Peristaltic pumping in a circular cylindrical tube is analyzed. The problem is a viscous fluid flow induced by an axisymmetric traveling sinusoidal wave of moderate amplitude imposed on the wall of a flexible tube. A perturbation method of solution is sought. The amplitude ratio (wave amplitude/tube radius) is chosen as a parameter. The nonlinear convective acceleration terms in the Navier-Stokes equation is retained. The governing equations are developed up to the second order in the amplitude ratio. The zeroth-order terms yield the classical Poiseuille flow, the first-order terms yield the Sommerfeld-Orr equation. If there is no pressure gradient in the absence of wall motion, the mean flow and mean pressure gradient (averaged over time) are both shown to be proportional to the square of the amplitude ratio. Numerical results are obtained for this simple case by approximating a complicated group of products of Bessel functions by a polynomial. The results show that the mean axial velocity is dominated by two terms. One term corresponds to a parabolic profile which is due to the mean pressure gradient set up by the wall motion. The other term arises from satisfying the no-slip boundary condition at the wavy wall rather than at the mean position of the wall. In addition, there are perturbations arising from the convective acceleration. If the mean pressure gradient set up by the wall motion itself reaches a certain positive critical value, the velocity becomes zero on the axis. Values of the mean pressure gradient larger than the critical value will induce backward flow in the fluid. Values of the critical pressure gradient for several cases are presented.