My standing wave animation (above) is meant to show what we would see if the pipe were transparent and the air were not. Higher pressure would be a darker color and lower pressure would be a lighter color. This best shows the resultant standing wave, but if you save it and play it in a program that lets you speed it up, you will be able to see the forward and backward waves that drive the process. They aren't there (above) but your eye will add them since light waves and sound waves share many characteristics. The obverse is true in reality: the standing wave is an additive creation of the forward traveling wave and its backward reflection. (sine wave graph below, see UNSW physics site for many excellent resources)
To see changes of both motion and pressure simultaneously, unless you are a Tesla-oid savant, it helps to look at an animation. While pressure changes can be shown by dot density (above) or graph height (below), to show motion changes at the same time, animation is the best tool since it shows relative changes of position with time. The back-and-forth motion of the air particles—caused by a wave moving through the air—is a longitudinal vibration, back-and-forth along the length of the wave's travel, not transverse to the wave motion as suggested by the sine graph. So the above animation is more realistic, but the animation below best shows how the standing wave is created by the compounding amplitudes of coinciding peaks.
The three animations below show the difference between what these three inventors accomplished with waves in fluids. There has been no attempt in these three animations to show anything to scale, in tune, at the right frequency, the right length, etc. The focus here is on the moving parts, so just assume that the most important moving part—the air—is in a resonant state, inflow and outflow are in the right place in relation to nodes, etc.
CONSTANTINESCO usually used liquid in his vibrating circuit, and if he used his system to compress air he would have had a prime mover (not shown) generating a resonant condition with a short-stroke piston in a fluid circuit. Then a wave motor, depicted here as a second piston, would have powered a conventional air compressor (not shown). His work was transmission of energy from one place to another. The animation shows two driven pistons: one is driven by a prime mover and the other is driven by the wave and in turn drives an air compressor. There is no free energy involved.
BELLOCQ, if he had designed Neal's compression unit as an analogy of his own acoustically driven water pump, would have done it like this. A vibrating piston generates resonance in a tuned fluid circuit, and correct positioning of the hardware causes the fluid to be pumped to a higher pressure. All the work in this kind of hypothetical compressor would be done by acoustic energy. Again there is probably no free energy involved, but Bellocq's water pump did overcome a limitation of pumping water from above since it was a machine to which Torricelli's limitation did not apply.
NEAL went a step further by using acoustic power only to operate three check valves. This is the answer to Maxwell's riddle of 1870, "Maxwell's Demon." The work of getting the air moving was done mechanically by a prime-mover-driven conventional compressor. The compressor Neal built had an unconventionally large number of pistons (see US Patent 2030759) and was capable of compressing air to the same pressure as any single-stage compressor, but once the machine's output reached the correct frequency of the pipeline it was delivering air to, the only work of acoustic power was to operate three valves. So the work of keeping the air moving, and (in my opinion) the work of increasing the air's pressure did not have to be done by acoustic power. The advantage of moving the air mechanically was so great that having the big part of the job—compression—done by ambient energy already residing in the atmosphere, yielded free energy.
Many conflicting opinions exist about how Neal put air in a tank without resisting the air already in the tank. I change my mind about this more often than I change this website, so if you want to discuss Neal, please post on the forum.