The concept of wave revisited

The concept of wave in the various fields of physics ranging from classical mechanics to general relativity, together with the physical reality and the mathematical structure for the physical quantity of concern in given context, the latter including both the governing equation and the function for the description of wave, are revisited. Wave has been taken both as the form of motion and as the form of being itself, yet it may be essentially just the form of a suitable mathematical tool that we can offer.