bbbeard said:
The physical picture you are promoting, that (a) a shock wave is the result of the coalescence of small perturbations, and that therefore (b) the shock wave retains some kind of information from the source of the perturbations, is incorrect.
Consider that the typical sonic boom is the result of the passing of the oblique shocks attached to a supersonic object. In the frame of the object, the atmosphere (quiescent in the Earth frame) is approaching uniformly at the flight speed of the object. The oblique shock arises from the need to turn the flow in a way that is compatible with the physical requirement that pressure waves can only travel at the speed of sound. Shocks are nature's way of matching upstream and downstream boundary conditions when the upstream flow is supersonic. There is also typically an oblique shock at the tail of the object, as the flow has to straighten out again.
Shock layer thickness is equal to only a few mean free paths. In practical terms this means that the shock is a step function. The Fourier transform of a step function goes like -i/omega, so it has contributions from all frequencies. There is no substructure that incorporates tiny oscillations emanating from the supersonic body. In fact shock are highly dissipative; they destroy information and generate entropy.
Sonic booms incorporate the pressure jumps from both the tip and tail of the flying object. Consider the shock structure for the paradigmatic diamond-wing airfoil. If the flight Mach number is sufficiently high, there is an attached oblique shock at the tip. There will be a Prandtl-Meyer expansion fan at the mid-chord corner, which allows the flow to accelerate around the corner. And then there is a compression shock at the tail as the flow has to straighten out again. On the ground, the timing of the shocks will depend on the length of the object, its altitude, and its flight speed. The typical sonic boom is the result of the passing of this "N wave". Larger objects create a bigger lag between tip and tail and may be experienced as a lower-pitched boom, although the shocks themselves are step functions.
bbbeard said:
The animation you linked from Wikipedia is mislabeled. I have informed the author and I hope he corrects this mistake.
The pretty picture labeled "sonic boom" is not a boom and is not really even a shock wave. The oblique wave you see in the picture is properly called a "http://en.wikipedia.org/wiki/Mach_wave" " and is equivalent to an oblique shock in the limit of zero strength (i.e. no turning of the flow and no rise in pressure). You can create a Mach wave in a wind tunnel by roughening the surface of a test specimen and the perturbations in flow will propagate and coalesce, very much like your description of shock waves. However, you would hardly notice the passage of a Mach wave, because there is no static pressure rise such as the rise you get across a true shock.
However, shock waves do not require the coalescence of perturbations to form. An oblique shock will form in a supersonic flowfield pretty much whenever the flow has to turn compressively (i.e. into itself). A normal shock will form when a supersonic flow needs to change speed abruptly without turning. The F-16, famously, has a normal-shock inlet which slows the flow abruptly from supersonic to subsonic to enable it to be further slowed down in the subsonic diffuser. These "true" shocks -- as opposed to Mach waves -- have nothing to do with coalescence of tiny perturbations.
I already admitted in a previous post that some of both what I said and the way I said it was inaccurate. I take back what I said about a shock retaining frequency information from the sound wave generated by the object. That was misguided.
However, your definition of a shock has some holes. If your definition was true, then a normal shock would not be possible. For example, there is no turning of the flow in a shock tube. Similarly, in the case of a bow shock, the flow physically can't turn, but the shock forms anyway detached from the surface and then the flow turns once it is slowed by the bow shock. In other words, shocks can form without the need for turning.
Now I will admit that before, I misspoke in equating Mach lines and the Mach angle with the shock angle. You are correct in saying that I was describing Mach lines (waves). However, I still maintain that the shock itself (of any sort) is a coalescence of compression waves. The difference between what I said earlier and what is correct, is that I was attributing the shock to sound waves rather than compression waves. If you look at a flow in terms of a series of discrete compressions (as illustrated by Seitzman at Georgia Tech
http://soliton.ae.gatech.edu/people/jseitzma/classes/ae3450/shocks.pdf), you can see that strong compression waves form and eventually catch up to one another, forming a shock. With a continuous compression as in the case of an actual shock tube or a plane or anything else in practice, you would get a continuous compression wave that eventually bunches up, or coalesces, into a nearly discontinuous step.
In other words, yes, you are right that I misspoke in saying it was sound waves. However, it is the coalescence of waves, and while the necessary turning of the flow determines the angle for an oblique shock, it is not the physical basis of the shock. The need to turn into itself would be one reason for the formation of the compression wave that forms a shock, but it is still that wave that coalesces into the shock.
I also feel it necessary to point out that it is actually the shock itself that is on the order of several mean free paths in thickness (about 6 if I remember correctly). The
shock layer is a different concept where, at very high mach numbers in the hypersonic regime, the shock comes so close to the surface of the object that you end up with strong entropy gradients close to the wall and in the boundary layer.
Also, it seems you are correct on the Wikipedia thing. Serves me right for hastily pulling something off of Wikipedia without fact checking; a cardinal sin to be sure. Even in the Doppler effect article you linked it had incorrect captions since the equation used is invalid for M \geq 1. Good catch.
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Just as a side note, I have to say I am happy to have this discussion. It is, in fact, why I browse these forums: to educate and be educated. This has certainly helped sort some thoughts in my own head out. I have to say, thanks and good day, sir.
A.T. said:
Maybe you don't have to interpret the circles in the animation as perturbations, bet merely as distance markers that show how fast a perturbation would advance. Would this be a correct description of a shock wave, in contrast of a Machian wave? However, the mention of Doppler effect in the labeling would have to go, because it refers to actual perturbations.
It would be the correct description of Mach lines (or waves, same thing), but not a shock. bbbeard is correct in saying that the sound waves do not coalesce into shocks, and the compression waves that do do not propagate in spheres from the objects (roughly) as do sound waves.
