The dynamics of turbulent velocity fluctuations in and somewhat outside the viscous sublayer are examined by linearizing the equations of motion around the known mean velocity profile. The rest of the boundary layer is assumed to drive the motion in the layer by means of a fluctuating pressure which is independent of distance from the wall. The equations, which are boundary-layer approximations to the Orr-Sommerfeld equations, are thus treated as a non-homogeneous system and solved by convergent power series. The solutions which exhibit the strong role of viscosity throughout the layer considered provide a model endowed with many of the known features of turbulence near a wall. In particular, the phase angle between streamwise and normal fluctuations is found to be in plausible agreement with experiments. An important role is ascribed by the solutions to the displacement of the mean velocity by the normal fluctuations. The impedance of the layer is found to be anisotropic in that it favours fluctuations with a much larger scale in the streamwise than in the spanwise direction. For such disturbances, the ratio of turbulent intensity to the intensity of the pressure fluctuations approximates the experimental ratio. According to the solutions it is primarily the spanwise component of the pressure gradient which is responsible for the intense level of turbulence very near the wall. The model apparently underestimates the amplitude ratio of normal to streamwise components of the velocity.