So here we go;
pressure/velocity/... phase vs pressure/velocity gradient...
I think this is what confuses people, and I mention this in my video too. Signals and waves are not the same. But they are often conflated into one thing by both students and engineers.
(Steady-state conditions are assumed throughout, so all field variables are complex.)
Signal view
A tube is excited at one end by a piston. Starting out, waves are
propagating only down the tube, with no reflections are present. The tube is infinitely long (abstraction). The acoustic impedance Za is equal to rho*c/S. This quantity is real. Since acoustic impedance is pressure divided by (volume) velocity, the present impedance is real. Pressure and velocity are thus in-phase in this situation. (If they were not, they would have different phases, and thus their division would not be real). If we plot this situation right as the piston has moved outwards to cross the rest position, we get the following picture (and the piston is shifted to the right to better see the pressure):
Red is positive pressure, blue is negative pressure. The piston is green and we are showing the DISPLACEMENT. We see that at the piston the displacement is zero, and that the pressure is positive to its highest amplitude. As velocity is 90 degrees out of phase with displacement, this all makes sense. If we show the piston VELOCITY, it should be at its highest positive value when the pressure is at the highest positive, and zero velocity should follow zero pressure. Let us check, and make the piston another color to indicate that it is now piston VELOCITY shown via its position:
The piston plotted has zero velocity in this instant (be careful; its placement at the rest position now indicates that VELOCITY is zero), and the pressure is also zero. So the pressure and the velocity peak at the same TIME, and we are looking at a single point in SPACE. That pressure and that velocity we evaluate at the piston position, or any other point in the tube, constitute SIGNALS.
Different phase relationships are found for different situations. Instead of the infinitely long tube, we can look at a closed tube. We now get
standing waves. In that situation, we will find that the pressure at velocity (signals) are 90 degrees out of phase with each other; they are in so-called
quadrature. In that situation, the pressure will be zero when the velocity is at its highest and at its lowest. Or as we show below, the pressure will be zero, when the displacement is zero:
Now, the piston in a baffle is yet another situation. In that case, the piston sees a mass-like impedance, and the analysis shows that acceleration will follow pressure, or negative displacement goes with positive pressure. Acoustic radiation is more than just a mass, but this is for the elevator pitch.
Wave view
So here is where it can go wrong. Looking at plots of waves and not being aware that this is not signals. The waves are plotted in space and changes with time, so you can animate it, but a signal varies in time only, and can only be animate as a phasor rotating.
A static plot of pressure and velocity can be misleading, as you are now typically looking at the real part only of complex values. First the propagation example.
Pressure is blue, (particle) velocity is green, the numerical values are not important, and to the left is the piston, to the right is the rest of the tube. The peaks align. Which seemingly makes. We said that for propagation pressure and velocity are in phase. But there is something else lurking, and that is that there some governing equations that have to hold. On is sometimes called Euler's equation, although that means different equations to different people.
Here we assumed steady-state conditions, so the velocity components and the pressure are complex values phasors. So at first glance there seems to be something wrong with the figure above. When the pressure peaks, it has a zero spatial derivative. But we need to remember that there is an "i" in the equation, which shift the phase on one side 90 degrees compared to the other side. And so looking at the plot of the real part only and deducing something from there is very tricky in general. Also, it is only when you animate it that you see the wave propagation.
Similarly, it is tricky just to look at the standing wave situation with the real parts plotted and understanding the field in its entirety:
The pressure varies down the tube, but the velocity is seemingly zero everywhere? Well, we need to remember the complex-ness of the fields, and that static images can be misleading. So we animate:
We see that in fact the velocity is not zero, we were just looking at a point in time, where it was. Also, the fact the peaks are a quarter period apart does not always translate to signals being in quadrature(!). If we animate pressure, velocity, acceleration, and displacement, we this:
Displacement, velocity, and acceleration all peak at the same places in SPACE for this standing wave situation, but not at the same TIME. Mixing up what is coming from the signal and what is coming from the wave is something I see all the time.
The issues mainly stem from not understanding complex numbers:
- Complex number (magnitude, phase, real part, imaginary part) are a prerequisite for understanding signal processing (transfer functions, filters, crossovers)
- Signal processing is a prerequisite for understanding field and physics in general (wave propagation, transductance principles, speaker drivers)
This is also why there are so many posts on group delay and phase delay that mix up these terms. They start from some fixed plot of a sinusoidal signal and then from there, delays are deduced from phase. That is the wrong way to go about it.
A more hidden issue, but very interesting, is that in many situations acoustical engineering students are taught about the plane wave tube, and how you can calculate the frequency by looking at the difference between two pressure peaks in the tube. I think the majority of people will think this is universally true. But it is not. There are so-called tube modes. Each have a characteristic frequency. If the mode is excited, it cannot propagate but dies out exponentially near the excitation source. Above it, it can propagate. But only part on the frequency of the excitation is linked to the propagation in the axial direction of the die. The rest, goes to the mode pattern in the cross-sectional direction. So slightly above the mode's cut-on frequency where it can propagate, there is very little axial frequency. So, there is a very long axial wavelength, any length you want, pretty much. So you would with two microphones measure a very long way between two pressure peak, calculate a very low frequency, and think that this is the applied signal frequency. But it is not. So certain things that are engrained in engineers are only correct in certain situations. For this case, the link between frequency and pressure peaks only works for plane waves, as the plane wave can always propagate.
In conclusion, do not learn acoustics purely by experimentation and intuition. You cannot see the piston moving as it moves to fast, and you can certainly not see the air particles move. So you have to build your intuition via the mathematics and simulations, and not via analogy where you experience something else and then link it to your situation.
Pressure and velocity are linked the acoustic impedance. These are all complex, and can have any relation, not just the extremes of mass, pure real resistance, or inductance. And the waves that occur can be a mix of standing and propagating ones. You have to look at each situation individually. As the variables are complex, it will not suffice to only look at real parts. And animate your variables for added insight.
