A Question About Shock Waves From an Airplane

[BrandonInFlorida]

The elementary treatments I've seen show the shock wave spreading out in spheres centered on the plane and growing in radius at the speed of sound. So, clearly, the shock wave is sound, but what sound? What is it the sound of? In order for the plane to give off sound, it has to be making a sound. The only sound I know is the sound of the engines and the sound of the plane cutting through air.

 

[DrClaude]

Have you read the description on Wikipedia?
https://en.wikipedia.org/wiki/Shock_wave#In_supersonic_flows

 

[sophiecentaur]

What is it the sound of?

It's the sound of the plane's body hitting the air faster than the speed of sound - i.e. the air doesn't have time to slip round the side without disturbing the air in front of it. Air builds up in front and moves out of the way very quickly (faster than the speed of sound). That is the so called shock wave which is moving faster than sound but, within a short distance, things settle down, the region of air slows down and the disturbance then propagates at the regular speed of sound through the air.

The shock wave itself is a fast impulse (sounding like a crack). By the time the sound reaches the ground, there has been dispersion and it has the form of a single impulse of positive pressure (which is what blows out windows etc.) followed by a wide region of negative pressure behind which we don't hear because it is infrasonic.

You would still hear it if the engine was turned off - same as when some meteorites (no engines) enter the atmosphere.

 

[BrandonInFlorida]

Have you read the description on Wikipedia?
https://en.wikipedia.org/wiki/Shock_wave#In_supersonic_flows

The only part I see in that article which seems relevant is the sentence:

"Shock waves are not conventional sound waves; a shock wave takes the form of a very sharp change in the gas properties."

The diagrams show the wave propagating from the plane in spheres. I still don't quite understand why or how the wave is created in the first place, especially if it's not sound.

 

[sophiecentaur]

The only part I see in that article which seems relevant is the sentence

It's possible that you are seeking an answer to your question in your terms only. Wiki often offers you a lot of words and you may need to read them all to get an answer and you may need to look at the information in another way.

I tried, in the above post, to point out that the shock wave only exist close up to the origin. The images of the bow wave of a boat give a clue to what I meant. The wake of a boat consists of just a short sequence of waves and they travel at the natural speed over the surface. If the boat is going fast, it's bow will cause a 'shock' wave but, once it has a wave front that's a single line, the wake is indistinguishable from a wave that's been launched from a slower boat .

I found this passage in the Wiki article which says it all:
"Shock waves are not conventional sound waves; a shock wave takes the form of a very sharp change in the gas properties. Shock waves in air are heard as a loud "crack" or "snap" noise. Over longer distances, a shock wave can change from a nonlinear wave into a linear wave, degenerating into a conventional sound wave as it heats the air and loses energy. The sound wave is heard as the familiar "thud" or "thump" of a sonic boom, commonly created by the supersonic flight of aircraft."
I have heard many sonic booms from aircraft at operating height and they are not describable as a "crack" so (as Mr Wiki says) - we don't hear a shock wave.

 

[BrandonInFlorida]

I think I just didn't see your previous response. You've made it clear to me. It's the sound of air getting out of the way. The boom is a region of constructive interference. Thank you.

 

[A.T.]

The elementary treatments I've seen show the shock wave spreading out in spheres centered on the plane and growing in radius at the speed of sound.

Those spheres are just a visual help, to indicate how the disturbance of the air pressure propagates. They are not necessarily wavefronts of some constant frequency sound that the plane produces. Moving through air at any speed procedures some noise.

 

[sophiecentaur]

how the disturbance of the air pressure propagates.

That's very reasonable. The forward velocity of the apparent wavefront may not be the velocity of the air near the source but the resultant of all displacements.
However, the displaced air must lose energy at a high rate as it leaves the source and cools down(?). Where (how close) does this happen?
The shock wave produced by an aircraft involves very little net displacement, compared with the shock / blast wave from a nuclear explosion in which there is a huge change in mass of expanding gases which would last a long period of time. I could believe that would cause a wave to travel a long way as a shock wave.

 

[A.T.]

However, the displaced air must lose energy at a high rate as it leaves the source and cools down(?). Where (how close) does this happen?

