My apologies to nervous fliers (believe it or not some read this blog), but there are lots of things in aviation with morbid names. We do take our jobs seriously so when we talk of a "graveyard spiral" or the "coffin corner" it's because we know the associated risks. They aren't theoretical risks. We know the names and families of people who have taken those risks and lost. But, that said, the term coffin corner summons to my mind the shape of a graph, not of the box that probably wouldn't even be needed to dispose of the remains of those who disregarded the meaning of that graph.
The graph in question shows airspeed on the x-axis, the horizontal, and altitude on the y-axis, the vertical. It depicts two lines, one showing the minimum flying speed and the other showing the maximum flying speed. I'll work up to explaining their slope and intersection, but I'll explain stall speed, and Mach in order to get there.
An airplane is supported in flight by the pressure difference that develops between the upper and lower surfaces of the wing because of its forward motion through the air. The slowest forward speed that develops sufficient pressure difference to counteract the weight of the airplane is called the stall speed. Attempt to fly slower than that and the airflow over the top of the wing starts to break away, resulting in ineffective controls, altitude loss, and usually (but not necessarily) a nose down pitch. This is undesirable in air transport operations, so pilots try to maintain a good margin above the stall speed.
As the air becomes thinner with altitude or high temperature, the speed the airplane must travel through the air mass to get the same airflow over the wing increases. Two days ago I said that that same pressure over the wing represents constant indicated airspeed. That is true for slow airplanes (200 kts is slow in this context!), because for them we can ignore the effect of compressibility of air. When your airplane gets above 10,000 feet and 200 kts, that is no longer true. The ram effect of the speed compresses air inside the pitot tube, causing a higher indication than would otherwise be seen. You can subtract a correction factor to get equivalent airspeed (EAS), but even quoted in EAS stall speed does not remain constant with altitude because of the changes that occur near the speed of sound. So when we are talking about high speed, high altitude aircraft, the speed at which the aircraft stalls actually increases with altitude, whether the speed is given as indicated, equivalent or true. That's the first line on the graph: stall speed. Starting from sea level at the bottom, it slopes slightly towards the right, more so at higher altitudes.
Speed of Sound
The speed of sound in air depends pretty much solely on the temperature of the air: the colder the air, the slower the speed of sound. Air gets steadily colder with altitude, thus the higher you go, the slower the speed of sound. That means without going any faster, an airplane gets closer to the speed of sound as it gets higher. It's sort of like you get closer to the speed limit as you drive towards town, not because you're accelerating, but because the speed limit decreases towards your speed.
While there are airplanes designed to fly at and beyond the speed of sound, most airplanes become unstable as the airflow over the control surfaces approaches supersonic speeds. Because of the way the air travels over surfaces, the airflow in some places is faster than the airplane, so adverse effects may start at an airspeed of about eight tenths the speed of sound, expressed as a "Mach number" of about 0.8. For safety, transport aircraft comply with a maximum operating Mach number (Mmo) specific to that airplane. For example, a Gulfstream V jet has an Mmo of 0.885, and if I keep getting distracted by looking up specs of other aircraft I'll never get this entry written, so that's the only example you're getting. At sea level, the speed of sound is so high that most airplanes would exceed structural limitations based on the airframe before they approached Mmo. So at low altitudes, the maximum operating speed (Vmo) is not related to the speed of sound.
Vmo does not change with altitude, so as you go up in altitude, the maximum operating speed line is vertical, pretty much paralleling the previously mentioned indicated stall speed line. But eventually, the vertical Vmo line intersects the negative slope of the speed of sound with altitude. From that altitude up, the governing maximum speed becomes the Mmo, the safe margin for that aircraft below the speed of sound. The higher you go, the lower the speed of sound, and that Mmo line slopes all the way back to meet the stall speed line.
Understandably, a pilot is always trying to maintain a safe airspeed above the stall and a safe airspeed below the Vmo/Mmo. But as the altitude increases, the difference between the stall and the max narrows into the space inside the pointy apex of that graph. And that is coffin corner. The pilot must fly accurately because pitching down may increase speed towards Mmo, pitching up may decrease speed towards stall, and banking actually increases the stall speed.
As usual, I've discussed a graphical topic without a graph, and you can't see me waving my hands around. The only one I could find online was in this this thread. The thread itself is alternately intriguing and amusing, as the one poster keeps interjecting the fact that the decrease in air density with altitude is the reason for the increase in stalling speed with altitude. He would be correct if the graph were in true airspeed, but it's in indicated airspeed, which already depends on density. I don't have information on the other poster's theory on momentum.
For the sake of completeness, I should mention that in aviation coffin corner also refers to the top left hand corner of an approach plate, where warnings such as mountainous terrain all quadrants or add 200' to all altitudes when using Dog River altimeter setting are located. It's an easy alliterative phrase that turns up on sports, other industries, and is probably the namesake of many a black diamond ski run. It might originate as the name of an alcove in Victorian staircase landings that allowed one to maneuver a coffin down the stairs. I'm not sure why they wouldn't bring the deceased down the stairs sans coffin, but then there's a lot about the Victorians that doesn't make a lot of sense.
Here's a final footnote on the stall speed. A graph I found in Handling the Big Jets by D.P. Davies plots low speeds against the probability of achieving those speeds in the course of an air transport jet flight. (The data comes from flight data recorders and other research). The probability of flying at 1.25 times the stall speed is 1:1, i.e. always, as that is the speed at which the airplane lifts off the runway. The jet quickly accelerates to a higher margin above the stall and usually does not return to that region until just before landing. But the graph shows that this is not always the case. There's a slightly better than one in ten chance of reaching 1.2 times stall speed, a one in a thousand chance of 1.1 times the stall speed, and one in one hundred thousand transport jet flights for some reason hit the actual stall speed of the airplane. Apparently these are almost the same odds as losing an engine near V1, a crucial take off speed. Davies uses the data to demonstrate the need for air transport pilots to know the handling characteristics of their aircraft at the stall.