pretty close to the real one, and so is the general picture of motion which he discovered. However, Riemann's equations do not fully satisfy the law of energy conservation. Hence we have to regard them as erroneous.
It is interesting that in the 1925 edition of the well-known book "Partial Differential Equations in Mathematical Physics", compiled by Weber on the basis of Riemann's lectures [97], even after the problem had been entirely clarified, he (Weber) expresses peculiar doubts as to whether or not Riemann's equations may still hold when considering disturbance.
The conclusion by Hugoniot [56], with whose name Eq. (VIII-7) is usually associated, has been dealt with in the preceding Chapter.
We shall now take a look at Rankine's book [78], which is most interesting from the viewpoint of physical gas dynamics because the author has a deep understanding of the phenomena occurring in a shock wave.
Rankine examines a motion which could propagate ad infinitum without changing its form, i.e., he studies a disturbance that propagates steadily in a gas. He establishes two control planes (like we did when deriving Hugoniot's adiabatic curve) and sets up the law of conservation of matter and the law of conservation of momentum. Rankine studies a substance which has thermal conductivity but no viscosity. He formulates principles of self-modelling which are of the utmost importance for shock waves. Specifically, he emphasizes that numerically the coefficient of thermal conductivity of a substance may be infinitesimal, but we may not neglect it in a shock wave because the width of a shock wave as well as the magnitude of the gradients are not pre-assigned. The smaller the coefficient of thermal conductivity, the greater we may expect the gradients to be in a shock wave, so that the product of the temperature gradient times the coefficient of thermal conductivity (equal to the amount of heat transferred by thermal conductivity in a unit of time) can remain finite as the coefficient itself approaches zero. This makes us thoroughly understand when we can ignore dissipation forces, in particular thermal conductivity, which is when the magnitude of the gradients is preassigned by the equations of motion without thermal conductivity. It also makes us thoroughly understand why we cannot ignore thermal conductivity when the magnitude of the gradient
