  We numerically and experimentally investigate the phononic loss for
superconducting resonators fabricated on a piezoelectric substrate. With the
help of finite element method simulations, we calculate the energy loss due to
electromechanical conversion into bulk and surface acoustic waves. This sets an
upper limit for the resonator internal quality factor $Q_i$. To validate the
simulation, we fabricate quarter wavelength coplanar waveguide resonators on
GaAs and measure $Q_i$ as function of frequency, power and temperature. We
observe a linear increase of $Q_i$ with frequency, as predicted by the
simulations for a constant electromechanical coupling. Additionally, $Q_i$
shows a weak power dependence and a negligible temperature dependence around
10$\,$mK, excluding two level systems and non-equilibrium quasiparticles as the
main source of losses at that temperature.
