Something I Suspect But Can't Prove
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Something I Suspect But Can't Prove
EDIT: My corrected version is now posted below, under the title 'A Frontal Assault'. Rewrites of the other two articles will appear in the next day or two. L Cottrill 23 SEP 2008
EDIT: Well, according to my first try at testing this with UFLOW1D, this hypothesis CANNOT be correct. I'll describe the test when I have time to write it up. I'll do more research and publish a corrected hypothesis, or some more reasonable explanation as deemed appropriate. Meanwhile, I'll get rid of the derivative descriptions below pending revision (though the graphics for those should still be OK, since they don't try to indicate any exact speeds). L Cottrill 21 SEP 2008
In going through the various "thought experiments" that I need for my theory paper on how mass motion relates to pressure, I "discovered" something that might be important, or at least interesting. Of course, I am an uneducated man, and my discovery might already appear in dozens of classic textbooks and I'd never know it. Or, it may in fact not even be true! Or, it may be so obvious that I will come off once again as a bumbling oaf. Anyway, I'd like any comments anyone would care to offer. I don't have what it takes to offer either a mathematical proof or an experimental one. There's probably a way I could use UFLOW1D, but I think it will take me a while to come up with a method I'd really trust for reasonable accuracy.
Anyway, here it is:
We know that (more or less by definition) a pressure wave moves through air at precisely 1.0 Mach at the local air condition. But consider: WITHIN the pressure wave the condition is different! For example, in a positively pressurized zone, pressurization has created higher density AND raised the air temperature. This altered condition means that the measured speed (meaning ft/sec or metres/sec) defined as Mach 1 will be lower than in normal air, due to the increased density. Since my experimental process tends to be Platonic rather than Aristotelian (i.e. I observe a lot just by thinking), I have come to the following assertion, which I have attempted to illustrate in the figure attached below:
If we view a pressure wave front as stationary, at a time T=0 we see an air mass M approaching the front at speed 1.0 Mach under some original temperature and pressure (which I have called called Condition A). After moving through the pressure front (i.e. at some later time T=n), that same air mass has some new temperature, density and pressure (called Condition B), but it ALSO has a new speed, receding from the front, of 1.0 Mach under that new condition. In other words, what I'm saying is: Assuming a pressure wave front changes the air condition, the advancing front will see air approaching at 1.0 Mach under the original condition and will see air leaving it at 1.0 Mach under the new condition.
Can you prove me wrong?
L Cottrill
EDIT: Well, according to my first try at testing this with UFLOW1D, this hypothesis CANNOT be correct. I'll describe the test when I have time to write it up. I'll do more research and publish a corrected hypothesis, or some more reasonable explanation as deemed appropriate. Meanwhile, I'll get rid of the derivative descriptions below pending revision (though the graphics for those should still be OK, since they don't try to indicate any exact speeds). L Cottrill 21 SEP 2008
In going through the various "thought experiments" that I need for my theory paper on how mass motion relates to pressure, I "discovered" something that might be important, or at least interesting. Of course, I am an uneducated man, and my discovery might already appear in dozens of classic textbooks and I'd never know it. Or, it may in fact not even be true! Or, it may be so obvious that I will come off once again as a bumbling oaf. Anyway, I'd like any comments anyone would care to offer. I don't have what it takes to offer either a mathematical proof or an experimental one. There's probably a way I could use UFLOW1D, but I think it will take me a while to come up with a method I'd really trust for reasonable accuracy.
Anyway, here it is:
We know that (more or less by definition) a pressure wave moves through air at precisely 1.0 Mach at the local air condition. But consider: WITHIN the pressure wave the condition is different! For example, in a positively pressurized zone, pressurization has created higher density AND raised the air temperature. This altered condition means that the measured speed (meaning ft/sec or metres/sec) defined as Mach 1 will be lower than in normal air, due to the increased density. Since my experimental process tends to be Platonic rather than Aristotelian (i.e. I observe a lot just by thinking), I have come to the following assertion, which I have attempted to illustrate in the figure attached below:
If we view a pressure wave front as stationary, at a time T=0 we see an air mass M approaching the front at speed 1.0 Mach under some original temperature and pressure (which I have called called Condition A). After moving through the pressure front (i.e. at some later time T=n), that same air mass has some new temperature, density and pressure (called Condition B), but it ALSO has a new speed, receding from the front, of 1.0 Mach under that new condition. In other words, what I'm saying is: Assuming a pressure wave front changes the air condition, the advancing front will see air approaching at 1.0 Mach under the original condition and will see air leaving it at 1.0 Mach under the new condition.
Can you prove me wrong?
L Cottrill

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Re: Something I Suspect But Can't Prove
Putting my rusty physics head on again...
Question 1, is it a standing wave or a travelling wave? This is a difficult one because of the relationship between the speed of the travelling wave, the desity of air which also defines this and the frequency and wavelength. Change one, and the others tend to change aswell, so there is a good chance that the advancing and retreating air may be moving at the same speed.
It may be better to more clearly define the boundary conditions, ie what is and is not moving, or even do a test experiment such as set up a loudspeaker & microphone. Send out a continuous tone signal and watch the trace on an oscilloscope. blow compressed air towards the speaker to create your moving airmass intercepting your pressure waves and look for a change in phase or frequency of your signal. By understanding a simpler version, it may be easier to then add on the complications to get you to your actual answer.
Or I could be talking complete mince!
