Quarter wave resonators used in independently phased linacs have a wide incident velocity acceptance above the velocity of about0.5&hthinsp;&bgr;opt,where&bgr;optis the optimum incident velocity for the resonators. Their transit time factor (TTF) crosses zero at the velocity of0.5&hthinsp;&bgr;optand has a small negative peak at0.35&hthinsp;&bgr;opt.An acceleration of ions across the zero-crossing velocity starting from a velocity in the negative peak is discussed. For taking into account of velocity changes in the resonators, a numerical calculation was carried out and it was found that a very slow acceleration can occur if the rf phase is properly changed over the transitional region. In the calculation,Cl10+ions and the shape of the quarter wave resonators of&bgr;opt=0.1in the JAERI Tandem Superconducting Booster were chosen. ©1999 American Institute of Physics.