When two compact objects, such as black holes or neutron stars, orbit each other and emit gravitational waves, the waves they produce do not create a gravitational wave singularity at the barycenter (the center of mass of the system). The strength of gravitational waves decreases with distance from the source, following the inverse square law.
Gravitational waves carry energy away from the orbiting objects, causing their orbits to gradually decay. As the objects lose energy, they spiral inward and the frequency and amplitude of the emitted gravitational waves increase. However, the waves do not become infinitely strong at the barycenter or any other point.
At the barycenter, the gravitational waves produced by the orbiting objects have a complex pattern that depends on the masses, velocities, and distances involved. The waves interfere with each other, resulting in a varying gravitational wave signal at different locations. However, this does not imply an infinite strength or singularity at the barycenter.
Gravitational waves are typically described in terms of strain, which is a measure of the fractional change in length caused by the passing wave. The strain decreases with distance from the source, so while the gravitational waves can be detectable and significant closer to the objects, they become weaker as one moves further away.
It's worth noting that gravitational wave singularities can arise in other contexts, such as the theoretical concept of a gravitational wave being strong enough to create a spacetime singularity. However, in the scenario of two orbiting compact objects emitting gravitational waves, there is no infinite strength or singularity associated with the waves at the barycenter.