Cheers
Steve
Question 1, is it a standing wave or a travelling wave? This is a difficult one because of the relationship between the speed of the travelling wave, the desity of air which also defines this and the frequency and wavelength. Change one, and the others tend to change aswell, so there is a good chance that the advancing and retreating air may be moving at the same speed.
It may be better to more clearly define the boundary conditions, ie what is and is not moving, or even do a test experiment such as set up a loudspeaker & microphone. Send out a continuous tone signal and watch the trace on an oscilloscope. blow compressed air towards the speaker to create your moving airmass intercepting your pressure waves and look for a change in phase or frequency of your signal. By understanding a simpler version, it may be easier to then add on the complications to get you to your actual answer.
Or I could be talking complete mince!
Cheers
Steve

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Re: Something I Suspect But Can't Prove
Steve 
What we have here is the wave traversing a straight pipe full of normal air (assumed stationary before the wave front arrives). For theory purposes, we could think of it as an infinitely long pipe of constant cross section. The wave moves from left to right at speed 1.0 Mach, and the speed behind the wave front is some subsonic speed (relative to the pipe), also from left to right. Behind the wave front we no longer have normal air, because the pressure, density and temperature are higher. If my assertion is true, the speed of this air (again, relative to the pipe) would be exactly the difference between the speed of sound in normal air and the speed of sound in air at the modified condition. Thus, if we imagine our viewpoint moving along with the wave front as described, we will see air approaching at 1.0 Mach (normal air condition) and leaving behind us at some lower speed, as suggested in the graphic.
I have serious doubts about an attempt to prove this with audio testing, just because what we want to measure exactly is the internal work of the wave. Since any audio wave will, almost by definition, always return to zero pressure, it would be extremely difficult to determine (with precision) what happens to Mach speed inside the envelope. A mic and oscilloscope will show the pressure variations, of course, and I have that kind of instrumentation available, but I don't see how there's anything I could do to compare the remaining internal gas conditions (density, temp) that wouldn't be some kind of interpretation. Another way to say this is that I think the internal condition of the wave is too transient to actually measure meaningfully (except for the pressure).
L Cottrill
What we have here is the wave traversing a straight pipe full of normal air (assumed stationary before the wave front arrives). For theory purposes, we could think of it as an infinitely long pipe of constant cross section. The wave moves from left to right at speed 1.0 Mach, and the speed behind the wave front is some subsonic speed (relative to the pipe), also from left to right. Behind the wave front we no longer have normal air, because the pressure, density and temperature are higher. If my assertion is true, the speed of this air (again, relative to the pipe) would be exactly the difference between the speed of sound in normal air and the speed of sound in air at the modified condition. Thus, if we imagine our viewpoint moving along with the wave front as described, we will see air approaching at 1.0 Mach (normal air condition) and leaving behind us at some lower speed, as suggested in the graphic.
I have serious doubts about an attempt to prove this with audio testing, just because what we want to measure exactly is the internal work of the wave. Since any audio wave will, almost by definition, always return to zero pressure, it would be extremely difficult to determine (with precision) what happens to Mach speed inside the envelope. A mic and oscilloscope will show the pressure variations, of course, and I have that kind of instrumentation available, but I don't see how there's anything I could do to compare the remaining internal gas conditions (density, temp) that wouldn't be some kind of interpretation. Another way to say this is that I think the internal condition of the wave is too transient to actually measure meaningfully (except for the pressure).
L Cottrill
wave thoughts
Early in the forum there was some talk about two different types of waves. The type that would bob mass about, or the type that would carry mass along.
From my simple thoughts, mass displacement is what mass leaves the engine at velocity and doesn't return.
Joe
From my simple thoughts, mass displacement is what mass leaves the engine at velocity and doesn't return.
Joe

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Re: wave thoughts
Hmmm ... you might be thinking of the contrast between the pressure and mass flow waves. Once the system reaches resonant operation, they would continue in a 90degree phase difference mode, like current and voltage in an RF tank circuit (coil and capacitor).PyroJoe wrote:Early in the forum there was some talk about two different types of waves. The type that would bob mass about, or the type that would carry mass along.
Yes, that's true enough in terms of net thrust, but it doesn't help much with grasping the internal theory, which is what I'm interested in communicating. There is mass displacement everywhere in the engine that pressure changes occur. Velocity changes everywhere, too, as you know.From my simple thoughts, mass displacement is what mass leaves the engine at velocity and doesn't return.
L Cottrill

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A Frontal Assault
All right, here's the heavily revised version  more truth, less speculation!
We know that (by definition) a pressure wave moves through air at precisely 1.0 Mach at the particular temperature and pressure of that air mass. The temperature, pressure and density of the air at a particular location is known as the local air condition. Consider, however, that once the pressure wave has begun to act on the air, the local condition is already changing! For example, once the air lies in a positively pressurized zone, that pressurization has forced the air into higher density, and even raised the air temperature to some extent. This altered condition means that the measured air speed (such as feet/sec or metres/sec) defined as Mach 1 will be higher than in normal air, mostly due to the higher temperature. If we imagine a very simple form for the pressure wave (a plateau of high pressure with sloping leading and trailing edges), the behavior of air under a changing local condition can be easily visualized. Both the leading and trailing edges are called "wave fronts", but we can use just the leading (rising pressure) front as an example; we can assume for now that the trailing (falling pressure) front would show an exactly opposite effect.
