Waves are characterized as spatially propagating phenomenon. Maybe you mean oscillation? Sure time varying values exist. What work is the word inherently doing? An how does that relate to consciousness?
The concept of propagation depends on the concept of duration. The point is to demonstrate an entity that is constituted by duration. The relation to consciousness is that some take it to be also inherently constituted by duration. In other words, consciousness is inherently a process/dynamic. E.g. if you suspend the movement of molecules in a brain such that it contains no dynamics, is the brain undergoing any conscious experience during this period of suspended animation? Some think the answer is no.