Tensor-scalar theory of gravity allows the generation of gravitational waves from astrophysical sources, like Supernovæ, even in the spherical case. That motivated us to study the collapse of a degenerate stellar core, within tensor-scalar gravity, leading to the formation of a neutron star through a bounce and the formation of a shock. We discuss in this paper the effects of the scalar field on the evolution of the system, as well as the appearance of strong non-perturbative effects of this

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... lar field (the so-called "spontaneous scalarization"). As a main result, we describe the resulting gravitational monopolar radiation (form and amplitude) and discuss the possibility of its detection by the gravitational detectors currently under construction, taking into account the existing constraints on the scalar field. From the numerical point of view it is worthy to point out that we have developed a combined code which uses pseudo-spectral methods, for the evolution of the scalar field, and High Resolution Shock-Capturing schemes, for the evolution of the hydrodynamical system. Although this code has been used to integrate the field equations of that theory of gravity, in the spherically symmetric case, a by-product of the present work is to gain experience for an ulterior extension to multidimensional problems in Numerical Relativity of such numerical strategy.