Pretend for a moment that we can move along with the advancing pressure wave; we would then view the pressure wave front as "stationary". Pretend also that we can see a small sample of air mass and view what it does over some space of time. At some time we'll again call T=0 suppose we observe a small parcel of air we'll call air mass M approaching us (and the wave front), before it begins to be affected by the increasing pressure. To keep things simple, let's say that the unaffected air out in front of the wave is "normal" air (i.e. standard sea level air pressure at a comfortable temperature)  we can say that all the approaching air (including the parcel we've chosen to watch) is at the "normal condition", which we'll call Condition A for purposes of comparison later. If we measure the speed of the approaching parcel of air, we quite naturally find it to be equivalent to 1.0 Mach at Condition A.
Now, as we watch our selected air mass parcel move through the pressure wave front, we can observe changes to its condition. Its mass never changes, of course, but its size (meaning, its volume) does! As the parcel moves through the zone of increasing pressure, its volume is squeezed smaller and smaller; in fact, the volume is changing inversely to the pressure acting on it (that is, doubling the pressure would squeeze it to approximately half its original volume!). So, as we see the parcel travel through the front, we see it gradually shrink. Since the mass remains constant, this means the density of the parcel is gradually increasing. If we observe carefully, we will note one other interesting phenomenon: the speed of the parcel is gradually decreasing!
After moving completely through the pressure front (at some later time T=n), our selected air mass parcel has arrived at some new temperature, density and pressure (in other words, a new "local condition" we'll call Condition B), but it ALSO has a new, much lower, speed. What has happened is that some of the wave energy has been applied to the mass, changing both the heat contained in the mass parcel AND its speed. I've attempted to show all this in the graphic below. The bold rightpointing arrows are called vectors; they obviously show direction of motion, but in addition, their lengths are set to illustrate the measurable speeds. Similarly, the width of the parcel at the two locations shows the volume change, and the color is intended to roughly communicate the change in density. The "departure speed" of the parcel behind the wave is designated by the vector Udep:
(See attached graphic, below)
Now, we might feel justified in restating our observation as a simple rule:
Assuming a pressure wave front which changes the air condition, the advancing front will see air approaching at a speed of 1.0 Mach under the original condition and will see air departing from it in the original direction but at a significantly lower speed, with the change in mass temperature and the difference in approach and departure speeds representing the total energy applied to this mass by the passage of the wave front.
It is crucial to understand the preceding assertion. From it, the entire theoretical basis for the physical action of the pressure wave on the air mass can be derived. From the standpoint of pure physics, this model already shows an important principle related to changing the velocity of masses: the slowing of the mass as its density increases is actually a demonstration of the conversion of external kinetic energy (the energy of motion) into internal kinetic energy (gas pressure and temperature). This kind of conversion and its reverse are absolutely fundamental to virtually all jet engine dynamics.
To help illustrate the effect of the pressure wave on the air condition and departure speed Udep, I offer the following table. Note that valveless pulsejets are low compression engines, and a pressure of 2.0 bar will cover all internal pressures normally encountered in design:
The kind of pressure wave shown in this graphic, a "ramp" of increasing pressure leading up to a "plateau" of constant high pressure, is not something we will ever see in a pulsejet, and was chosen basically because it simplifies the illustration of wave front action. This does not mean that there can't be a "real life" wave like this, though  it actually is a fairly good representation of the pressure wave that moves out in front of a piston moving in a cylinder. Another example would be the pressurization of air in a gun barrel in front of a moving bullet.
Hope you all agree that this is much better!
L Cottrill
We know that (by definition) a pressure wave moves through air at precisely 1.0 Mach at the particular temperature and pressure of that air mass. The temperature, pressure and density of the air at a particular location is known as the local air condition. Consider, however, that once the pressure wave has begun to act on the air, the local condition is already changing! For example, once the air lies in a positively pressurized zone, that pressurization has forced the air into higher density, and even raised the air temperature to some extent. This altered condition means that the measured air speed (such as feet/sec or metres/sec) defined as Mach 1 will be higher than in normal air, mostly due to the higher temperature. If we imagine a very simple form for the pressure wave (a plateau of high pressure with sloping leading and trailing edges), the behavior of air under a changing local condition can be easily visualized. Both the leading and trailing edges are called "wave fronts", but we can use just the leading (rising pressure) front as an example; we can assume for now that the trailing (falling pressure) front would show an exactly opposite effect.
Pretend for a moment that we can move along with the advancing pressure wave; we would then view the pressure wave front as "stationary". Pretend also that we can see a small sample of air mass and view what it does over some space of time. At some time we'll again call T=0 suppose we observe a small parcel of air we'll call air mass M approaching us (and the wave front), before it begins to be affected by the increasing pressure. To keep things simple, let's say that the unaffected air out in front of the wave is "normal" air (i.e. standard sea level air pressure at a comfortable temperature)  we can say that all the approaching air (including the parcel we've chosen to watch) is at the "normal condition", which we'll call Condition A for purposes of comparison later. If we measure the speed of the approaching parcel of air, we quite naturally find it to be equivalent to 1.0 Mach at Condition A.
Now, as we watch our selected air mass parcel move through the pressure wave front, we can observe changes to its condition. Its mass never changes, of course, but its size (meaning, its volume) does! As the parcel moves through the zone of increasing pressure, its volume is squeezed smaller and smaller; in fact, the volume is changing inversely to the pressure acting on it (that is, doubling the pressure would squeeze it to approximately half its original volume!). So, as we see the parcel travel through the front, we see it gradually shrink. Since the mass remains constant, this means the density of the parcel is gradually increasing. If we observe carefully, we will note one other interesting phenomenon: the speed of the parcel is gradually decreasing!
