Bernoulli's equation works beautifully at low speed — until the flow gets close to Mach 1. Then the physics changes qualitatively, not just quantitatively, and a shock wave appears.
At low speed, air is remarkably forgiving to model: treat its density as constant and standard incompressible aerodynamics — simple Bernoulli, in particular — gets you close enough for most purposes. That approximation holds up well below roughly Mach 0.3. Push the flow faster, and it stops holding. As local flow speed approaches and passes the local speed of sound, density changes become large enough that they can no longer be ignored, and the flow starts doing something incompressible theory has no way to describe at all: forming a shock wave.
In subsonic flow, pressure disturbances travel outward at the speed of sound in every direction, including forward — ahead of whatever is generating them. That lets the air "feel" an approaching obstacle and adjust smoothly before it arrives: streamlines curve gradually, pressure and density change continuously, and nothing changes abruptly. Once local flow speed exceeds the local speed of sound, that forward-warning mechanism breaks. A disturbance can no longer outrun the flow that's already moving faster than sound. The flow has no way to adjust smoothly ahead of time — so it adjusts nearly instantaneously instead, in an extremely thin region (often just a few molecular mean free paths thick) called a shock wave, where pressure, density, and temperature all jump discontinuously.
Every moving object continuously radiates small pressure disturbances outward at the speed of sound, like ripples from a pebble dropped in a pond. Below the speed of sound, the object is always slower than its own ripples — they spread out ahead of it, giving the surrounding air advance notice to start adjusting smoothly. Above the speed of sound, the object is moving faster than the ripples it just created. It outruns them. Instead of spreading freely ahead, the disturbances pile up on top of each other, stacking into a cone-shaped shock front that trails the object rather than warning the air ahead of time.
Ordinary subsonic pressure variation — around a car, over a low-speed wing, through a slow duct — is always smooth because the flow has, in effect, advance notice: pressure disturbances outrun it and let it adjust continuously. A shock wave exists specifically because that advance-notice mechanism fails once local flow speed exceeds the local speed of sound. The flow physically cannot spread the adjustment out over distance, because the very thing that would carry that adjustment forward — a pressure wave — cannot travel upstream fast enough to get there first. The result is a region only a few molecular mean free paths thick where pressure, density, and temperature complete their entire jump almost instantly. It is a genuinely different flow regime, not just a steeper version of the same one.
Incomplete, and misleading in practice. Compressibility effects start becoming significant well before Mach 1 — typically noticeable above about Mach 0.3, in what's called the transonic regime, which can begin even lower for some airfoil shapes. More importantly, shock waves can and do form locally over parts of an airfoil — commonly over the upper surface, where local flow accelerates well above the freestream speed — even while the aircraft's overall flight Mach number is still comfortably subsonic. That local supersonic pocket and its terminating shock is exactly what causes transonic wave drag rise: a real, measurable drag penalty that shows up before the aircraft itself ever reaches Mach 1, not at some single threshold speed.
Explains why incompressible aerodynamics (constant density, simple Bernoulli) breaks down as flow speed approaches the speed of sound, and why supersonic flow forms shock waves — extremely thin regions where pressure, density, and temperature jump nearly discontinuously — instead of adjusting smoothly the way subsonic flow does.
Below roughly Mach 0.3, density changes in the flow are small enough to neglect, so treating air as an incompressible fluid — constant density, simple Bernoulli's equation relating pressure and velocity — gives accurate results with far less mathematical complexity than full compressible flow analysis requires.
As local flow speed approaches and exceeds the local speed of sound, density changes become large and central to the flow's behavior — the flow physics changes qualitatively, not just quantitatively. A shock wave forms: an extremely thin region, often only a few molecular mean free paths thick, where supersonic flow abruptly transitions to a lower velocity with a sudden jump in pressure, density, and temperature, rather than the smooth, continuous variation seen in subsonic flow.
Pressure disturbances propagate at the speed of sound. In subsonic flow, they travel ahead of the object generating them, letting the surrounding air adjust smoothly and continuously in advance. In supersonic flow, the object moves faster than the disturbances it creates — it outruns its own pressure waves, which pile up into a cone-shaped shock front instead of spreading out ahead of time. That is why the flow adjustment becomes nearly instantaneous rather than gradual.
Noticeably above about Mach 0.3, in what is called the transonic regime — well before Mach 1. The exact onset depends on the geometry; some airfoil shapes see local compressibility effects even lower, especially where local flow accelerates well above the freestream speed.
Yes. Local flow over parts of an airfoil, typically the upper surface, can accelerate above the freestream Mach number even while the aircraft's overall flight speed is subsonic. If that local flow exceeds Mach 1 and then has to slow back down, a local shock wave can form and terminate that supersonic pocket — this is exactly what drives transonic wave drag rise before the aircraft itself reaches Mach 1.
Because its thickness is on the order of a few molecular mean free paths — physically about as abrupt as a change in a continuum fluid can be. That is fundamentally different from ordinary subsonic pressure variation, which changes gradually over distances vastly larger than the molecular scale.
The simple incompressible form (constant density) is wrong once density changes become significant, but a compressible form of the energy equation still applies — it just has to account for density and temperature changes along the flow, which is why compressible-flow analysis (isentropic relations, shock relations) becomes necessary instead of the simple constant-density version.
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