The sound barrier and the Mach cone

On 14 October 1947 Chuck Yeager flew the Bell X-1 to Mach 1.06 at 43,000 feet over Muroc Dry Lake. Nothing broke. In Anderson's account, "the flight was smooth; there was no violent buffeting of the airplane and no loss of control as feared by some engineers." The sound barrier was never a wall in the air. What does exist past Mach 1 is a cone of pressure waves, and one short equation gives its angle.

Where the wall came from

Anderson dates the myth to 1935. The British aerodynamicist W. F. Hilton was showing a newsman a plot of wing drag and said: "See how the resistance of a wing shoots up like a barrier against higher speed as we approach the speed of sound." The next morning the leading British papers were writing about "the sound barrier".

The real problems were compressibility effects. Air speeds up as it flows over a wing, so part of that flow reaches the speed of sound before the aircraft does. On Yeager's flight a pocket of supersonic flow formed over the wing at Mach 0.87. A shock standing almost straight up from the wing closed off the back of that pocket, and Anderson calls this shock "the culprit which made flight through Mach one such a harrowing concern." Drag rises steeply just past the point where the flow first goes sonic. Most engineers knew there was no wall. What they did not know was how far the drag would rise, and the engines of the day had little thrust to spare. Anderson also notes that beyond Mach 0.85 there was no wind-tunnel data on transonic flight in 1947.

Why a cone, and why that angle

The Mach number is "the ratio of the speed of a gas to the speed of sound in that gas" (Anderson). Picture the aircraft as a point that sends out a faint pressure pulse at every instant. Each pulse spreads as a sphere at the speed of sound, a, from where the aircraft was when it sent the pulse. Seitzman's lecture slides work out the geometry from there.

Take a pulse sent a time t ago. Its sphere now has radius at, and the aircraft, flying at speed v, is a distance vt from that sphere's centre. If v is less than a, the sphere has overtaken the aircraft, so the waves run on ahead of it. If v is greater than a, the aircraft is outside every sphere it has made. Draw a line from the nose that just touches one sphere. It meets the radius there at a right angle, so the sine of its angle to the flight path is at/vt. The time t cancels, so one line touches every sphere. Turn that line around the flight path and you have the cone. Its half-angle μ is the Mach angle:

sin μ = a/v = 1/M,  so  μ = sin⁻¹(1/M)

NACA Report 1135 defines the Mach angle in exactly this form and lists it against Mach number in its Table II. Carlson's sonic-boom paper gives the same definition. Seitzman calls the inside of the cone the zone of action and the outside the zone of silence. A listener outside the cone has not yet been reached by any wave the aircraft has made. When the cone sweeps over the ground, that listener hears the sonic boom.

What the numbers say

At Mach 1 the sine is 1 and the angle is 90°. The cone has opened into a flat front across the nose, where the waves bunch up. At Yeager's Mach 1.06 the angle is 70.63°, so his cone was still very wide. At Mach 2 it is 30°, and at Mach 3 it is 19.47°. We computed these, and they match the rows of NACA Table II.

A Mach number is not a fixed speed, because the speed of sound depends on the temperature of the air. Anderson records Hugh Dryden's round figure for the standard sea-level speed of sound: 1,117 feet per second, or about 762 miles per hour (our arithmetic). Anderson gives Yeager's speed as 700 miles per hour, and it was Mach 1.06 only because the air at 43,000 feet is cold. Dividing 700 by 1.06 puts the speed of sound up there at about 660 miles per hour. At sea level, 700 miles per hour would be about Mach 0.92, still below the speed of sound (our arithmetic).

What the equation leaves out

The Mach angle holds for weak waves far from the aircraft. Close to the nose, the real bow shock is stronger. Anderson describes it as "curved and more oblique to the flow" than the near-vertical shock over the wing. Near the nose it stands steeper than μ, and it bends toward the Mach angle farther out, where it weakens. The boom is also not one bang at the moment an aircraft passes Mach 1. Maglieri and his co-authors describe a primary carpet of booms that is dragged along the ground for the whole supersonic flight. Only focus booms, made while accelerating past Mach 1 or in a sharp turn or dive, are in their words "a one-time occurrence".

In short

The sound barrier was a steep rise in drag and a set of shock problems near Mach 1, not a wall. Past Mach 1 an aircraft outruns its own waves, and they gather into a cone with sin μ = 1/M. That gives 90° at Mach 1, 70.63° for Yeager and 30° at Mach 2.