After moving completely through the pressure front (at some later time T=n), our selected air mass parcel has arrived at some new temperature, density and pressure (in other words, a new "local condition" we'll call Condition B), but it ALSO has a new, much lower, speed. What has happened is that some of the wave energy has been applied to the mass, changing both the heat contained in the mass parcel AND its speed. I've attempted to show all this in the graphic below. The bold rightpointing arrows are called vectors; they obviously show direction of motion, but in addition, their lengths are set to illustrate the measurable speeds. Similarly, the width of the parcel at the two locations shows the volume change, and the color is intended to roughly communicate the change in density. The "departure speed" of the parcel behind the wave is designated by the vector Udep:
(See attached graphic, below)
Now, we might feel justified in restating our observation as a simple rule:
Assuming a pressure wave front which changes the air condition, the advancing front will see air approaching at a speed of 1.0 Mach under the original condition and will see air departing from it in the original direction but at a significantly lower speed, with the change in mass temperature and the difference in approach and departure speeds representing the total energy applied to this mass by the passage of the wave front.
It is crucial to understand the preceding assertion. From it, the entire theoretical basis for the physical action of the pressure wave on the air mass can be derived. From the standpoint of pure physics, this model already shows an important principle related to changing the velocity of masses: the slowing of the mass as its density increases is actually a demonstration of the conversion of external kinetic energy (the energy of motion) into internal kinetic energy (gas pressure and temperature). This kind of conversion and its reverse are absolutely fundamental to virtually all jet engine dynamics.
To help illustrate the effect of the pressure wave on the air condition and departure speed Udep, I offer the following table. Note that valveless pulsejets are low compression engines, and a pressure of 2.0 bar will cover all internal pressures normally encountered in design:
Code: Select all
Departure Speed vs Condition B Pressure
(Condition A = 1.0 bar air at 300 °K)
PressCondB TempCondB Udep MachDep
1.1 bar 308 °K 297.9 m/sec .916 Mach
1.2 bar 316 °K 275.9 m/sec .840 Mach
1.3 bar 323 °K 254.7 m/sec .768 Mach
1.4 bar 330 °K 236.7 m/sec .708 Mach
1.5 bar 337 °K 218.3 m/sec .648 Mach
1.6 bar 343 °K 202.1 m/sec .595 Mach
1.7 bar 349 °K 185.6 m/sec .542 Mach
1.8 bar 355 °K 169.2 m/sec .490 Mach
1.9 bar 360 °K 155.8 m/sec .449 Mach
2.0 bar 366 °K 140.7 m/sec .402 Mach
Hope you all agree that this is much better!
L Cottrill
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 Corrected graphic showing wave passage. Drawing Copyright 2008 Larry Cottrill
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Putting the Pressure On
The revised version:
Now we can get back to looking at things in a more conventional way, seeing the wave front as moving at Mach 1 speed through something resembling an engine pipe. What we want to see now is a magnified view of exactly how the changing pressure actually stirs air mass into motion. Let's recast the graphic we used above as the rather more complex drawing, below.
Some explanation is needed of the added "features": The drawing is vertically divided into three zones; a large upper region showing the pressure wave graph, a crude representation of a length of straight pipe containing a row of adjoining air mass parcels as affected by the wave, and a narrow "ruler" mapping how the parcels were laid out before the arrival of the wave. The parcels in the pipe (and on the ruler) are alternately shown blue and white for clarity  it must be kept in mind that whether a parcel is colored or white means absolutely nothing in terms of its behavior; they are all created equal, and assumed to all have exactly the same mass. A straight pipe (represented by the thick black bars) is used because it allows us to show changes in volume simply and precisely as changes in the thickness (i.e. length) of the parcels. Three of the parcels have been individually "marked" with colored dots; this is purely to clarify their motions, and has no other meaning  they are exactly like all other parcels shown.
A couple of final details: the drawing we used before showed the same parcel at two different moments in time; it was a kind of "before and after" shot. This graphic, on the other hand, shows all the adjoining parcels as they appear at a particular instant in time; a true "freeze frame" of the action. And, while the heavy black 1.0 Mach vector indicated the approach speed of air in the previous drawing, from now on it will indicate the speed of the wave front we're modeling:
(See drawing attached below)
To see how pressure actually works to move air around, lets see from the picture above what happens to a particular air mass parcel. We'll choose the parcel "marked" with the yellow dot (shown in the "pipe" segment of the drawing). Each parcel of air has a finite length in the pipe; that is, it has a left face and a right face which adjoin neighboring parcels (the neighboring "white" parcels in the drawing). If we project the locations of these faces upward through the pressure wave front graph, we see that the left face cuts across a lower position on the pressure curve than the right face. We designate the height at which the left face crosses the curve as PLO and the height at which the right face crosses as PHI, representing two slightly different pressures. These are the actual static pressures on the two faces of the parcel, and they represent forces acting inward on it (because pressure x area = force).
