  Detecting neutral hydrogen structures in the intergalactic medium (IGM)
during cosmic reionization via absorption (21 cm forest) against a background
radiation is considered independent and complementary to the three-dimensional
tomography and power spectrum techniques. The direct detection of this
absorption requires very bright ($\gtrsim 10$-100 mJy) background sources at
high redshifts ($z\gtrsim 8$), which are evidently rare; very long times of
integration; or instruments of very high sensitivity. This motivates a
statistical one-dimensional (1D) power spectrum approach along narrow
sightlines but with fainter background objects ($\sim 1$-10 mJy), which are
likely to be more abundant and significant contributors at high redshifts. The
1D power spectrum reduces cosmic variance and improves sensitivity especially
on small spatial scales. Using standard radiative transfer and fiducial models
for the instrument, the background sources, and the evolution of IGM structures
during cosmic reionization, the potential of the 1D power spectrum along
selected narrow directions is investigated against uncertainties from thermal
noise and the chromatic synthesized point spread function (PSF) response.
Minimum requirements on the number of high-redshift background sources, the
telescope sensitivity, and the PSF quality are estimated for a range of
instrumental, background source, and reionization model parameters. The 1D
power spectrum is intrinsically stronger at higher redshifts. A $\sim 1000$ hr
observing campaign targeting $\sim 100$ narrow sightlines to radio-faint,
high-redshift background objects with modern radio telescopes, especially the
Square Kilometre Array, can detect the 1D power spectrum on a range of spatial
scales and redshifts, and potentially discriminate between models of cosmic
reionization.
