as nozzles and pipes. Again, gas dynamics becomes indispensable if and when the velocity of the gas attains or exceeds the speed of sound. In this field, the nature of the flow, and the dependence of velocity and flow rate on pressure drop, are subject to qualitative changes. This group of problems is of great significance for the theory of turbines, jet engines and missiles.
A peculiar field of gas dynamics based on the consideration of the compressibility of the moving medium is the teaching on sound - acoustics. The velocity of the medium and the amplitude of pressure changes under the effect of sound are very small. Nevertheless, consideration of compressibility becomes indispensable when studying the initial stages of any motion, and when studying rapidly changing, especially periodical motion.
Shock waves are of particular interest from various points of view, and they will be one of the main subjects of the present book. On the one hand, wherever the attempts of integrating equations without introducing discontinuities (i.e., shock waves) lead to paradoxes which make it impossible to solve these equations, the theory of shock waves eliminates the paradoxes and makes it possible to design a regime of motion under any conditions.
On the other hand, the shock waves themselves are a paradoxical phenomenon. They are paradoxical in that, without introducing any assumptions regarding dissipative forces (viscosity and thermal conductivity), from elementary considerations we can derive shock wave laws which include the increase in entropy, i.e., laws which include the irreversibility of the processes occurring in shock waves.
From this point of view shock waves afford a considerable logical and scientific interest, irrespective of their application.
It is worth noting that all basic relations and fundamental concepts have been established from the study of the general equations of gas dynamics some 50 years ago, at a time, that is, when there existed no experimental material, and long before shock waves were investigated by researchers.
As Emile Jouguet once said in a very poignant figure of speech, "the shock waves