The relatively small pressure difference (PHI  PLO) is shown as PDIFF. I have brought PDIFF down into the pipe part of the drawing and shown it as a pressure vector acting on the yellowdotted parcel. So, in effect, we see a net pressure (and therefore, a net force) acting as a leftward influence on the parcel. From elementary physics, we know that an object with a force acting on it accelerates in the direction the force is applied, and this is true of our selected parcel; I've tried to show this with the velocity vector labelled ACCELERATING FLOW  the parcel indeed is moving leftward at increasing speed.
The volume and density of the parcel are also affected by these pressures; however, it is a convenient and very close approximation to simply use the average pressure, which I have labelled PAV, as the effective pressure applied to the parcel as a whole. Note the little blue and white detail I've shown at about midheight on the vertical projection lines. This shows how the volume (and length) of the parcel have been affected by applying pressure PAV. (Again, remember that because of the constant pipe crosssection, the volume of the parcel is perfectly represented by its length.) The original volume of the parcel is represented by VORIG and the modified volume by VNEW. The change in volume is inversely related to the change in pressure, so we can say that
VNEW / VORIG = 1.00 bar / PAV (since 1.00 bar was the original pressure)
Again, I have used the darker blue color as a crude way of showing the higher density of the parcel due to its compressed state. As a mass of air is compressed, density essentially rises linearly with pressure. It should be remembered that the temperature of the parcel is increased somewhat by the compression, as well. But, unlike the relationship between pressure and density, the change in temperature is a highly nonlinear function.
From all of the above, we can formulate the following rule:
The acceleration of a finite air mass comes from a force that is the result of a pressure difference. This pressure difference comes from the finite size of the mass and the nonuniform pressure represented by the wave front traversing the mass.
Because "acceleration" is defined as a change in velocity (from basic physics), we can generalize the above (perhaps with some danger of oversimplification) as:
CHANGES in air pressure cause corresponding CHANGES in air velocity.
If we now move to the right side of the drawing, we find a region where the parcels in the pipe are highly compressed and are of perfectly uniform volume, density and temperature (that is, they are identical in terms of "local condition"). This can be seen to correspond to the flat top (or "plateau") of the pressure curve shown. The velocity vectors are given the label HIGH SPEED FLOW, and these vectors are exactly equal  this is a region where the air speed is constant (i.e. no longer accelerating). As you may have guessed, it is no accident that constant speed flow corresponds to the constant pressure portion of the wave graph. In this part of the wave, the PHI, PLO and PAV are exactly equal, while PDIFF = 0. This means that, while pressure PAV still acts to compress each parcel, there is no longer any net force acting leftward on the parcels! Again looking at the situation in terms of basic physics, a mass with no net force acting on it keeps moving at unchanged velocity.
That may seem counterintuitive if you have ever used compressorpowered air tools, or other high air pressure equipment  after all, aren't you supplying pressure to maintain constant air flow through the hose to the tool? The real answer to that is, basically, no! What you are really doing is using compressor power to force the air up to its working pressure so the tool can recover that power at the other end. The movement of air through the hose would require NO power at all, if it weren't for friction losses in the hose and turbulence and shear losses in the fittings (and these effects account for only a tiny fraction of the total energy used). The actual motion of the air through the hose is essentially free.
We should still note one more observation from the constant pressure / constant velocity region (the right end of the drawing): It is important to understand that the uniform air parcels in this region are not moving at the forward speed of the wave front. The forward speed of each parcel is basically determined by the compression of all those directly ahead of it!. Note the three "dotted" parcels shown in the pipe, then find the same three "dotted" parcels on the "ruler" below. Note that the "reddotted" parcel is unmoved, the "yellowdotted" parcel (being accelerated by increasing pressure) has moved a little, while the "greendotted" parcel is already heavily displaced. Again, remember that this drawing is a "snapshot" of an instant in time; at this moment, the distances are closing between the "dotted" parcels, but in a moment the wave front will be completely past the "yellow" one, and the distance between it and the "green" one will then stay constant  the "yellow" parcel will be at the velocity labelled HIGH SPEED FLOW. It is critically important that this concept be fully understood and accepted.
So, what speed is our constant HIGH SPEED FLOW, anyway? We already have our answer, if we can measure the local air condition within the flat "plateau" part of the pressure wave (in other words, the condition after the front has completely passed by). From our previous discussion (where our viewpoint tracked right along with the wave), we observed that once a mass parcel of air has gotten to the final air condition region, it is traveling away from the front at a reduced speed we called UDEP; so, taking into account the actual speed of the wave front (1.0 Mach at the original condition, which is about 321.5 metres/sec), the actual reduced flow speed under the plateau of the curve (we'll call this speed UCondB) is easily derived:
Warning: These values should be usable for areas of the pressure curve that are level (like the top of our our "plateau") or gradually changing (long smooth curves or slopes)  they will not be accurate (and in many cases, not even close!) under rapidly changing zones of the wave such as pressure "spikes", or even within the "ramps" at the start and end of our "plateau" region! Fortunately, most of the time in pulsejets, the pressure is changing fairly gradually during the cycle.
Again, we have assumed stationary air before the wave passes, i.e. UCondA = 0; in most reallife examples, we would not be starting with stationary air, but instead we would have some initial velocity at the region of interest in the pipe. This can be positive (in the same direction the wave front is moving) or negative (flowing against the advancing wave front) or (rarely) zero, as seen in our graphic example here. If nonzero, this initial velocity would simply be added algebraicly to the value in the table as selected for the local pressure. Thus, the flow velocity within the pressure wave can be readily determined even if all we know is the initial flow speed and the initial and final air conditions. When the actual speed for 1.0 Mach under any air condition is needed, it can be referenced in the table above (interpolating if necessary for greater accuracy).
