This research is concerned with the mysteries of neutron stars and the quest for gravitational waves. Neutron stars are anticipated sources of gravitational waves, and are expected to be detectable within the next decade using kilometre-scale laser interferometry especially the aLIGO. We estimate the range of amplitudes that the waves may have if a neutron star spirals inside a giant star in the end phase of binary evolution. The signal of the calculated values with a peak gravitational-wave strain of h ~ 5.749 x 10−22 – 3.992 x 10-22 matches the strain amplitude sensitivity range h ~ 10-23 - 10-18 of LIGO and VIRGO for the inspiral and merger of a pair binary systems. The results obtained in this research as compared to aLIGO’s values of GW detection has proved beyond reasonable doubt that NS are promising candidates; which imply that there are more neutron stars (binary systems) out there than expected.