The acoustic boundary-layer structure is investigated in a cylindrical tube where steady sidewall injection is imposed upon an oscillatory ow. Culick's steady, rotational, and inviscid solution is assumed for the mean ow. The time-dependent velocity is obtained by superimposing the acoustic (compressible, inviscid, irrotational) and the vortical (incompressible, viscous, rotational) velocity vectors. A multiplescales perturbation technique that utilizes proper scaling coordinates is applied to the axial momentum equation by retaining the viscous terms and ignoring the axial convection of vorticity. A closed-form expression for the time-dependent axial velocity is derived that agrees well with the corresponding numerical solution, cold-ow experimental data, and Flandro's near-wall analytic expression. A similarity parameter that controls the thickness of the rotational region is identi ed. The role of the Strouhal number in controlling the wavelength of rotational waves is established. An accurate assessment of the amplitude and phase relation between unsteady velocity and pressure components is obtained. Increasing viscosity is found to reduce the depth of penetration of the rotational region. Nomenclature a 0 = mean chamber speed of sound, m/s f = oscillation mode frequency, Hz k = dimensionless wave number or frequency, vR/a 0 L = internal tube length M b = blowing Mach number, V b/ a 0 P 0 = mean chamber pressure p = dimensionless pressure, p*/P 0 R = effective radius, volume/half of porous area, m Re = Reynolds number based on sound speed, a 0 R/n Re a = acoustic Reynolds number, k/d 2 = vR 2 /n = 2 2 R /d s r = dimensionless radial position, r*/R r 1 = radial scale, magni ed or compressed S p = penetration number, = 3 2 2 2 2 3 2 2 2 1 2 1St = Strouhal number, k/M b = vR/Vb t = dimensionless time, t*a 0/ R =t/k U = Culick's steady ow velocity vector, (U r , U z ) U r = Culick's steady radial velocity, 2r 2 1 sin u u = dimensionless velocity, u*/a0 u 9 z = acoustic velocity, sin(kz)exp(ikt) V b = injection velocity at the porous boundary, m/s Y = penetration control parameter, S p