We have computed the gravitational wave signal from supernova core collapse using the presently most realistic input physics available . We start from state-of-the-art progenitor models of rotating and non-rotating massive stars , and simulate the dynamics of their core collapse by integrating the equations of axisymmetric hydrodynamics together with the Boltzmann equation for the neutrino transport including an elaborate description of neutrino interactions , and a realistic equation of state . Using the Einstein quadrupole formula we compute the quadrupole wave amplitudes , the Fourier wave spectra , the amount of energy radiated in form of gravitational waves , and the signal-to-noise ratios for the LIGO I and the tuned Advanced LIGO ( “ LIGO II ” ) interferometers resulting both from non-radial mass motion and anisotropic neutrino emission . The simulations demonstrate that the dominant contribution to the gravitational-wave signal is produced by neutrino-driven convection behind the supernova shock . For stellar cores rotating at the extreme of current stellar evolution predictions , the core-bounce signal is detectable ( S / N \gtrsim 7 ) with LIGO II for a supernova up to a distance of \sim 5 kpc , whereas the signal from post-shock convection is observable ( S / N \gtrsim 7 ) with LIGO II up to a distance of \sim 100 kpc , and with LIGO I to a distance of \sim 5 kpc . If the core is non-rotating its gravitational wave emission can be measured with LIGO II up to a distance of \sim 15 kpc ( S / N \gtrsim 8 ) , while the signal from the Ledoux convection in the deleptonizing , nascent neutron star can be detected up to a distance of \sim 10 kpc ( S / N \gtrsim 8 ) . Both kinds of signals are generically produced by convection in any core collapse supernova .