Waves in Classical Physics

If I say the word “wave” in no particular context, the image which most probably springs to your mind is one of a roughly periodic undulation on the surface of a pond or, perhaps, the sea. But the concept of a wave in classical physics is actually much more general, and in particular the periodicity is not really an essential aspect. At least in classical (i.e. pre-relativistic as well as pre-quantum) physics, a wave always occurs in (or on) some medium. Examples: surface waves on water, transverse waves on a plucked string, sound waves (pressure, or density waves in air). We will assume that there exists a state where the medium is undisturbed (string stretched straight and taut, water flat, air still . . . ) and until further notice we will choose an (initial) frame of reference in which the undisturbed medium is at rest. In general, we can describe a disturbance in the medium by specifying some quantity (call it generically q) which tells us how far the medium is displaced from its undisturbed (equilibrium) state at a given point at a given time. (By contrast, in the case of a particle we ask: what is its position at a given time? I.e. we write x=x(t).) Thus, for a stretched string, q would be the transverse displacement from the original straight configuration and would be a function of one space variable, the distancex along the string, and of time: