The time structure and integrated diffraction profile of cold neutrons of wavelength λ = 6.27 Å transmitted through a longitudinally vibrating silicon crystal were calculated by Monte Carlo simulations and measured on the backscattering spectrometer IN10 at the Institut Laue-Langevin. Neutrons of velocity vN = 630 ms-1 require tT = 158.5 us for direct transit through the 100 mm-long silicon resonator. This time is long compared with the vibration period TP = 22.3 μs and most of the neutrons experience multiple Bragg reflections in the oscillating Doppler-strain field. Monte Carlo calculations predict that neutrons will be stored in the crystal and released with a time structure determined by the vibration period and by the energy width of the incident beam. For a continuous beam, the usual time modulation of the diffracted neutrons with twice the vibration frequency is expected. If a neutron pulse much shorter than the vibration period impinges on the crystal, the transmitted signal should be a decaying sequence of pulses separated by the vibration period. For pulses which are long compared with the vibration period, the effect of neutron storage should be manifest as a delayed staircase-like intensity variation on both pulse edges. A silicon crystal was set with the (111) lattice planes in backreflection and excited in a λ/2 resonance at 44.78 kHz. A typical deformation amplitude was u0 = 1 μm corresponding to a scattering range of ΔE = ±1.9 μeV. The response of the λ/2 resonator to a quasicontinuous beam and to neutron pulses of lengths ΔtP = 3 ms and 33 μs FWHM was measured. Experiments were performed with neutron beams of two energy widths ΔE= ±0.35 and ±1.23 μeV. In agreement with the Monte Carlo simulations, neutron storage in the silicon crystal was observed. The storage time was 250 μs for both long and short incident pulses. This time equals about 11 vibration periods of the crystal resonator and is in good agreement with the calculations for the Si 111 reflection and chosen vibration parameters. A first indication of the dependence of the time structure on the energy width of the neutron beam was seen in the experiments. The predicted pulsed structure of the transmitted signal in response to neutron pulses shorter than the vibration period could not be resolved. With a shortest incident pulse width of ΔtP = 33 μs, the condition ΔtP ≪ TP was not fulfilled. The measured transmission profile is in good agreement with the calculations for the λ/2 resonator. Compared with vibrating crystals excited into higher harmonics at the same applied strain, the λ/2 resonator has a lower reflectivity.