Do you mean where it stops propagating faster than the speed of sound? In the image below, the strongly curved shockwave right in front of of the supersonic object is obviously also supersonic. But in the outer areas where it approaches a straight line (or cone in 3D) it also approaches the speed of sound (from above).

From: https://www.tat.physik.uni-tuebingen.de/~kley/lehre/theoast/background/fig266.html

 

[sophiecentaur]

But in the outer areas where it approaches a straight line (or cone in 3D) it also approaches the speed of sound (from above).

Nice image and it indicates that, for the actual shock wave to hit you, you'd need to be very near an aircraft. At normal heights, a plane would be much further away than that. I imagine a meteorite landing nearby could cause some damage from its shock wave before any impact explosion would be involved.

Edit: I guess there would be a lot of supersonic debris flying outwards too. . . . . in fact I just made a false dichotomy, I think.

 

[BrandonInFlorida]

Those spheres are just a visual help, to indicate how the disturbance of the air pressure propagates. They are not necessarily wavefronts of some constant frequency sound that the plane produces. Moving through air at any speed procedures some noise.

Helpful. Thank you.

 

[A.T.]

for the actual shock wave to hit you, you'd need to be very near an aircraft.

The distinction between an actual shock wave & sound wave is not very sharp, given that you have a gradual transition from one to the other. But there are examples with people very close to the shock wave source:

Supersonic propeller tips.
https://en.wikipedia.org/wiki/Republic_XF-84H_Thunderscreech

Unlike standard propellers that turn at subsonic speeds, the outer 24–30 inches (61–76 cm) of the blades on the XF-84H's propeller traveled faster than the speed of sound even at idle thrust, producing a continuous visible sonic boom that radiated laterally from the propellers for hundreds of yards. The shock wave was actually powerful enough to knock a man down; an unfortunate crew chief who was inside a nearby C-47 was severely incapacitated during a 30-minute ground run.[17] Coupled with the already considerable noise from the subsonic aspect of the propeller and the T40's dual turbine sections, the aircraft was notorious for inducing severe nausea and headaches among ground crews.[11] In one report, a Republic engineer suffered a seizure after close range exposure to the shock waves emanating from a powered-up XF-84H.[18]

The whip crack.


Supersonic bullet. If the bullet passes you some distance away from the shooter, you might hear an initial crack (conical shock wave), before hearing the gun noise (spherical shock wave).

 

[sophiecentaur]

The distinction between an actual shock wave & sound wave is not very sharp, given that you have a gradual transition from one to the other.

That's a good observation; I'm wondering when you can say the shock wave has actually stopped being a shock wave. Velocity measurements would have surely been straightforward enough??
This was in the Wiki article on shock waves:

"A sonic boom produced by an aircraft moving at M=2.92, calculated from the cone angle of 20 degrees. Observers hear nothing until the shock wave, on the edges of the cone, crosses their location."
They claim that the angle of cone tells you the Mach number but doesn't that assume the speed of the wave is the normal wave speed?

 

[A.T.]

They claim that the angle of cone tells you the Mach number but doesn't that assume the speed of the wave is the normal wave speed?

The cone shape itself assumes constant propagation speed, which is reached when it drops to the normal speed of sound. Only the curved shock front near the source is propagating faster than the normal speed of sound.

 

[sophiecentaur]

Only the curved shock front near the source is propagating faster than the normal speed of sound.

I was just confirming that. So I am quite justified in complaining when people describe the sonic boom as a shock wave. The shock wave occurred a long way away.
The examples of actual exposure of humans to shock waves were interesting. It seems to me that the result of exposure to big aircraft propellors could be as much due to infrasound as to pressure levels. But that's another topic.

 

[boneh3ad]

The cone shape that everyone keeps showing is not a shock wave. It's a Mach cone (conical Mach wave). The angle $\mu = \arcsin 1/M$ is the Mach angle, which is not the same as the shock angle that arises at a given Mach number and flow deflection.

In reality, the distinction between a shock wave and a sound wave is quite sharp, pun intended. Usually, sound waves are assumed to be small enough that their effect on gas properties are negligible, but what if the disturbance amplitude and its effects on gas properties are no longer infinitesimal? Any time you have a compression, the density and temperature increase and, thus, so does the speed of sound. The opposite is true of rarefaction. This means the propagation speed of a compression and rarefaction increase and decrease respectively.

