions gives a uniformly valid first approximation to the solution for a momentum source referred to as the rotational oscillatory point source of momentum; the linear combination is chosen to remove the dipole singularity in the velocity distribution. This composite solution gives also the Green's function for the time periodic linearised momentum equation. A different approximation to the solution in the inner region (r « (ѵ/ω)1/2 ) is obtained by making use of the Landau solution for a submerged jet. A proof of the equivalence of momentum flux (divided by density) and the flux of the density of moment of circulation is also included.
To set the results in a physical context, the problem of a translationally oscillating sphere is discussed in Chapter III. For low Reynolds' numbers R, and large values of the frequency parameter (or Strouhal number) Λ with RΛ = 0(1) the solution is again a linear combination of the dipole solution and the solution for a ring source of circulation, i.e., in this approximation at least, the sphere is equivalent to the two point sources of momentum. As is well known, in potential flow the sphere is equivalent to an oscillating mass-dipole and therefore to the irrotational oscillatory point source of momentum.