L Cottrill
Now we can get back to looking at things in a more conventional way, seeing the wave front as moving at Mach 1 speed through something resembling an engine pipe. What we want to see now is a magnified view of exactly how the changing pressure actually stirs air mass into motion. Let's recast the graphic we used above as the rather more complex drawing, below.
Some explanation is needed of the added "features": The drawing is vertically divided into three zones; a large upper region showing the pressure wave graph, a crude representation of a length of straight pipe containing a row of adjoining air mass parcels as affected by the wave, and a narrow "ruler" mapping how the parcels were laid out before the arrival of the wave. The parcels in the pipe (and on the ruler) are alternately shown blue and white for clarity  it must be kept in mind that whether a parcel is colored or white means absolutely nothing in terms of its behavior; they are all created equal, and assumed to all have exactly the same mass. A straight pipe (represented by the thick black bars) is used because it allows us to show changes in volume simply and precisely as changes in the thickness (i.e. length) of the parcels. Three of the parcels have been individually "marked" with colored dots; this is purely to clarify their motions, and has no other meaning  they are exactly like all other parcels shown.
A couple of final details: the drawing we used before showed the same parcel at two different moments in time; it was a kind of "before and after" shot. This graphic, on the other hand, shows all the adjoining parcels as they appear at a particular instant in time; a true "freeze frame" of the action. And, while the heavy black 1.0 Mach vector indicated the approach speed of air in the previous drawing, from now on it will indicate the speed of the wave front we're modeling:
(See drawing attached below)
To see how pressure actually works to move air around, lets see from the picture above what happens to a particular air mass parcel. We'll choose the parcel "marked" with the yellow dot (shown in the "pipe" segment of the drawing). Each parcel of air has a finite length in the pipe; that is, it has a left face and a right face which adjoin neighboring parcels (the neighboring "white" parcels in the drawing). If we project the locations of these faces upward through the pressure wave front graph, we see that the left face cuts across a lower position on the pressure curve than the right face. We designate the height at which the left face crosses the curve as PLO and the height at which the right face crosses as PHI, representing two slightly different pressures. These are the actual static pressures on the two faces of the parcel, and they represent forces acting inward on it (because pressure x area = force).
The relatively small pressure difference (PHI  PLO) is shown as PDIFF. I have brought PDIFF down into the pipe part of the drawing and shown it as a pressure vector acting on the yellowdotted parcel. So, in effect, we see a net pressure (and therefore, a net force) acting as a leftward influence on the parcel. From elementary physics, we know that an object with a force acting on it accelerates in the direction the force is applied, and this is true of our selected parcel; I've tried to show this with the velocity vector labelled ACCELERATING FLOW  the parcel indeed is moving leftward at increasing speed.
The volume and density of the parcel are also affected by these pressures; however, it is a convenient and very close approximation to simply use the average pressure, which I have labelled PAV, as the effective pressure applied to the parcel as a whole. Note the little blue and white detail I've shown at about midheight on the vertical projection lines. This shows how the volume (and length) of the parcel have been affected by applying pressure PAV. (Again, remember that because of the constant pipe crosssection, the volume of the parcel is perfectly represented by its length.) The original volume of the parcel is represented by VORIG and the modified volume by VNEW. The change in volume is inversely related to the change in pressure, so we can say that
VNEW / VORIG = 1.00 bar / PAV (since 1.00 bar was the original pressure)
Again, I have used the darker blue color as a crude way of showing the higher density of the parcel due to its compressed state. As a mass of air is compressed, density essentially rises linearly with pressure. It should be remembered that the temperature of the parcel is increased somewhat by the compression, as well. But, unlike the relationship between pressure and density, the change in temperature is a highly nonlinear function.
From all of the above, we can formulate the following rule:
The acceleration of a finite air mass comes from a force that is the result of a pressure difference. This pressure difference comes from the finite size of the mass and the nonuniform pressure represented by the wave front traversing the mass.
Because "acceleration" is defined as a change in velocity (from basic physics), we can generalize the above (perhaps with some danger of oversimplification) as:
CHANGES in air pressure cause corresponding CHANGES in air velocity.
If we now move to the right side of the drawing, we find a region where the parcels in the pipe are highly compressed and are of perfectly uniform volume, density and temperature (that is, they are identical in terms of "local condition"). This can be seen to correspond to the flat top (or "plateau") of the pressure curve shown. The velocity vectors are given the label HIGH SPEED FLOW, and these vectors are exactly equal  this is a region where the air speed is constant (i.e. no longer accelerating). As you may have guessed, it is no accident that constant speed flow corresponds to the constant pressure portion of the wave graph. In this part of the wave, the PHI, PLO and PAV are exactly equal, while PDIFF = 0. This means that, while pressure PAV still acts to compress each parcel, there is no longer any net force acting leftward on the parcels! Again looking at the situation in terms of basic physics, a mass with no net force acting on it keeps moving at unchanged velocity.
That may seem counterintuitive if you have ever used compressorpowered air tools, or other high air pressure equipment  after all, aren't you supplying pressure to maintain constant air flow through the hose to the tool? The real answer to that is, basically, no! What you are really doing is using compressor power to force the air up to its working pressure so the tool can recover that power at the other end. The movement of air through the hose would require NO power at all, if it weren't for friction losses in the hose and turbulence and shear losses in the fittings (and these effects account for only a tiny fraction of the total energy used). The actual motion of the air through the hose is essentially free.
