A shock represents a discontinuity in fluid properties that occurs in a very thin region, meaning there are significant temperature and pressure gradients, where viscosity and dissipative effects are strongly felt. As a result, the process is not isentropic, and the relations in Equation 6 are no longer valid. As no heat is added or removed from the flow, the process is adiabatic. Equations 13, 14, 15, and 16 establish the properties of the flow across the normal shock through an adiabatic and irreversible process. M2 2 = 1 + γ−1 2 M2 1 γM2 1 − γ−1 2 (13) ρ2 ρ1 = (γ + 1)M2 1 2 + (γ − 1)M2 1 (14) P2 P1 = 1 + 2γ γ + 1(M2 1 − 1) (15) T2 T1 = [1 + 2γ γ + 1(M2 1 − 1)][ 2 + (γ − 1)M2 1 (γ + 1)M2 1 ] (16) Nonetheless, these equations admit that the flow before the shock can be subsonic (M<1). There- fore, the second law of thermodynamics must be taken into account through Equation 17. For an unitary Mach number, an isentropic (s2 − s1 = 0) infinitely weak normal shock is obtained, and for a subsonic Mach number, a negative variation of entropy is predicted, which is physically impossible. s2 − s1 = cpln(T2 T1 ) − Rln( P2 P1 ) (17) Since shocks are adiabatic processes, the stagnation temperature remains the same. Nevertheless, the stagnation pressure drops, and it is related to the entropy gain as shown in Equation 18. s2 − s1 = −Rln( P02 P01 )

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A shock wave represents a fundamental phenomenon in fluid dynamics - a thin discontinuity region where fluid properties undergo dramatic changes. Within this extremely narrow zone, we observe significant temperature and pressure gradients that create intense viscosity and dissipative effects. Unlike isentropic processes, shock waves are non-isentropic due to these irreversible effects, yet they remain adiabatic since no heat is added or removed from the flow. The upstream flow approaches with lower pressure, temperature, and density, while the downstream flow exhibits substantially higher values of these properties. The normal shock relations are governed by four fundamental equations that describe how fluid properties change across the shock discontinuity. Equation 13 determines the downstream Mach number M2 based on the upstream Mach number M1 and specific heat ratio gamma. Equation 14 gives the density ratio, showing how the fluid becomes more compressed downstream. Equation 15 provides the pressure ratio, indicating the pressure jump across the shock. Finally, equation 16 describes the temperature ratio, accounting for both pressure and density changes. These equations are interconnected and depend critically on the upstream Mach number M1 and the gas properties through gamma. Notice that for supersonic upstream flow, the downstream flow becomes subsonic, demonstrating the shock's ability to decelerate supersonic flow. The normal shock equations present a fascinating paradox when we examine subsonic upstream flow conditions. While equations 13 through 16 mathematically permit solutions for Mach numbers less than one, these solutions lead to a fundamental physical impossibility. When we calculate the entropy change using equation 17 for subsonic upstream conditions, we discover that the entropy change becomes negative, which directly violates the second law of thermodynamics. This graph illustrates the entropy change as a function of upstream Mach number. The red region shows where subsonic flow would produce negative entropy change, making such shocks physically impossible. At exactly Mach one, we have the limiting case of an infinitely weak shock with zero entropy change. Only in the green region, where upstream flow is supersonic, do we obtain positive entropy change, satisfying thermodynamic principles and confirming that normal shocks can only occur with supersonic upstream flow. The second law of thermodynamics provides the crucial constraint that resolves the subsonic shock paradox. Equation 17 allows us to calculate entropy change across the shock using temperature and pressure ratios. Let's examine three specific cases. For subsonic upstream flow with Mach 0.8, the calculated entropy change is negative 0.15R, violating the second law and confirming this scenario is physically impossible. At the critical Mach number of exactly 1.0, both temperature and pressure ratios equal unity, resulting in zero entropy change, representing an infinitely weak isentropic shock. For supersonic upstream flow with Mach 2.0, we obtain positive entropy change of 0.38R, satisfying thermodynamic requirements. This analysis demonstrates that only supersonic upstream conditions produce the necessary positive entropy increase, establishing Mach 1 as the minimum threshold for physically realizable normal shocks. Stagnation properties provide crucial insights into shock wave behavior and energy considerations. The stagnation temperature, which represents the temperature achieved when the flow is brought to rest isentropically, remains constant across the shock. This constancy reflects the adiabatic nature of the shock process - no heat is added or removed, so total energy is conserved. However, the stagnation pressure tells a different story. Unlike stagnation temperature, stagnation pressure decreases across the shock, and this decrease is directly related to the entropy increase through equation 18. The relationship shows that entropy gain equals negative R times the natural logarithm of the stagnation pressure ratio. This stagnation pressure loss represents the irreversible nature of the shock process and quantifies the reduction in available work that can be extracted from the flow. While total energy is conserved, the quality of that energy - its ability to do useful work - is degraded across the shock.