  The phonon dynamics of the low-temperature superconductor Sr$_{2}$RuO$_{4}$
is calculated quantitatively in linear response theory and compared with the
structurally isomorphic high-temperature superconductor La$_{2}$CuO$_{4}$. Our
calculation corrects for a typical deficit of LDA-based calculations which
always predict a too large electronic $k_{z}$-dispersion insufficient to
describe the c-axis response in the real materials. With a more realistic
computation of the electronic band structure the frequency and wavevector
dependent irreducible polarization part of the density response function is
determined and used for adiabatic and nonadiabatic phonon calculations. Our
analysis for Sr$_{2}$RuO$_{4}$ reveals important differences from the lattice
dynamics of $p$- and $n$-doped cuprates. Consistent with experimental evidence
from inelastic neutron scattering the anomalous doping related softening of the
strongly coupling high-frequency oxygen bond-stretching modes (OBSM) which is
generic for the cuprate superconductors is largely suppressed or completely
absent, respectively, depending on the actual value of the on-site Coulomb
repulsion of the Ru4d orbitals. Also the presence of a characteristic
$\Lambda_{1}$-mode with a very steep dispersion coupling strongly with the
electrons is missing in Sr$_{2}$RuO$_{4}$. Moreover, we evaluate the
possibility of a phonon-plasmon scenario for Sr$_{2}$RuO$_{4}$ which has been
shown recently to be realistic for La$_{2}$CuO$_{4}$. In contrast to
La$_{2}$CuO$_{4}$ in Sr$_{2}$RuO$_{4}$ the very low lying plasmons are
overdamped along the c-axis.
