The Wikipedia defintion of a shock wave pretty much sums up all I've found online about what a shock wave is:
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium (solid, liquid, gas or plasma) or in some cases in the absence of a material medium, through a field such as an electromagnetic field. Shock waves are characterized by an abrupt, nearly discontinuous change in the characteristics of the medium. Across a shock there is always an extremely rapid rise in pressure, temperature and density of the flow.... A shock wave travels through most media at a higher speed than an ordinary wave.
To me, however, this doesn't seem to provide a very rigorous definition that would allow me to look at a bunch of propagating disturbances and be able to clearly classify it as being a shock wave or (as Wikipedia puts it) a "normal" wave. Although this definition provides a qualitative definition of what sets a shock wave apart from a normal wave, I am wondering if there is a definite difference between a shock wave and normal waves that would allow me to definitively classify a wave as one or the other or if there is a continuous spectrum of wave properties between normal waves and shock waves with no clear boundary between the two (like the electromagnetic spectrum, with only arbitrary boundaries being drawn between the various classes of EM waves).