A Mathematical Analysis of the Wind Triangle Problem and an Inquiry of True Airspeed Calculations in Supersonic Flight

Abstract

In the first half of this paper, we present a fresh perspective toward the Wind Triangle Problem in aerial navigation by deriving necessary and sufficient conditions, which we call "go/no-go conditions", for the existence/non-existence of a solution of the problem. Although our derivation is based on simple trigonometry and basic properties of quadratic functions, it is mathematically rigorous. We also offer examples to demonstrate how easy it is to check these conditions graphically. In the second half of this paper, we use function theory to re-examine another problem in aerial navigation, namely, that of computing true airspeed — even in supersonic flight — from only three instrument readings obtainable from a basic flight instrument panel: calibrated airspeed, pressure altitude, and total air temperature. We present the first known mathematically rigorous analysis of the use of fixed-point iteration to compute true airspeed and Mach number from the Rayleigh Supersonic Pitot Equation. Our analysis comes with error estimates that are not found anywhere in the literature.