We have investigated a copper-mesh grid oscillating at its fundamental (0, 1) Bessel mode in superfluid 4He for temperatures 10< T <1500 mK at a pressure of P = 5 bar. The high quality factor (Q ∼ 10 to power 5) of the oscillator allowed us to observe new features of its response to a periodic drive which, at the lowest T, was found to depend on the prehistory of the helium. The experiments have confirmed the existence of two critical velocities, and we discuss whether these critical velocities are associated with quantized vortices.