  Understanding the nature of charge density waves (CDW) in cuprate
superconductors has been complicated by material specific differences. A
striking example is the opposite doping dependence of the CDW ordering
wavevector in La-based and Y-based compounds, the two families where charge
ordering is strongest and best characterized. Here we report a combined
resonant soft X-ray scattering (RSXS) and neutron scattering study of charge
and spin density waves in isotopically enriched La$_{1.8-x}$ Eu$_{0.2}$
Sr$_{x}$ CuO$_{4}$ over a range of doping $0.07 \leq x \leq 0.20$. For all
dopings studied by RSXS, we find that the CDW amplitude is approximately
temperature-independent and develops well above experimentally accessible
temperatures. Surprisingly, the CDW ordering wavevector shows a non-monotonic
temperature dependence, with a sudden change occurring at temperatures near the
SDW onset temperature. We describe this behavior with a Landau-Ginzburg theory
for an incommensurate CDW in a metallic system with a finite charge
compressibility and CDW-SDW coupling. Our Landau-Ginzburg analysis suggests
that the ordering wavevector at high temperatures decreases with increased
doping. This behavior is opposite to the trend at low temperatures and highly
reminiscent of the doping dependence seen in YBa$_2$ Cu$_3$ O$_{6+\delta}$ ,
suggesting a common origin of the CDW in hole-doped cuprate superconductors.
