The transverse vibration of an axially accelerating string is investigated. The equation of motion is developed using Hamilton's principle. The resulting partial differential equations are discretized using Galerkin's method. Assuming the axial velocity to be periodic, a stability analysis is performed using Floquet theory. One-, two-, three-, four-, six- and eight-term series approximations are considered in the Galerkin's method. The one-term approximation leads to a Mathieu equation, the solution of which is well known. The numerical results for one term are compared with the analytical solution for the Mathieu equation, and they are in full agreement. The two-term approximation leads to gyroscopically coupled equations, and the solutions differ significantly from that of the one-term approximation, whereas the three-term approximation solutions look similar to the one-term approximation solutions. The analysis is carried out for higher order even approximations and the solutions are in qualitative agreement. The two- and four-term approximation solutions are compared with the analytical results from Hsu's method, and are in reasonable agreement. The results show that instabilities occur at much higher amplitudes and frequencies of the periodic axial velocity than that of typical devices such as tape machines and band saws.