  The most intensively star-forming galaxies are extremely luminous at
far-infrared (FIR) wavelengths, highly obscured at optical and ultraviolet
wavelengths, and lie at $z\ge 1-3$. We present a programme of ${\it Herschel}$
FIR spectroscopic observations with the SPIRE FTS and photometric observations
with PACS, both on board ${\it Herschel}$, towards a sample of 45
gravitationally lensed, dusty starbursts across $z\sim 1-3.6$. In total, we
detected 27 individual lines down to 3-$\sigma$, including nine $[\rm C{\small
II}]$ 158-$\mu$m lines with confirmed spectroscopic redshifts, five possible
$[\rm C{\small II}]$ lines consistent with their far-infrared photometric
redshifts, and in some individual sources a few $[\rm O{\small III}]$
88-$\mu$m, $[\rm O{\small III}]$ 52-$\mu$m, $[\rm O{\small I}]$ 145-$\mu$m,
$[\rm O{\small I}]$ 63-$\mu$m, $[\rm N{\small II}]$ 122-$\mu$m, and OH
119-$\mu$m (in absorption) lines. To derive the typical physical properties of
the gas in the sample, we stack all spectra weighted by their intrinsic
luminosity and by their 500-$\mu$m flux densities, with the spectra scaled to a
common redshift. In the stacked spectra, we detect emission lines of $[\rm
C{\small II}]$ 158-$\mu$m, $[\rm N{\small II}]$ 122-$\mu$m, $[\rm O{\small
III}]$ 88-$\mu$m, $[\rm O{\small III}]$ 52-$\mu$m, $[\rm O{\small I}]$
63-$\mu$m, and the absorption doublet of OH at 119-$\mu$m, at high fidelity. We
find that the average electron densities traced by the $[\rm N{\small II}]$ and
$[\rm O{\small III}]$ lines are higher than the average values in local
star-forming galaxies and ULIRGs, using the same tracers. From the $[\rm
N{\small II}]/[\rm C{\small II}]$ and $[\rm O{\small I}]/[\rm C{\small II}]$
ratios, we find that the $[\rm C{\small II}]$ emission is likely dominated by
the photo-dominated regions (PDR), instead of by ionised gas or large-scale
shocks.