We should still note one more observation from the constant pressure / constant velocity region (the right end of the drawing): It is important to understand that the uniform air parcels in this region are not moving at the forward speed of the wave front. The forward speed of each parcel is basically determined by the compression of all those directly ahead of it!. Note the three "dotted" parcels shown in the pipe, then find the same three "dotted" parcels on the "ruler" below. Note that the "reddotted" parcel is unmoved, the "yellowdotted" parcel (being accelerated by increasing pressure) has moved a little, while the "greendotted" parcel is already heavily displaced. Again, remember that this drawing is a "snapshot" of an instant in time; at this moment, the distances are closing between the "dotted" parcels, but in a moment the wave front will be completely past the "yellow" one, and the distance between it and the "green" one will then stay constant  the "yellow" parcel will be at the velocity labelled HIGH SPEED FLOW. It is critically important that this concept be fully understood and accepted.
So, what speed is our constant HIGH SPEED FLOW, anyway? We already have our answer, if we can measure the local air condition within the flat "plateau" part of the pressure wave (in other words, the condition after the front has completely passed by). From our previous discussion (where our viewpoint tracked right along with the wave), we observed that once a mass parcel of air has gotten to the final air condition region, it is traveling away from the front at a reduced speed we called UDEP; so, taking into account the actual speed of the wave front (1.0 Mach at the original condition, which is about 321.5 metres/sec), the actual reduced flow speed under the plateau of the curve (we'll call this speed UCondB) is easily derived:
Code: Select all
Duct Air Speed vs Condition B Pressure & Temperature
(Condition A = 1.0 bar air at 300 °K, stationary)
PressCondB TempCondB 1.0 MachCondB UDEP UCondB MachCondB
1.1 bar 308 °K 325 m/sec 298 m/sec 24 m/sec .073 Mach
1.2 bar 316 °K 329 m/sec 276 m/sec 46 m/sec .139 Mach
1.3 bar 323 °K 332 m/sec 255 m/sec 67 m/sec .202 Mach
1.4 bar 330 °K 334 m/sec 237 m/sec 85 m/sec .254 Mach
1.5 bar 337 °K 337 m/sec 218 m/sec 103 m/sec .306 Mach
1.6 bar 343 °K 340 m/sec 202 m/sec 119 m/sec .352 Mach
1.7 bar 349 °K 342 m/sec 186 m/sec 136 m/sec .397 Mach
1.8 bar 355 °K 345 m/sec 169 m/sec 152 m/sec .442 Mach
1.9 bar 360 °K 347 m/sec 156 m/sec 166 m/sec .477 Mach
2.0 bar 366 °K 350 m/sec 141 m/sec 181 m/sec .517 Mach
Again, we have assumed stationary air before the wave passes, i.e. UCondA = 0; in most reallife examples, we would not be starting with stationary air, but instead we would have some initial velocity at the region of interest in the pipe. This can be positive (in the same direction the wave front is moving) or negative (flowing against the advancing wave front) or (rarely) zero, as seen in our graphic example here. If nonzero, this initial velocity would simply be added algebraicly to the value in the table as selected for the local pressure. Thus, the flow velocity within the pressure wave can be readily determined even if all we know is the initial flow speed and the initial and final air conditions. When the actual speed for 1.0 Mach under any air condition is needed, it can be referenced in the table above (interpolating if necessary for greater accuracy).
L Cottrill

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Letting Down
And yet another ...
The odds are strongly against getting air pressure to stay above normal indefinitely in an acoustic device like a pulsejet, so now it's time to look at the "trailing" pressure front, i.e. descending pressure. This is a case where "reversability" applies, and it turns out that this is, indeed, exactly the opposite action to what we just saw at the "leading" wave front. Of course, this means we are coming to the problem with a very different starting condition: the air is already compressed and moving forward at the speed established by the action of the "leading" front. We see this in the bottom "original locations ruler", which now shows up as a series of highly compressed parcels that are moving leftward (note the HIGH SPEED FLOW vector just above the ruler)  again, the meaning is that this is where the parcels would have been if the wave front had not just traversed this region:
(See graphic attached, below)
Again, we have "marked" selected air mass parcels with colored dots; note that these are not meant to be the same parcels marked in the earlier graphic. Looking at the yellowdotted parcel, we see that as the falling pressure front passes, a pressure difference is again developed, due to the parcel length in the pipe. This time, however, the higher pressure PHI is acting on the left side of the mass and the lower pressure PLO is acting on the right. Because of this, the vector PDIFF is shown acting rightward on the parcel (in the pipe section of the drawing), so the acceleration of the parcel is toward the right end. This results in slowing motion of the parcel as the wave passes, as indicated by the vectors labelled SLOWING FLOW, and also the displacement of the parcels rightward (note the locations of the yellow and greendotted parcels relative to their "original" positions on the ruler). In fact, in this case, the masses have returned to a stationary state (relative to the pipe wall)!