If you look above, you can see the effect on a simple finite sinusoidal wave with both a compression and rarefaction component included. As it propagates, the compression portion catches up with the rest of the wave (the snapshot shown in purple). If you let this continue, you end up with a multivalued function in space (green and light blue). Mathematically, the characteristic curves of the wave equation cross. Clearly this can't happen in reality.

In the real world, as that wavefront approaches vertical (gradients approach $\pm\infty$), the assumptions in how we approximate sound waves break down. Viscosity and heat conduction are no longer negligible at that point, each of which are dissipative and prevent the wavefront from "breaking." The nearly vertical wavefront that is sustained under those conditions is a shock, and features sharp changes in properties across it. We usually assume with no real loss of accuracy for most purposes that the phenomenon is infinitely thin and a step change, but in reality it is just a very sharp change with small but finite thickness.

When an aircraft passes through air, it compresses the air in front of it and, if that compression wave cannot find a region of the flow field where it is relieved before it breaks, a shock forms. In practice, this happens any time the flow relative to the vehicle is supersonic. The angle depends on the Mach number and the so-called "turning" angle of the flow as it moves to get around the vehicle.

I should also note that the shock wave is not just the curved portion near the front. They can propagate a fairly large distance from the source: 10s of thousands of feet. Sonic booms are absolutely the result of shock waves and not simply sound waves. If all it took was a simple sound wave, a subsonic aircraft could generate one, which does not happen.

 

[sophiecentaur]

If all it took was a simple sound wave, a subsonic aircraft could generate one, which does not happen.

It would need a distributed source which would make it hard but not impossible to simulate a sonic boom over a limited region. I was thinking of a wave machine / ripple tank which can produce a flat(tish) wave front over a limited range.
I would imagine the wash from a fast moving non-displacement hull would be similar. It forms a linear wave front which seems to travel almost for ever until it reaches the shore when the conditions are right. I remember watching the waves from passing 'steamers' on Lochness (Scotland) which is very deep and there was no wind that day. The shape of the wave appeared to consist of moving 'humps' just like the Monster is said to look like. No apparent attenuation until they finally lapped onto the shore.

You can, of course, record what a sonic boom would sound like at a single point but as you say, that's not the same thing.

 

[boneh3ad]

It would need a distributed source which would make it hard but not impossible to simulate a sonic boom over a limited region. I was thinking of a wave machine / ripple tank which can produce a flat(tish) wave front over a limited range.
I would imagine the wash from a fast moving non-displacement hull would be similar. It forms a linear wave front which seems to travel almost for ever until it reaches the shore when the conditions are right. I remember watching the waves from passing 'steamers' on Lochness (Scotland) which is very deep and there was no wind that day. The shape of the wave appeared to consist of moving 'humps' just like the Monster is said to look like. No apparent attenuation until they finally lapped onto the shore.

You can, of course, record what a sonic boom would sound like at a single point but as you say, that's not the same thing.

The bow wave generated by the bow of a ship moving at high speed through the water is actually a nonlinear phenomenon and a direct analog to shock waves in a gas.

 

[sophiecentaur]

The bow wave generated by the bow of a ship moving at high speed through the water is actually a nonlinear phenomenon and a direct analog to shock waves in a gas.

OK but I am considering the situation a long way from the source.
The propagation of gravity waves on water in very dispersive so perhaps the analogy with a tilted(?) plane (conical) sound wave doesn't go very far. Low level sound waves (or the distant result of a shock wave) will surely have linear propagation. So it seems you could synthesise identical disturbances to a sonic boom over a large area by using a linear array of closely spaced loudspeakers.

Bottom line is why do people call the "boom" that's heard as a shock wave?

 

[boneh3ad]

OK but I am considering the situation a long way from the source.
The propagation of gravity waves on water in very dispersive so perhaps the analogy with a tilted(?) plane (conical) sound wave doesn't go very far. Low level sound waves (or the distant result of a shock wave) will surely have linear propagation. So it seems you could synthesise identical disturbances to a sonic boom over a large area by using a linear array of closely spaced loudspeakers.

