Waves, in general, do not move with the Earth itself. Waves are disturbances that propagate through a medium, whether it's a solid, liquid, or gas. The motion of waves is relative to the medium in which they travel, not the motion of the Earth.
For example, if you observe ocean waves at the beach, you'll notice that the waves move towards the shore and crash on the beach. However, the movement of the waves is primarily due to the energy transferred from wind to the water surface. The waves themselves are not directly influenced by the Earth's motion or rotation.
Similarly, other types of waves, such as sound waves or light waves, propagate through air or other mediums, and their motion is determined by the properties of the medium itself and the source that generates them. The Earth's rotation or movement does not directly affect the propagation of these waves.
However, it is worth noting that the Earth's rotation and motion can have indirect effects on waves in certain situations. For example, the rotation of the Earth causes the Coriolis effect, which can influence the behavior of large-scale atmospheric and oceanic waves. Additionally, the relative motion of the Earth and an observer can affect the perceived frequency or wavelength of waves due to the Doppler effect. But these effects are not inherent to the waves themselves and depend on the specific circumstances.