There are two different waves of the gravitational field (GF):
1. Gravitational field transverse wave. What it reflects is the disturbance of the surrounding GF, and the transmission speed of this disturbance is equal to the slow light speed c. For example, the motion of the sun disturbs the GF generated by the center of the galaxy, causing transverse waves of GF around the sun.
2. Gravitational field longitudinal wave. It is generated by the gravitational source itself, and GF will transfer energy quickly, and this speed is much greater than the speed of light c.
When the gravitational source changes (position, mass), this change will first be reflected on the longitudinal wave of GF, and distant objects will feel the change of GF soon. At the same time, the disturbance of the gravitational source to other surrounding GFs will propagate to the surroundings at a slow speed c in the form of transverse waves.
To make an inappropriate analogy: when you throw a stone into a calm lake, you will observe slow water waves spreading around, which is the disturbance of the stone to the water surface, thus generating water waves. But in addition to water waves, there are sound waves in the water. The speed of the sound waves is much faster than that of the water waves, and the sound waves in the water arrive long before the slow water waves reach the shore.
A brief summary: the longitudinal wave of GF is produced by the gravitational source itself, and the transverse wave of GF is produced by the disturbance of the gravitational source to other surrounding GFs.
Newtonian gravity studies "sound waves in water", the longitudinal waves of GF.
Einstein GR studies the "water wave", that is, the transverse wave of GF.
I hope that you can understand the whole gravity from my simple narrative. You can also download my two papers on gravity here:
https://www.researchgate.net/publication/364635795_The_Effect_of_Gravitational_Waves_on_Planetary_Orbits
https://www.researchgate.net/publication/364262469_Exploring_the_Accelerating_Expansion_of_the_Universe
Kind regards,
Tony