Bottom line is why do people call the "boom" that's heard as a shock wave?

Because it is a shock wave? Or, maybe more precisely, is caused by the sudden pressure change generated by one. As the Fourier transform of a step function (a good approximation of a shock) contains many frequencies concentrated at the low end, it sounds to the ear like a low, rumbling boom.

 

[sophiecentaur]

Because it is a shock wave?

By the time the wave it reaches the ground, what is actually moving faster than the speed of sound? Is that not what defines a shock wave?

It was a shock wave but that isn't my point. Why shouldn't it change as it travels?

 

[boneh3ad]

It most certainly is still moving faster than the speed of sound if it reaches the ground as a shock since it is still attached to and traveling with the plane that produced it. If it dissipates prior to reaching the ground, it will no longer result in a sonic boom. This is precisely why altitude and shape are so important in determining the strength of a sonic boom to a ground observer.

 

[sophiecentaur]

It most certainly is still moving faster than the speed of sound if it reaches the ground as a shock since it is still attached to and traveling with the plane that produced it. If it dissipates prior to reaching the ground, it will no longer result in a sonic boom. This is precisely why altitude and shape are so important in determining the strength of a sonic boom to a ground observer.

The wave spreads out as a cone. The angle of the cone, once the shock wave has dissipated (look at the pictures higher up) will depend on the ratio of the Mach Number and a, perhaps modified version of the 'small signal' speed of sound in air.

In reality, the distinction between a shock wave and a sound wave is quite sharp, pun intended. Usually, sound waves are assumed to be small enough that their effect on gas properties are negligible, but what if the disturbance amplitude and its effects on gas properties are no longer infinitesimal?

What sort of 'disturbance amplitude" would be significant? I would suggest that the place where it ceases to be significant would be when the curve around the apex of the cone has become a straight line. I have only heard sonic booms which are no more disturbing than what you can hear on your hifi or with a door being slammed in the house. Can you really describe that as a shock wave?

 

[boneh3ad]

The wave spreads out as a cone. The angle of the cone, once the shock wave has dissipated (look at the pictures higher up) will depend on the ratio of the Mach Number and a, perhaps modified version of the 'small signal' speed of sound in air.

The pictures higher up are not pictures of shock waves. Shock waves absolutely do spread out in cones around the tip of of an aircraft but they are not the same as the cones referenced earlier. Those are Mach waves. See, for example, this schlieren visualization of shock waves around supersonic T-38 Talon jets.

These shocks will propagate a considerable distance, easily several miles, before they weaken enough to no longer be shocks. In fact, a supersonic aircraft flying at 30,000 ft can create a boom carpet 30 miles in width due to its shock waves (source).

What sort of 'disturbance amplitude" would be significant? I would suggest that the place where it ceases to be significant would be when the curve around the apex of the cone has become a straight line. I have only heard sonic booms which are no more disturbing than what you can hear on your hifi or with a door being slammed in the house. Can you really describe that as a shock wave?

Shocks can travel in straight lines. The straightness of the lines are not in any way directly related to the shock strength or boom strength. In fact, the majority of shocks are straight. It is only in the immediate vicinity of curved or insufficiently sharp surfaces that you deal with highly curved shocks.

The actual volume of a sonic boom depends on a number of factors, but they can range in intensity form a fairly mild "thump" to being loud enough to shatter windows.

 

[sophiecentaur]

In fact, a supersonic aircraft flying at 30,000 ft can create a boom carpet 30 miles in width due to its shock waves (source).

You have managed to confuse me all over again with some of the things you've been writing. Yes. "Due to its shock waves" but the source doesn't appear to say that the boom carpet is a shock wave. My problem with what you say is that the angle of the cone (not sure what to call it after your comment) shows that the speed of the wave front must be less than the plane speed. Wiki tells us the speed of the wave is the speed of sound so is that a shock wave? On the rare occasions (once a year, here) that we hear a boom, the likely offender is a very long way away. That confirms the claims that the speed is not excessive.

