  We develop a configuration of radio-frequency (rf) cavities to dramatically
improve the performance of slip-stacking. Slip-stacking is an accumulation
technique used at Fermilab to nearly double proton intensity by maintaining two
beams of different momenta in the same storage ring. The two particle beams are
longitudinally focused in the Recycler by two 53 MHz 100 kV rf cavities with a
small frequency difference between them. We propose an additional 106 MHz 20 kV
rf cavity with a frequency at the double the average of the upper and lower
main rf frequencies. We show the harmonic rf cavity cancels out the resonances
generated between the two main rf cavities and we derive the relationship
between the harmonic rf voltage and the main rf voltage. We find the area
factors that can be used to calculate the available phase space area for any
set of beam parameters without individual simulation. We establish Booster beam
quality requirements to achieve 99\% slip-stacking efficiency. We measure the
longitudinal distribution of the Booster beam and use it to generate a
realistic beam model for slip-stacking simulation. We demonstrate that the
harmonic rf cavity can not only reduce particle loss during slip-stacking, but
also reduce the final longitudinal emittance.
