strict discontinuity [46,47]. Emile Jouguet, a pupil of Duhem, followed Rankine and pointed out that dissipation forces result in an exceedingly small width. If one disregards it, then one can speak of a discontinuity or a shock wave. Not only did Jouguet clarify Duhem's error, but he greatly contributed to an advance in the theory of shock waves and detonation waves [58,59,60]. Yet, to this day French authors, probably on account of Duhem's remarks, frequently speak of "quasi-waves", with a view on the finite width of the front.
Essentially we are dealing here with the general problem of the value and significance of approximate methods or approximate solutions in physics (see the remarkable paper by V. A. Foch [29]). This involves also the question as to when as approximate realization of some formulas or relations justifies the creation of new qualitative concepts.
Rankine also touches upon the problem of expansion waves, and refers to an oral communication by Thomson according to which an expansion wave must be mechanically unsteady. In point of fact, however, Rankine already implies the impossibility of a expansion wave (and not its unsteadiness or instability). In fact, if we study the processes of thermal conductivity inside the wave then, besides the conservation equation written by Rankine, , which states that in a process of thermal conductivity the amount of heat received by one layer's equal to the amount of heat released by other layers, we must take account, at least qualitatively, of the elementary fact according to which in the process of thermal conductivity heat always passes from a hotter body to a cooler one. Hence, of course, we get that in a shock wave entropy can only increase. Thus, were we to type to plot a expansion wave by inverting in a shock wave all the velocities, then inside the shock wave front, inside the "discontinuity" we would also run into the necessity of inverting the heat flow and achieve a transfer of heat from cooler gas layers to hotter ones -- which is impossible. We cannot but regret that these elementary considerations are sometimes ignored even in the contemporary literature (see Chapter 1 of Vlasov's book [3], which is otherwise quite valuable).