Blowing out windows is a very hit and miss phenomenon, depending on the angle, the volume of the building and the way the wave couples to it. I remember a Readers' Digest article (there's a good reference, lol) describing how, during a very early supersonic flight, all the windows on a long, single story airfield building blew out as the wave found its way into a corridor, parallel with the flight path. But you don't need a lot of pressure to do that. We don't have many mechanical sources that are as big as a shock wave from a plane so it's hard to think of examples of Booms that are from any other source but a low volume sound alike can be produced from a sub woofer.

 

[boneh3ad]

You have managed to confuse me all over again with some of the things you've been writing. Yes. "Due to its shock waves" but the source doesn't appear to say that the boom carpet is a shock wave. My problem with what you say is that the angle of the cone (not sure what to call it after your comment) shows that the speed of the wave front must be less than the plane speed. Wiki tells us the speed of the wave is the speed of sound so is that a shock wave? On the rare occasions (once a year, here) that we hear a boom, the likely offender is a very long way away. That confirms the claims that the speed is not excessive.

Blowing out windows is a very hit and miss phenomenon, depending on the angle, the volume of the building and the way the wave couples to it. I remember a Readers' Digest article (there's a good reference, lol) describing how, during a very early supersonic flight, all the windows on a long, single story airfield building blew out as the wave found its way into a corridor, parallel with the flight path. But you don't need a lot of pressure to do that. We don't have many mechanical sources that are as big as a shock wave from a plane so it's hard to think of examples of Booms that are from any other source but a low volume sound alike can be produced from a sub woofer.

Shock waves travel faster than the speed of sound normal to their length. That's about as fundamental to shocks as it gets. If the Mach number based on the velocity component of the flow approaching normal to the wave is not greater than unity, you cannot have a shock. It follows, then, that the Mach number of the flow approaching any oblique shock (such as those emanating from an aircraft) must also be supersonic (triangle inequality). Thus, any shock wave propagates faster than the speed of sound relative to the gas approaching it (or, alternatively, relative to the gas into which it is propagating).

The cone you shared above in #13 is not a shock wave. It is, in fact, a Mach cone illustrating the concept of Mach waves and the Mach angle. It is essentially an illustration/extension of the Doppler effect when the sound source is moving at a supersonic speed.

The wave shown in #9 is a shock wave. Near the leading edge, it is what we call a bow shock (or detached oblique shock). That curved region eventually transitions into a more standard oblique shock. Those shocks will propagate a considerable distance from their point of origin.

The Mach angle is defined very simply and is based on the speed of sound waves propagating relative to a supersonic source. It is
$$\mu = \arcsin\frac{1}{M_1}$$
where $M$ is the inflow Mach number. In contrast, the shock angle is defined very differently. For a simple 2D wedge, it is common to use the $\theta$-$\beta$-$M$ equation, which is quadratic in $M_1^2$ and depends on $\theta$ (the flow turning angle or wedge angle) and $\beta$ (the shock angle).
$$\tan\theta = 2\cot\beta\frac{M_1^2\sin^2\beta - 1}{M_1^2\left( \gamma + \cos 2\beta \right) + 2}.$$
Clearly, $\beta\neq\mu$. Additionally, $\beta > \mu$ and $\mu + \theta > \beta$. Finally,
$$\lim_{\theta\to 0}\beta = \mu.$$
The schematic below from Wikimedia commons (and the oblique shock Wikipedia page) lays out the variables. Note that $\gamma = c_p/c_v$ is the ratio of specific heats of the gas.

I will note that the $\theta$-$\beta$-$M$ does not work for conical flows, so you have to use the more complicated Taylor-Maccoll equations to solve for $\beta$ in that case, though the answers are similar to those for a wedge.

A typical supersonic aircraft might be simplified just to the nose cone. For simplicity, let's just assume it's a blunted right circular cone. A bow shock will form at the tip which will transition to a conical shock wave with angle $\beta$ according to the Taylor-Maccoll equations. This shock will propagate for a great distance (for as long as there needs to be a corresponding change in the direction of the flow) and will travel with the speed of the aircraft since it is attached to and continuously generated by it.

 

[A.T.]

$$\lim_{\theta\to 0}\beta = \mu.$$

Is $\theta$ decreasing with increasing distance from the supersonic object? If yes, at what distance does it roughly become negligible, so $\beta \approx \mu$?

