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Let's say Bob is flying at Mach 1 and then John, in another aircraft, overtakes him at Mach 3. They are flying parallel to each other, say 500 meters apart. And there is no wind. Their relative speed is Mach 2 of course. Does Bob hear a sonic boom from John, and why or why not?

Please explain it sufficiently enough so that I can apply it to other situations. For example, would it still work if their speeds were more like Mach 1 and Mach 1.25?

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To answer the question:

Yes, and even more, both would be able to notice the sonic boom of the other aircraft.

Why

To understand why, you have to understand how sonic booms are created.

Sound is nothing more than waves of different air pressure travelling through the air. Once released by an aircraft or other object, they start to propagate in all directions with the speed of sound in the resting air. (This means the waves do not travel faster than sound, even when the aircraft does!) If we mark e.g. the regions of high pressure, we get growing circles, the center of each being the point where it was created by the aircraft.

If the aircraft is moving, the circles will not be concentric, like in this animation for an object traveling at half the speed of sound:

If the aircraft is moving with the speed of sound, the circles start to touch each other on a line perpendicular to the track of the aircraft, just where the aircraft is. This is not so clear on the next animation, as there are simply not (big) enough circles.
However, on this line, all the regions of high pressure sum up to a region of 'very high' pressure. This is the shock wave, which is often referred to as sound barrier. (The term also describes just the insurmountable speed limit encountered by non-supersonic aircraft.) If this shock wave passes an observer on the ground, he will hear a sonic boom.

As the aircraft accelerates more, the shock wave isn't perpendicular any more, instead, you get two lines. In reality (3D), you get a cone with the aircraft at the tip, and the opening angle is defined by the speed of the aircraft.

Now, to answer your question: As the tip of the cone is where the aircraft is, this cone travels with the aircraft, and with its speed.

So, if an aircraft is overtaken by a supersonic aircraft, it will be overtaken by the cone-shaped shock wave of that aircraft a little bit later, too. Also, the faster aircraft will pass the shock wave of the slower aircraft.

Remarks

  • An aircraft itself is very loud, but also dampens every noise from outside, and headsets and helmets may contribute to the dampening, too. So, I don't know if a pilot would hear something from the boom. Maybe he does not, but the aircraft definitely crosses the shock wave.
  • My explanation does not describe where the sound comes from. While the sound of a subsonic aircraft mostly comes from the engines, it comes from the compression of air at the surfaces of supersonic aircraft.
  • This cone sometimes is visible, as on the following picture of an F18, stolen from air-attack.com. This is because the high pressure region is followed by a low pressure region, where air expands and moisture may condensate. The picture also shows that there are usually two shock waves, each where the cross section of the aircraft changes significantly. One is generated near the canopy and one near the wings. EDIT: As commented by @PeterKämpf, this picture does NOT show an F18 at supersonic speed. It is at transsonic speed, which is close to, but still below mach 1. The fog forms due to other effects.

However, here is an other picture of a Northrop T-38 at mach 1.1, showing the shock waves, from wiki commons: foo

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    $\begingroup$ Nice explanation, but the F-18 picture doesn't quite fit in, because the plane is flying close to, but still slower than Mach 1. The picture shows a condensation cone which is only visible below Mach 1. $\endgroup$ – Peter Kämpf Jul 26 '15 at 20:13
  • $\begingroup$ Also planes cannot travel at Mach 1 they must punch through it. If they stay at Mach 1 the pressure wave keeps building until it rips the plane apart. $\endgroup$ – Mark Aug 11 '15 at 4:12
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Both planes would create a sonic boom. The boom is a pressure wave. The pressure wave is not aware of the relative speed of the two planes. The slower plane would hear the boom when then cone overtakes it.

The more curious question is will the faster plane actually hear the boom from the slower as it approaches. I think the answer there there is the faster plane will hear the sonic boom of the slower when it crossed the cone - it just starts in the cone.

Sonic_boom

I just have one year of freshman physics so don't take this as authoritative.

I spoke with a fighter pilot (and instructor) and he said the trailing plane would feel the shock wave going in a out but not that big a deal. Mainly the air pitot tube speed indicator would go off for a bit. Since you are moving with the percussion it speeds out the force. The individual cone when they meet would have minimal interaction as relative to each other they are subsonic.

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This should be asked in the physics part of stack exchange, but here are some thought about sonic boom:

  • To answer your question, you must be clear about frame of reference (this is one reason why you should ask your question in physics.SE)
  • A sonic boom is created because an aircraft is going faster than the speed of sound. This speed is measured relative to the airmass the aircraft is flying in.
  • A sonic boom is heard only in a specific place relative to the supersonic aircraft. The boom is due to the shock wave genereted by the aircraft.
  • In the case you ask for, if any traveller cross the shock wave of the slower supersonic aircraft (whatever the direction), he can hear a sonic boom. This sentence is true as long as you consider a supersonic aircraft (either the slower or the faster)
  • Any sound is less strong when you are far away from the source of this sound. This include shock wave (and that's why you won't hear a supersonic aircraft flying few thousands of miles away from you)
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    $\begingroup$ How is the frame of reference relevant? We're not talking about anything even close to the speed of light, so Newtonian relativity is a near-perfect approximation. And, in any case, the frame of reference is explicitly stated in the question, since it asks about the experiences of the two pilots. $\endgroup$ – David Richerby Jul 28 '15 at 10:05
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    $\begingroup$ @DavidRicherby My answer don't use anything else than Newtonian relativity. Speed cannot be defined without speaking of frame of reference. The question shows some confusion about the definition of speed of sound. I think it is important to specify that speed of sound is relevant only relative to the frame of reference of the airmass. $\endgroup$ – Manu H Jul 28 '15 at 12:45

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