The key here is that to produce gravitational waves, the supernova explosion must not have perfect spherical symmetry. Technically, what you need is an "accelerating" gravitational quadrupole moment.
It is quite unlikely that supernovae explosions will be symmetric. Asymmetries can be produced by binary companions, magnetic fields, rotation...
The characteristics of a "burst" source of gravitational waves will be quite unlike the signature of coalescing binary black holes. The magnitude of the signal is expected to follow the collapsing core of the star and reach a peak at something called the "core bounce", where the equation of state of the gas "stiffens" and a shock wave propagates outwards. The signal is thus likely to gradually increase in amplitude for some tens of milli-seconds, followed by an abrupt spike at the bounce, followed by a "ring-down" as the core (a proto-neutron star) settles down towards a spherically symmetric configuration over the course of a further 10 milli-seconds or so.
The frequencies of the waves will have a broad range governed by the characteristic frequency implied by $(Grho)^{1/2}$, which is the inverse of the freefall timescale, where $rho$ is the density at the time when the GWs are emitted. For a typical core collapse, the densities could range from $10^{14}$ to a few $10^{17}$ kg/m$^3$ over the course of the event, producing GW frequencies of tens to thousands of Hz - exactly in the range at which aLIGO is sensitive.
Because GWs pass through the envelope of the star unimpeded then the GW signature of a supernova probes the very heart of a supernova explosion and should be seen some hours before the visible signature is apparent. On the other hand any neutrino emission from a core collapse supernova should be detected at a very similar time to the GW waves. Any delay here could fix the currently poorly known neutrino mass.
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