 

[sophiecentaur]

It follows, then, that the Mach number of the flow approaching any oblique shock (such as those emanating from an aircraft) must also be supersonic (triangle inequality).

The straight definition of a shock wave (>c) is fine and no one can argue with that. So, what would a probe, a few hundred metres away from a ss aircraft 'see'? Where would the supersonic flow be and in what direction? From the angle, the velocity of apparent propagation of the conical wave appears to be sonic.

The Mach angle is defined very simply and is based on the speed of sound waves propagating relative to a supersonic source.

If what you have told me is true then where does this come from? When / where is the transition from shock to sound? Everything I have written in this thread has been to do with the sound wave. Are there two waves hitting the ground.

Your post seems to deal with just the formation of the shock wave near the craft. I now understand that the shock wave will be oblique. That is interesting. Does it imply that the resulting wave will always have that tilt? Different planes will have a different Mach Angle? Or does it mean that the Mach angle will be established sooner?

This shock will propagate for a great distance (for as long as there needs to be a corresponding change in the direction of the flow) and will travel with the speed of the aircraft since it is attached to and continuously generated by it.

This seems do deal with my question but what constitutes a great distance and what would the 'velocity', rather than 'speed' be?
Intuitively, I would think that ss air flow would dissipate energy in a short distance and leave you with a sonic speed. The power flux would follow an inverse law (square / linear?) as the cone widens.

If I try to apply these ideas to ship bow waves, I know that they propagate a long way but the 'shape' of the disturbance formed by the hull soon changes from 'peaky' (as you get when waves break on the shore) to smooth (but high) ripples. The wake is a very straight line so nothing seems to be happening during its journey to the shore. That seems to indicate that it has settled down to a constant wave speed very soon on its journey. I've seen this with various ships at (importantly) different speeds; you still end up with a straight sided V.

Bottom line is can you answer my question about the probe at a distance. How will the air be moving? Will any of it be supersonic and in what direction? I think all this is so obvious to you that you can't see how a bear of vey little brain could have a problem with it. ;-)

 

[jack action]

How will the air be moving? Will any of it be supersonic and in what direction?

The air doesn't move, only its pressure, temperature and density are changing. Though, this "change" is moving.

When your boat creates a wave in the middle of the ocean, the particular drop of water hit by the boat doesn't necessarily travel with the wave all across the ocean until the wave hits the shore, thousands of kilometers away.

The wave moves (i.e. change in pressure), not the matter.

Pressure waves travel at the [local] speed of sound. A shock is basically a region where there is an accumulation of pressure waves superposing one over another.

What is confusing is that "shock wave", "shockwave" and "shock" are synonyms (Wikipedia). A shock can be stationary (like in a nozzle). I guess "shock wave" sounds more appropriate when talking about a moving object where the shock moves with the object. But its apparent velocity depends solely on the velocity of the object.

 

[sophiecentaur]

The air doesn't move,

Now that is totally confusing me. A shock wave is, according to pretty much every source I can find, is generated when the air (/medium) travels faster than the local speed of sound. The first thing we learn about regular waves in school is that no material actually moves from A to B (no net displacement). The shock wave happens due to an explosion , a whip 'cracking and a supersonic aircraft which is displacing the air it travels through and causes it to travel fast (a net displacement). My problem with this is not the fact that a sonic boom is caused by the shock wave of an aircraft. At some distance from the generator of the shock, the wave is conventional and net flow will be zero.
Is there some analogy with Solitons here?
The only replies to that question I have received are that where the shock wave merges into a sound wave 'can be' a long distance away. The detailed images of shock waves are all of the local situation and not at a distance.
I read that the sonic boom is sound caused by constructive interference. That makes sense as the plane acts like a distributed and not a point source but what mechanism would cause the resultant disturbance to travel faster than sound. If the reason is obvious then why doesn't someone say so? (Words of one syllable or less could help.)
There is no point in anyone describing the local effect of a fast plane moving through the air. There have been plenty of references about that and they are a very reasonable and I have taken it on board.

I read that the shock wave from a nuclear explosion will travel six or seven km. Thereafter, the disturbance becomes sonic. Interesting and the distance must scale down for an aircraft quite significantly. Bearing in mind the relative volumes of gas and the energy involved, an aircraft shock wave wouldn't seem likely to reach the ground from normal altitudes.

