The Speed of Sound, a = √(γRT): Why Only Temperature Matters | AE 302

Clap your hands at one end of a long tube of air and the news races away at 340.2 metres a second. Take the same tube up to ten kilometres and the same clap manages only 299.5. Same tube, same gas: the air up there is thinner and the pressure far lower, and by the end of this video both of those suspects have been struck out. The derivation needs three ingredients and one borrowed number. We ride with the wave — hop onto the front and, instead of a wave running through still gas, you have gas rushing at you at speed a, getting squeezed as it passes, and leaving a touch slower, a touch denser, at a touch more pressure. Mass in equals mass out gives ρ da = a dρ. Newton's second law, rewritten as mass-per-second times change-in-speed, gives dp = ρa da. Put one into the other and a² = dp/dρ: the speed of sound squared is a slope — how hard the gas pushes back when you squeeze it. Then the catch the whole video exists for. Squeeze it how? Squeeze a gas slowly and let every scrap of heat leak out, and the pressure climbs a gentle curve. Slam it, so the heat has nowhere to go, and the gas warms as it compresses and climbs a curve steeper by exactly γ. A sound wave is feeble, so nothing about it is wasteful — reversible. Its front is thin and gone in a thousandth of a second, so no heat can cross it — adiabatic. Reversible plus adiabatic is isentropic — and that subscript s is worth about fifty metres a second. From there it is one line you already earned, p v^γ = constant, plus the power rule, plus pv = RT — and the slope becomes γRT. So a = √(γRT). Read what is in that formula and, more importantly, what is not: no density, no pressure. Both suspects innocent. Then the video makes that believable. Squeeze a box of gas at fixed temperature and the pressure doubles — but so does the density, and the ratio the formula cares about never moves. Temperature is how fast the molecules are flying: their typical speed is √(3RT), about 500 m/s at 288 K, the one number borrowed from kinetic theory rather than derived here. Sound, handed from molecule to molecule, comes out at √(γ/3) of it — about 68%, at any temperature. Pack the molecules closer and each hand-off happens more often but covers less ground; the two effects cancel. Then the arithmetic, and Newton. γR = 401.8 for air, fixed forever; times 288 K, square-rooted, gives 340.2 m/s. And since √401.8 is almost exactly 20, the speed of sound in air is about 20√T. Newton did this calculation in 1687 and got it wrong: he assumed the slow, isothermal squeeze, so his answer was √(RT) = 287.5 m/s — 15.5% low. Gunners timing cannon fire across a valley kept measuring a bigger number than the greatest physicist alive had predicted, and kept measuring it for 130 years. The whole error is one letter s: √(RT) over √(γRT) is 1/√γ = 0.845, precisely the gap they were seeing. It took Laplace to put it back. And the answer to the opening question: the news is slower at ten kilometres not because the air is thin but because it is cold — about −50 °C up there against about +15 °C down here. Mach number is a ratio, not a speed: the same aeroplane at the same 250 m/s airspeed sits at M 0.735 at sea level, 0.780 at 5 km and 0.835 at 10 km. The yardstick shrank. Sound is how a gas warns itself, so an aeroplane that outruns its own warning gets a shock wave instead — which is where the course goes next. Chapters: 0:00 The same clap, at sea level and at ten kilometres 0:17 Two suspects: thinner air and lower pressure 0:34 The plan: three ingredients and four results 0:55 Sound is not a wind — what actually moves 1:32 The stadium wave: the pattern travels, the crowd stays 1:57 The shove is feeble, and what a little d means 2:39 Standing on the front: gas in, gas out 3:51 Ingredient one: mass in equals mass out 5:44 Ingredient two: F = ma for a stream of gas 7:25 Putting the two together 8:05 The speed of sound squared is a slope 8:26 But squeeze it how? Slow versus slammed 9:44 Weak and fast — so the squeeze is isentropic 10:52 Ingredient three: p v^γ, and the power rule 12:13 The perfect gas law, and earning the factor γ 12:57 a = √(γRT) — and what is not in it 14:13 Why p and ρ cancel: molecules as messengers 16:01 The numbers: 340.2 m/s, and the 20√T shortcut 17:19 Newton, 1687, and the 15.5% he was missing 18:24 The answer: not thin air, cold air 18:41 Mach number is a ratio, not a speed 19:45 Sound is how a gas warns itself — and the Mach cone Part of AE 302 High-Speed Aerodynamics, Chapter 8, Aerospace Engineering, San Diego State University. Animated in manim. The numbers on screen — 340.2, 299.5, 401.8, 287.5, 0.845, the Mach numbers — are computed from the verified data for this chapter, not asserted. One thing is out of scale on purpose, and the video says so: da, the speed the gas itself picks up, is drawn far larger than life; at true scale it would be thinner than a pixel.