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This might help inform or confuse the topic:https://www.har-bal.com/wp-content/uploads/Speaker_theory.pdf
Well, I think you will continue to struggle with the article as long as you insist on starting from an incorrect premise. I pointed out the obvious contradiction in your reasoning... do you not see that contradiction?
and this followup explanation where it says that "presence of the i shows that the velocity is out of phase with the pressure" and rewrites in terms of acceleration:https://physics.stackexchange.com/questions/405191/why-is-this-equation-true-for-the-sound-pressure-a-loudspeaker-creates
I'm not sure pressure is a function of velocity at all as Pmax is at Vmin... right when the cone has stopped... and not at Vmax. From what you've wrote, you haven't changed your premise at all.
I do think V matters, but likely as a corrective factor rather than a main factor, at least within the range of human hearing. While the air will move the instant the cone starts moving, it is compressible and we can look at the simplified ideal that the air is acted on evenly and pressure built evenly by the cone while it's in motion. If this is the case, then when the cone is at Vmax it's still in the process of compressing the air and the pressure is still going up. It's not until the cone stops moving that it stops compressing the air, which is at Vmin=0 and Amax. Also, another real life factor may be that the Pmax is right before Vmin because as V approaches zero, at some point the velocity will become slow with respect to the speed of sound, and V will be effectively zero even though it's not quite there yet.
As far as the math, I can do math... but, well... I'd much rather not do math unless I really have to.
It really boils down to: Pressure at time T = (a bunch of complicated stuff) x Velocity of cone at time T. p(t)=iρckϕv(t)This is the far field reduction of the equation. The guy says the presence of i puts the pressure out of phase with the velocity. i x 0 = 0.
Ahhmm.... I don't think it works that way. i is the imaginary unit, v(t) is a complex sinusoid i.e. cos(wt) + isin(wt). When you multiply the i in front you then get i cos(wt) - sin(wt) ...
The cone never even gets remotely close to the speed of sound. Any pressure built up in front of the diaphragm escapes at the speed of sound in free space...
I'm not sure about that. The air is being compressed by a cone going under the speed of sound, and the air is moving less than the speed of the cone unless it's right next to the cone. This results in a pressure wave that propagates at the speed of sound, but the compressed air is not moving anywhere near that fast.... how can a cone moving much slower than the speed of sound propel the air faster than it's own velocity? There's a difference between a pressure wave propagating through the air vs the air actually moving as a result of cone motion. Both are obviously happening to some degree, as pressure requires the air molecules to move closer together. I misspoke when I said cone motion relative to speed of sound, it's cone motion relative to how fast the cone needs to move to cause the air to compress significantly.
It is interesting to compare this speed [of sound] with the speed of molecules as a result of their thermal energy.
ρ (rho) is the density of air. (From here.)
Back to the question about when during the cone's excursion is the pressure in front of the cone the greatest, I got to thinking about how to test it, and came up with an idea. Play a 100hz tone and bring the microphone up to the cone until it just barely starts getting tapped by the cone. Record that and see where the tap happens. That should tell you when the cone is at maximum excursion relative to the pressure.Here is the result I got.
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