You are all going to get cross with me (have already) but, if it's really so obvious, there must be a reference which gives actual figures about this. It should be much easier than reconciling Bernoulli with Newton in the flight explanation.

 

[russ_watters]

I read that the sonic boom is sound caused by constructive interference. That makes sense as the plane acts like a distributed and not a point source but what mechanism would cause the resultant disturbance to travel faster than sound. If the reason is obvious then why doesn't someone say so?

The obvious reason is that the plane is traveling faster than the speed of sound, so the cone that surrounds it and moves with it moves faster than the speed of sound. It has to.

It's probably also useful to look at the water droplets analogy again. The individual waves propagate from the source at the speed of "sound". The lines connecting them move parallel to the source at the speed of the source. I guess maybe the simple answer is geometry? Or maybe that the cone is in a way virtual (like the intersection point between scissors?)?

 

[sophiecentaur]

so the cone that surrounds it and moves with it moves faster than the speed of sound. It has to.

The sound from a passing sub sonic aircraft also follows a cone which also 'travels' with the aircraft. The wave that hits you is traveling at the speed of sound and not the speed of the aircraft so your argument is hardly conclusive. What the 'virtual cone' does is not relevant to the propagation speed of the wave as it goes past you; its apparent source (for a sub sonic aircraft) will be more or less where the plane was at a previous time d/c before. You can identify a direction if the DF of your hearing is up to it. For a supersonic craft the time for the sound of the shock wave will be different. My basic question is how different? The early part of the propagation will / may take less time.

 

[hutchphd]

You are all going to get cross with me (have already) but, if it's really so obvious, there must be a reference which gives actual figures about this.

I don't think there is a precise definition. For nondispersive media I think a reasonable definition requires that the the wavefront be supersonic. After it slows it is just sudden loud sound. By this definition I would be surprised if it propogated very far from the airplane.
This definition is of course not much use for dispersive media. The bow of a boat therefore always has some frequency components for which it is supersonic. But the wake angle is quite different from the Mach angle and is ~independent of speed.

 

[boneh3ad]

Is $\theta$ decreasing with increasing distance from the supersonic object? If yes, at what distance does it roughly become negligible, so $\beta \approx \mu$?

That depends on the shape of the object. If the object is, say, a cone, then $\theta$ won't change unless and until the flow comes under the influence of something else (e.g. another wave or solid object).

 

[jack action]

shock wave is, according to pretty much every source I can find, is generated when the air (/medium) travels faster than the local speed of sound.

We will be going to need a source for this. According to Wikipedia:

n physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium.

Here is how it works. The plane at time $t$ hits the air. This pushes the air in front of which tends to move out of the way. But there is nowhere to go - because it is surrounded by other molecules of air. So it pushes on every other molecule of air. This increases the local pressure. This is nature's way of sending a message: «Something's coming, find an exit path somewhere to make way.»

So the next molecules of air push on the next ones, and so, and so on, propagating the same message. The speed at which this message is displaced is the speed of sound.

If one molecule of air is next to a wall, it will be squished and won't like it. So it will push back on the previous molecule of air with the following message: «No way out here, find somewhere else to go.» We say the pressure wave is reflected and it will effectively go back to the source. If the molecule of air is instead close to a lower pressure point (better visualized when the pressure wave is inside a tube and you are close to an outlet to a larger opening like a divergent nozzle for example), the molecule will instead reflect a negative pressure wave (vacuum) with the message: «There is an outlet here, come this way.» The lower pressure will go back to the source, trying to tell which direction the air in front of the plane should go to move out of the way.

The whole process repeat itself at time $t + \Delta t$. Pressure waves and reflected pressure waves add up together as they cross each other. This is more true inside pipes since a plane is in the atmosphere with not much to reflect the pressure waves from the source.

So if the plane is subsonic, the second pressure wave just follows the first pressure wave. But if the plane is supersonic, the second pressure wave will reach the first one and they will add up. And then the third, and the fourth, and so on. At one point, something will have to give. The pressure will be so high that it will just "tumble" onto the surrounding air. That is the shock where you can see the air properties drastically change to requilibrate everything.