The mathematical approach to the sonic barrier
Cathleen Synge Morawetz · Bulletin of the American Mathematical Society · 1982
MORAWETZ 1. Introduction.For my topic today I have chosen a subject connecting mathematics and aeronautical engineering.The histories of these two subjects are close.It might however appear to the layman that, back in the time of the first powered flight in 1903, aeronautical engineering had little to do with mathematics.The Wright brothers, despite the fact that they had no university education were well read and learned their art using wind tunnels but it is unlikely that they knew that airfoil theory was connected to the Riemann conformai mapping theorem.But it was also the time of Joukowski and later Prandtl who developed and understood that connection and put mathematics solidly behind the new engineering.Since that time each new generation has discovered new problems that are at the forefront of both fields.One such problem is flight near the speed of sound.This one in fact has puzzled more than one generation.Everyone knows that in popular parlance an aircraft has "crashed the sonic *or sound barrier" if it flies faster than sound travels; that is, if the speed of the craft q exceeds the speed of sound c or the Mach number, M = q/c > 1.In simple, if perhaps too graphic, terms this means that if you are in the line of flight the plane hits you before you hear it.But it is not my purpose to talk about flight at the speed of sound.The mathematical and engineering problems today are centered around a different range called the transonic range where M is roughly between 0.6 and 0.9.We shall see that this is where the real sound barrier lies.But the main reason for working in this range is that we can expect, because of the energy shortage, to fly for a good many years at those speeds.We need to understand them and find the best way to do it.In the first part of my talk I want to explain where the energy benefits come from and something of the history of this type of flight.After that, I shall turn to the mathematics.People have always dreamed of flying.In ancient times-like the birds.In modern times like Buck Rogers.For a long time we have known that if you have a big enough engine, creating lots of thrust, you can fly anywhere and theoretically at any speed.Like Buck Rogers you do not need wings.It costs money but it can be done.But back in the Wright brothers' day there was noThe Josiah Willard Gibbs lecture was presented at the 783rd meeting of the American