Oscillations in a non-homogeneous viscous fluid

The method of matched asymptotic expansions is employed to derive formulae for viscous corrections occurring in small amplitude oscillatory disturbances in a nonhomogeneous fluid. Such formulae are given for quite general variations with depth of the equilibrium density and viscosity distributions in the following cases: (i) two-dimensional disturbances in a fluid of finite depth bounded above by a free surface, (ii) two-dimensional disturbances in fluid bounded above and below by two fixed horizontal planes. The derived expressions are seen to depend on the value of the viscosity at rigid bounding surfaces and not at all on its actual distribution. An additional correction for viscosity is given for case (ii), when the density has an exponential variation with depth and the kinematic viscosity is constant.