  The mixing of orbital and spin character in the wave functions of the $5d$
iridates has led to predictions of strong couplings among their lattice,
electronic and magnetic degrees of freedom. As well as realizing a novel
spin-orbit assisted Mott-insulating ground state, the perovskite iridate
Sr$_{2}$IrO$_{4}$ has strong similarities with the cuprate La$_{2}$CuO$_{4}$,
which on doping hosts a charge-density wave that appears intimately connected
to high-temperature superconductivity. These phenomena can be sensitively
probed through momentum-resolved measurements of the lattice dynamics, made
possible by meV-resolution inelastic x-ray scattering. Here we report the first
such measurements for both parent and electron-doped Sr$_{2}$IrO$_{4}$. We find
that the low-energy phonon dispersions and intensities in both compounds are
well described by the same nonmagnetic density functional theory calculation.
In the parent compound, no changes of the phonons on magnetic ordering are
discernible within the experimental resolution, and in the doped compound no
anomalies are apparent due to charge-density waves. These measurements extend
our knowledge of the lattice properties of (Sr$_{1-x}$La$_{x}$)$_{2}$IrO$_{4}$
and constrain the couplings of the phonons to magnetic and charge order.
