... by "tail will go away", i'm sure you mean the tail will not asymptotically approach zero, but rather will really hit zero ... after some period of time
In any case, the "reflective approximation" of the infinite line WILL have an impulse tail ... meaning, the impulse response will NOT, itself, be 'impulsive'. This is EXACTLY the same thing as saying that the frequency response will not be 'flat'.
Okay, so as we bring the ceiling and floor closer and closer together, the source starts to look more like a point source at audio frequencies and is bounded so as to constrain wavefront propagation increasingly close to laterally. Immediately close to the source, pressure can't escape, or dissipate by wave motion, as rapidly as it is building up (relative to frequency)?
I'm just trying to visualize the geometry a little is all.
I'm just trying to visualize the geometry a little is all.
Start with a finite line source, extending between perfectly reflective planes spaced 8 feet apart : the result (measured anywhere between the planes) is indistinguishable from an infinite line source.
Now extend a finite line source between perfectly reflecting planes spaced 4 feet apart : the result (measured anywhere between the planes) is indistinguishable from an infinite line source ... AND indistinguishable from the above case.
Diminish the spacing all you want ... as long as the spacing is finite, you STILL have an infinite line, rather than a point. Conclusion: reducing the spacing between the planes does not get you closer to a point-source .... but rather, remains indistinguishable from an infinite line.
Sure, mathematically. What is the physical process underlying the result? Sometimes if we look at the extremes, it can help to intuit the physical process better. Not always possible though, particularly when more than three dimensions are involved, but should be possible here. In higher dimensional problems one can only develop some intuition for the equations, at least according to Leonard Suskind.
i love analyzing extremes ... i'm an engineer, and i use that thought process ALL the time
But the problem in this case, is that we can reduce the spacing between the perfectly reflective planes all we want, and the acoustic field (between the planes) doesn't change AT ALL. We aren't gradually (nor rapidly, for that matter) approaching a point-source, as we reduce the spacing between the planes. So the thought process of "imagining extremes" doesn't work in this case because we don't "approach" the point-source "in the limit".So the thought process of "imagining extremes" doesn't work in this case
We approach a 2-dimensional cylinder. But, it should be enough to consider small enough spacing between surfaces that all audio frequencies wavelengths are much larger than the spacing. Also, use a listening point much closer to the source than any wavelength of interest. The impulse can be a finite approximation of sufficiently short time duration to be negligible relative to periods of audio frequencies. What do we see physically at the observation point as a function of frequency as time evolves using fast meteorological instrumentation? Which way does the wind blow and why as time passes?
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We don't see, or "approach" ANYTHING different than what we started with:
1. Start with a "finite" line extending between perfectly-reflecting planes, spaced, in the limit, infinitely far apart. Measure the acoustic field ... in frequency, or time .... anywhere you wish (between the planes).
2. Now place the reflecting planes 1 mile apart. Measure the field ... frequency, time, whatever you like ... anywhere between the planes. Result? IDENTICAL to step #1.
3. Keep moving the planes ... move the bottom one up, move the top one down ... and keep measuring : time, frequency, pink noise, gated pulses, whatever you like. Result? It's IMPOSSIBLE to tell that the planes have gotten any closer.
4. Our planes are now so close, they are only one quarter of an inch apart. What do we measure, anywhere between the planes? The EXACT same thing we measure in step #1 : same measurements in the time domain, same measurements in the frequency domain ... YES, even for quickly-spaced pulses followed by raised-cosine envelopes of female voices
ALL measurements between the planes are IDENTICAL to those from step #1.NOTHING has changed, as we move the planes closer ... no matter how close they get
No, it's coming DIRECTLY from the physical phenomenon ... of REFLECTIONS.
In reality, any reflective surface can be replaced by a "virtual source" behind (or above, or below) the reflective surface. All reflected waves behave IDENTICALLY to "interference" from a "virtual source". This observation comes directly from imagining, or mentally-visualizing, the physical phenomenon of reflections ... not from the equations. And this property holds for time as well as frequency, and it holds no matter how close or far you are from the reflective surface.
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YESMy question would be, take a point source, put it between two reflective planes...
If the space between the planes is infinitely small, just fitting the point source in between, do we get a line array result?
but it's only "observable" between the reflective planes ... a space which has now diminished, in dimension, to zero
Only if you can let go of the reality
But what happens if we space out those perfect reflective planes, and keep the point source in between them?
Say we have perfectly reflective planes spaced 8 feet apart, and an ideal point source half-way between. Between the planes, the field is identical (in all measurable ways) to a "spaced array" of an infinite number of point sources, 8 feet apart, extending (in space) without bound.
In THIS case, changing the distance between the planes will DEFINITELY change the measured field
(because we're changing the finite, non-zero distance between point sources ... real, and virtual).One really needs to just "draw" the scenarios ... no math involved!
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