Supersonic flow at M = 3.0 encounters a wedge. The θ-β-M relation yields two mathematically valid shock angles for the given deflection. In the absence of a downstream constraint forcing a specific solution, which shock will form?
AThe strong shock, because stronger shocks are more stable under supersonic conditions
BThe weak shock, because nature minimizes entropy production wherever possible
CA standing normal shock, since the wedge geometry forces the flow to decelerate fully
DNeither solution is stable; a detached bow shock always forms at M = 3.0
Nature selects the weak shock (smaller wave angle β) in the vast majority of practical cases. The weak shock leaves the downstream flow closer to (or at) supersonic speed and corresponds to a lower entropy rise. A strong shock solution exists mathematically but requires a specific downstream pressure condition — such as a closed duct — to be forced. Detached bow shocks only form when the deflection angle exceeds the maximum possible for the given Mach number (δ > δ_max), not for all wedge geometries.
Question 2 Multiple Choice
In an oblique shock analysis, how is the pressure ratio across the shock calculated?
AUse normal shock tables directly with the full upstream Mach number M₁
BDecompose M₁ into normal and tangential components; apply normal shock relations to the normal component M_{n1} = M₁ sin β
CThe pressure ratio across an oblique shock equals one, since the tangential component dominates
DMultiply the normal shock pressure ratio by cos²β to account for the oblique geometry
This is the key insight: the tangential velocity component is unchanged across the shock (no pressure gradient drives it), so only the normal component experiences the shock transition. By substituting M_{n1} = M₁ sin β into the standard normal shock pressure ratio formula, you recover the correct oblique shock result. This decomposition reduces the oblique shock problem to a normal shock problem — all the normal shock relations and tables you already know apply directly to the normal Mach number.
Question 3 True / False
Flow through an oblique shock typically decelerates to subsonic speed, just as flow through a normal shock does.
TTrue
FFalse
Answer: False
This is a critical distinction between normal and oblique shocks. A normal shock always decelerates supersonic flow to subsonic speed. An oblique shock (specifically the weak solution) often leaves the downstream flow supersonic — it merely reduces the Mach number. The downstream Mach number M₂ = M_{n2} / sin(β − δ) can be greater than 1 for weak shocks. This is why oblique shocks are useful in inlet design: a series of oblique shocks can gradually decelerate supersonic flow while keeping it supersonic throughout, until a final, weaker shock transitions to subsonic.
Question 4 True / False
An oblique shock can be fully analyzed using normal shock relations by substituting the component of the upstream Mach number normal to the shock wave.
TTrue
FFalse
Answer: True
Yes — this is the fundamental insight of oblique shock analysis. Because the tangential velocity is unchanged across the shock (no tangential pressure gradient), the shock only 'sees' the normal component of the incoming flow. All normal shock relations (pressure ratio, temperature ratio, density ratio, downstream Mach number) hold exactly when M₁ is replaced by M_{n1} = M₁ sin β. The downstream quantities are then reconstructed using the geometry of the deflection angle δ. This decomposition makes oblique shocks tractable with only normal shock theory.
Question 5 Short Answer
Why do supersonic inlet designers use multiple oblique shocks to decelerate flow rather than a single normal shock, and what is the theoretical optimum?
Think about your answer, then reveal below.
Model answer: Each oblique shock produces less entropy rise (less total pressure loss) than a normal shock decelerating flow by the same total amount. By using a series of increasingly weaker oblique shocks, each one carries a smaller entropy penalty, so the total pressure recovery is higher than from a single strong normal shock. The theoretical optimum is infinitely many infinitely weak shocks — isentropic compression — which approaches zero entropy production. In practice, engineers use two to four oblique ramps to approximate this, balancing total pressure recovery against mechanical complexity.
Total pressure recovery (p₀_exit / p₀_inlet) is a key performance metric for supersonic inlets; every point of recovery translates to more thrust from the engine. A single normal shock at M = 2.5 recovers roughly 50–60% of total pressure. A two-shock oblique system can recover 80–90%. This is why supersonic military jets and supersonic transports use variable-geometry inlet ramps rather than flat perpendicular inlets — the aerodynamic efficiency gain is substantial.