  Context: There is a need to improve the fidelity of SPH simulations of
self-gravitating gas dynamics. Aims: We remind users of SPH that, if smoothing
lengths are adjusted so as to keep the number of neighbours, ${\cal N}$, in the
range ${\cal N}_{_{\rm NEIB}}\pm\Delta{\cal N}_{_{\rm NEIB}}$, the tolerance,
$\Delta{\cal N}_{_{\rm NEIB}}$, should be set to zero, as first noted by Nelson
& Papaloizou. We point out that this is a very straightforward and
computationally inexpensive constraint to implement. Methods: We demonstrate
this by simulating acoustic oscillations of a self-gravitating isentropic
monatomic gas-sphere (cf. Lucy), using ${\cal N}_{_{\rm TOT}}\sim6,000$
particles and ${\cal N}_{_{\rm NEIB}}=50$. Results: We show that there is a
marked reduction in the rates of numerical dissipation and diffusion as
$\Delta{\cal N}_{_{\rm NEIB}}$ is reduced from 10 to zero. Moreover this
reduction incurs a very small computational overhead. Conclusions: We propose
that this should become a standard test for codes used in simulating star
formation. It is a highly relevant test, because pressure waves generated by
the switch from approximate isothermality to approximate adiabaticity play a
critical role in the fragmentation of collapsing prestellar cores. Since many
SPH simulations in the literature use ${\cal N}_{_{\rm NEIB}}=50$ and
$\Delta{\cal N}_{_{\rm NEIB}}\geq10$, their results must be viewed with
caution.
