voko said:
I do not think you have really tried to understand what I have been saying. I specifically do not discuss whatever happens during the steady supersonic flight.
I am talking about the brief period of formation of the shock system. There is no "cone" at that time, so I cannot see how your statement 1 could apply to this case.
The problem is that steady supersonic flows are not really all that different in this regard to the transient period at the beginning while accelerating. Right as the object passes Mach 1 there would be an locally spherical bow shock at the tip, but as soon as it passes Mach 1, even while still accelerating, that shock is now going to be locally conical. The only thing changing as it continues accelerating is the angle of the cone.
voko said:
Please cite any source demonstrating "conical shock waves" during the brief period of formation of the shock system. Please be specific.
On the contrary, please cite any source showing that they are not conical. I am contending that the rules governing shock structure are no different while accelerating than they are while moving steadily. Therefore, with the exception of the infinitesimal moment in time where the Mach number is identically 1, the shock is conical. In that brief moment where the object is exactly sonic, the "shock wave" is infinitesimally weak. If you plug Mach 1 into the normal or the oblique shock relations, you get that there is no shock. There is no overpressure, velocity change, temperature change or any other change across this locally spherical boundary, so it makes no sonic boom.
voko said:
And when that pressure wave's amplitude is equal to or greater than the pressure of ambient atmosphere, you would still call it a "sound wave"?
Do you mean its absolute pressure or its overpressure? Either way whether or not it is a shock wave depends on the character of the wave. Most of the time a wave that compresses the flow an entire 1 atm above ambient pressure will be a shock wave (at M=1.363, if normal). That isn't necessarily the case though, although any such sound wave would almost certainly very quickly degenerate into a shock wave.
voko said:
Frankly, you wrote so much that I am not sure at this stage how you define a "shock wave". Really, I am not nitpicking. Could you please provide a concise and unambiguous definition of a "shock wave"? Thank you.
A shock wave is a disturbance wave in a medium across which the disturbed properties jump discontinuously (or at least can be treated as such). In the microscopic sense, their formation in the context of supersonic objects moving through air typically results from the small disturbance waves generated by the object in motion constructively interfering until their amplitude is such that the speed of sound can no longer be assumed constant on all points of the wave and the wave distorts, reaches the point where it is multivalued, and becomes a shock wave (see the series of four graphs I posted earlier).
voko said:
Let's put this back into the original context. The question was whether an observer would hear a siren from a police car speeding away from the observer supersonically.
Two situations are possible.
1. The car has passed the observer while already supersonic.
2. The car becomes supersonic away from the observer.
In situation one, I think we all agree that the observer must have heard a sonic boom (provided the observer was close enough to the car).
Sure they would. They would hear the siren, too. I don't believe anyone has contended that they wouldn't hear a boom in that situation.
voko said:
In situation two, the sonic boom must have started propagating while the car was away. My claim is that initially it propagates omnidirectionally, so the observer will hear it (same condition). Whether it will be louder than the sound heard previously, I cannot say, but I do think the observer should detect some change in the character of the sound.
I hope that answers your questions.
And this is where we take issue. The sonic boom does not propagate omnidirectionally for the reasons stated earlier in this post. The siren would still be audible, but its frequency would be Doppler shifted significantly lower. As long as that shift doesn't bring the sound waves below the frequency of human hearing, it would be audible.
I see exactly why you are thinking the way you are, and it almost makes sense. The problem is that at exactly Mach 1, the small pressure waves are still spherical, but everywhere except at the very tip of the object, they are moving with enough speed to continue to outrun the wave emitted after them since the supersonic body is not moving fast enough to generate the waves before the previous wave can leave the area. If you think about it, at the very tip, where the vehicle is moving at the exact speed of sound, each wave will be generated on top of the last and you might expect a shock to form. If you move even a fraction of a degree (around the spherical wave front), then you have to factor in a cosine term to the velocity of the emitting body in that direction while the wave front still moves at the same constant speed of sound. In other words, anywhere except at the very tip, the waves are perfectly capable of keeping ahead of those emitted after them. So, at Mach 1, there is no shock, and therefore no sonic boom. Once you bump up to Mach 1.000000000001, a shock would form, albeit very weak and almost normal to the direction of travel, and this shock would be conical.
russ_watters said:
That never happens, even in a shock wave - and trying to split hairs on the definition does not change the science.
That does happen, and it happens all the time. For example, at work I run a Mach 6 wind tunnel. The ambient atmospheric pressure inside the undisturbed region of the test section is about 0.08 psia. If I put a 7-degree wedge into the flow, the resulting shock wave raises the ambient pressure behind it to 0.21 psia. You could get it a lot higher if you use other shapes so that the wave isn't so oblique, too. For example, we have a miniature Apollo capsule model, and at the very front of that in the same tunnel, the pressure would rise to 3.44 psia, more than 40 times the ambient pressure.
