The Machmeter — Page 92, Lesson 107

Let’s pick this up right where the Machmeter chapter gets into the real meat: what happens to Mach number, TAS, and CAS as you climb and descend through different temperature layers. First, I want to make sure we’re all speaking the same language. LSS is the local speed of sound. TAS is true airspeed — the actual speed of the aeroplane through the air. CAS is calibrated airspeed — what the airspeed indicator shows after instrument and position error corrections, but before density error is corrected. And Mach number is simply TAS divided by LSS. That ratio is the heart of everything we’re about to do. Now, the chapter walks us through two special atmospheric situations: an isothermal layer and an inversion. An isothermal layer is a layer of air where the temperature does not change with altitude. Because temperature is constant, the LSS does not change either. So if you hold a constant Mach number, the TAS will not alter — it stays the same. But the CAS will change, and it changes because of density error. In a climb through an isothermal layer at constant Mach, CAS reduces. In a descent at constant Mach, CAS increases. So even though the aeroplane’s true speed is constant, the indicated calibrated speed drifts because the air density is changing. Now, what if you climb at a constant CAS through an isothermal layer? Then both TAS and Mach number increase, and they increase at the same rate. That’s the clean, simple case. Let’s move to the inversion. An inversion is a layer where the air gets warmer as altitude increases. So in a climb, the LSS increases — sound travels faster in warmer air. If you hold a constant Mach number in a climb through an inversion, the TAS must increase, because Mach number equals TAS divided by LSS, and if LSS is going up, TAS has to go up to keep the ratio constant. Meanwhile, CAS will reduce as air density reduces. On the descent at constant Mach, the TAS reduces and the CAS increases. What about climbing at a constant CAS through an inversion? Then both TAS and Mach number increase, but here’s the key contrast: TAS increases at a greater rate than Mach number. That’s because the LSS is also increasing, so the Mach number, which is TAS divided by LSS, doesn’t rise as fast as TAS alone. Now the chapter gives us a summary of these relationships, and I want you to hold onto these as rules of thumb: - TAS will always increase when an aeroplane climbs at a constant CAS. - Climbing at a constant TAS, the CAS will always reduce. The reason given is that pressure has a greater effect on air density than temperature. - Climbing at a constant CAS, the Mach number will always increase. - Climbing at a constant Mach number, the CAS will always reduce. And the reason here is that the CAS/TAS density error dominates over the change in LSS due to temperature variation. So the density error is the big player — it swamps the temperature effect on the speed of sound. Now let’s put this into practice with the example problems the chapter works through. These are the kind of calculations you’ll be expected to do. Problem 1: What is the speed of sound at FL380 in ISA conditions? In the ISA atmosphere, FL380 is above the tropopause, so the temperature is a constant -56.5°C, which is 216.5 Kelvin. The formula for LSS is: LSS = 38.95 × √T where T is the temperature in Kelvin. So we plug in: LSS = 38.95 × √216.5 That gives us 573 knots. The chapter also mentions you can do this on the navigation computer. You place the Mach number index arrow against the temperature in °C, locate M 1.0 — that’s the blue 10 on the navigation computer — on the inner Mach number scale, and read off the TAS on the outer scale. Problem 2: Determine the TAS corresponding to M 0.70 at JSA MSL, which is +15°C or 288 K. Using the computer, you set the Mach number index against +15°C in the Airspeed window, then against 7 (for M 0.7) on the inner scale, read off 463 knots on the outer scale. Alternatively, calculate it directly: TAS = Mach number × LSS TAS = 0.7 × 38.95 × √288 LSS at 288 K is 38.95 × √288, which is 661 knots. So: TAS = 0.7 × 661 = 463 knots. Problem 3: Calculate, without a computer, the altitude in the JSA atmosphere at which a TAS of 450 knots corresponds to Mach 0.80. We start from the definition: Mach Number = TAS / LSS So: LSS = TAS / Mach Number LSS = 450 / 0.8 = 562.5 knots Now we use the LSS formula: LSS = 38.95 × √T So: √T = LSS / 38.95 = 562.5 / 38.95 = 14.44 Square that to get T: T = 14.44² = 209° Absolute, which is -64°C. And the chapter tells us that -64°C occurs at FL395 in the JSA, which has no tropopause — so the temperature keeps dropping with altitude all the way up. Problem 4: If a decrease of 0.12 in the Mach number results in a decrease of 80 knots in the TAS, what is the local speed of sound? Again, we use the definition: Mach Number = TAS / LSS So LSS = TAS / Mach Number. The change in Mach is 0.12, the change in TAS is 80 knots. Since LSS is constant in this scenario, the ratio of the change in TAS to the change in Mach gives us LSS: LSS = 80 / 0.12 = 666.7 knots. So the local speed of sound is about 667 knots. That’s the full sweep of the Machmeter’s behaviour through temperature layers, plus the calculation techniques you’ll use on the navigation computer and by hand. The key takeaway is always the same: Mach number is a ratio of TAS to LSS, and temperature drives LSS, while density error drives the difference between CAS and TAS.