No, the presence of compact objects orbiting each other does not produce a gravitational wave singularity at the barycenter. Gravitational waves are generated by the acceleration of massive objects that have asymmetrical motion, such as orbiting or merging compact objects like black holes or neutron stars.
In an orbiting binary system of compact objects, such as black holes or neutron stars, the gravitational waves are emitted as the objects move in their orbits and experience acceleration. As the objects orbit each other, they produce gravitational waves that carry away energy and angular momentum from the system. This energy loss causes the objects to gradually spiral inward and eventually merge.
At the barycenter, which is the center of mass of the binary system, the gravitational waves are not infinitely strong. The strength of gravitational waves diminishes with distance from the source, following the inverse square law. The waves become weaker as they propagate away from the source, just like how the intensity of any radiation decreases with distance.
While the gravitational waves may be strongest in the immediate vicinity of the compact objects, they do not form a singularity at the barycenter. Instead, the gravitational waves radiate outward from the system, carrying energy and momentum with them. The waves can be detected and measured by sensitive instruments on Earth, such as LIGO (Laser Interferometer Gravitational-Wave Observatory), which have successfully observed gravitational waves from merging black holes and neutron stars.
It's important to note that gravitational waves themselves are a prediction of Einstein's theory of general relativity and have been indirectly confirmed through various observations. They do not involve singularities like those found in the centers of black holes.