For this restoration to zero speed to be achieved, there also has to be a reexpansion of the masses; in fact, they need to be restored to exactly their original condition of pressure, temperature and density. This happens automatically as PAV decreases with the passage of the front across the parcel. At the end of the process, perfect restoration of the original air condition is complete, and the original velocity (in this case, zero) is attained. Hence, the initial speed SpeedcondA is what is denoted by the HIGH SPEED FLOW vectors in the drawing. Since the trailing wave front is moving along the pipe at 1.0 MachcondB and the pressure returns to its original value of exactly 1.0 bar, SpeedcondB will turn out to be zero, for this example. The trailing edge of the wave has taken up the kinetic energy that once appeared as forward motion and elevated temperatures of the masses that are now left behind.
The same table used above to show the relationships of pressure, temperature and speed for the increasing pressure wave front also applies here to the decreasing pressure front, except that we must switch the CondA and CondB subscripts, because in the present case Condition A is the compressed high velocity condition, and Condition B is the restored normal air condition (in this case, stationary in the pipe).
One other fact should be mentioned here, to make sure we're perfectly clear on the total concept: Even though the passage of the trailing wave front has restored the air to its original condition and velocity, it has not restored the masses to their original locations! The wave taken as a whole has in fact succeeded in displacing all the air mass it passed through significantly leftward. The magnitude of this displacement depends on the degree of compression achieved (i.e. the pressure difference between the original and pressurized conditions) and the duration of the pressure difference (essentially, the length of the wave, fronttofront).
The importance of this is to understand that it takes both pressure and time to perform the work of moving the air mass along. A lengthy lowpressure wave may move air mass more effectively than a much more intense highpressure pulse! This is a lot easier to see with a wave that is a smooth plateau between two short ramps than it is with the kind of wave shapes in a real life acoustic device, but the principle is the same. The wave really does displace mass, and that means actual work is accomplished by the wave passage (in other words, power is actually involved, just as it takes power for a fan or compressor to move some volume of air mass per unit time). Down the road, we'll see more "macroscopic" examples of how this works over large parts of the acoustic cycle time.
L Cottrill
The odds are strongly against getting air pressure to stay above normal indefinitely in an acoustic device like a pulsejet, so now it's time to look at the "trailing" pressure front, i.e. descending pressure. This is a case where "reversability" applies, and it turns out that this is, indeed, exactly the opposite action to what we just saw at the "leading" wave front. Of course, this means we are coming to the problem with a very different starting condition: the air is already compressed and moving forward at the speed established by the action of the "leading" front. We see this in the bottom "original locations ruler", which now shows up as a series of highly compressed parcels that are moving leftward (note the HIGH SPEED FLOW vector just above the ruler)  again, the meaning is that this is where the parcels would have been if the wave front had not just traversed this region:
(See graphic attached, below)
Again, we have "marked" selected air mass parcels with colored dots; note that these are not meant to be the same parcels marked in the earlier graphic. Looking at the yellowdotted parcel, we see that as the falling pressure front passes, a pressure difference is again developed, due to the parcel length in the pipe. This time, however, the higher pressure PHI is acting on the left side of the mass and the lower pressure PLO is acting on the right. Because of this, the vector PDIFF is shown acting rightward on the parcel (in the pipe section of the drawing), so the acceleration of the parcel is toward the right end. This results in slowing motion of the parcel as the wave passes, as indicated by the vectors labelled SLOWING FLOW, and also the displacement of the parcels rightward (note the locations of the yellow and greendotted parcels relative to their "original" positions on the ruler). In fact, in this case, the masses have returned to a stationary state (relative to the pipe wall)!
For this restoration to zero speed to be achieved, there also has to be a reexpansion of the masses; in fact, they need to be restored to exactly their original condition of pressure, temperature and density. This happens automatically as PAV decreases with the passage of the front across the parcel. At the end of the process, perfect restoration of the original air condition is complete, and the original velocity (in this case, zero) is attained. Hence, the initial speed SpeedcondA is what is denoted by the HIGH SPEED FLOW vectors in the drawing. Since the trailing wave front is moving along the pipe at 1.0 MachcondB and the pressure returns to its original value of exactly 1.0 bar, SpeedcondB will turn out to be zero, for this example. The trailing edge of the wave has taken up the kinetic energy that once appeared as forward motion and elevated temperatures of the masses that are now left behind.
The same table used above to show the relationships of pressure, temperature and speed for the increasing pressure wave front also applies here to the decreasing pressure front, except that we must switch the CondA and CondB subscripts, because in the present case Condition A is the compressed high velocity condition, and Condition B is the restored normal air condition (in this case, stationary in the pipe).
One other fact should be mentioned here, to make sure we're perfectly clear on the total concept: Even though the passage of the trailing wave front has restored the air to its original condition and velocity, it has not restored the masses to their original locations! The wave taken as a whole has in fact succeeded in displacing all the air mass it passed through significantly leftward. The magnitude of this displacement depends on the degree of compression achieved (i.e. the pressure difference between the original and pressurized conditions) and the duration of the pressure difference (essentially, the length of the wave, fronttofront).
The importance of this is to understand that it takes both pressure and time to perform the work of moving the air mass along. A lengthy lowpressure wave may move air mass more effectively than a much more intense highpressure pulse! This is a lot easier to see with a wave that is a smooth plateau between two short ramps than it is with the kind of wave shapes in a real life acoustic device, but the principle is the same. The wave really does displace mass, and that means actual work is accomplished by the wave passage (in other words, power is actually involved, just as it takes power for a fan or compressor to move some volume of air mass per unit time). Down the road, we'll see more "macroscopic" examples of how this works over large parts of the acoustic cycle time.
L Cottrill