From: dpierce@cartchunk.org   
      
   UnsteadyKen wrote:   
   > ChrisCoaster said...   
   >>My theory is that small rigid cones with high excursion move as much   
   >>air as effectively as a huge cone with less excursion and perhaps not   
   >>as structurally stiff.   
   >   
   > It might work at very low frequencies less than 10 hz perhaps.   
   >   
   > In the normal frequency range, 500hz for example if the cones were to   
   > use the full 1 inch travel yet still produce an accurate representation   
   > of the waveform presented to them they would have to accelerate and   
   > decelerate at values approaching infinity or thereabouts thus implying   
   > a cone with no mass would be required.   
      
   Uh, no. Not even remotely close.   
      
   Acceleration is easy to calculate: assume 500 Hz, with a peak   
   excursion (XMax) of 1.3 cm (that's a LARGE excursion for any   
   woofer).   
      
   Since   
      
        Xt = Xmax sin(wt)   
      
   where Xmax is the peak excursion (1.3 cm in our example), w   
   is radian frequency (2 pi F) and T is time. The result, Xt,   
   is the time-dependent position of the woofer cone.   
      
   Differentiating to get velocity, we get:   
      
       Vt = w Xmax cos(wt)   
      
   And, differentiate once more for acceleration:   
      
       At = w^2 XMax -sin(wt)   
      
   Select t for the point of maxiumum acceleration, such   
   that sin(wt) = 1, and we get:   
      
       At = |w^2 Xmax|   
      
   We're taking its absolute value because, regardless of   
   direction of acceleration, acceleration is acceleration   
   for our purposes here.   
      
   So, plugging in some real numbers, since 500 Hz = 3140 r/s,   
   we get:   
      
       At = (3140 r/s)^2 *.013m   
      
   or about 128,000 m/s^2. Yes, that's a LARGE acceleration,   
   but it is most assuredly NOT "infinite."   
      
   However, on a technical basis, you assertion is, well,   
   patently absurd for a number of reasons. Let's take the   
   rest of the parameters specified and see what drops out.   
      
   Assume a nominal 9" woofer moving with a peak excursion of   
   1.3 cm at 500 Hz. What's coming out of such a contrivance?   
      
   Well, from the other post, we know that   
      
       P = pc^c sqrt(Sd w^2 XMax)   
      
   And considering that the Sd for a nominal 9" woofer is   
   is about 0.032 m^2, at 500 Hz, this driver, IF it could   
   survive AND if enough electrical power were available to   
   drive it, would be producing a sound pressure level on   
   the order 152 dB SPL. which is about 1,000 time higher   
   than the threshhold of pain. That corresponds roughly to   
   an about of about 1,000 acoustic watts, or about a   
   horsepower and a third of pure sound.   
      
   And assuming such a driver had a sensitivity on the order   
   of 90 dB 1W @1m, you would need an amplifier capable of   
   on the order of 150,000 watts to get there. If such a speaker   
   had a typical magnet structure whose total thermal dissipation   
   was on the order of 5 C/W, given that 149,000 of those watts   
   end up being dissipated as heat, you'd have a magnet structure   
   whose temperatur would rise about 30,000 degrees C, hotter than   
   the surface of the vast majority of stars (though not hotter   
   than the surface of most neutron stars, at least).   
      
   Let's, if you don't mind, assume some more, well, realistic   
   conditions. Let's say we splurge and get a kilowatt of   
   amplifier power. Such a speaker under THOSE conditions,   
   BEST case, is going to produce a sound pressure level   
   at 500 Hz of about 123 dB SPL. A 9" nominal cone producing   
   123 dB SPL is going to have an excursion of about 0.05 cm,   
   about half a millimeter, about 0.02" or about the thickness   
   of 2 business cards.   
      
   Now, the acceleration under THOSE conditions is:   
      
       At = (3140 r/s)^2 * 0.0005 m   
      
   About 5000 m/s^2 to produce a sound level that's still,   
   well, physiologically dangerous.   
      
   Now, take it one more step: let's assume the guy wants   
   to play his 500 Hz music REALLY loud, 100 dB SPL (that's   
   REALLY loud). To get there would require our 9" woofer   
   to move a whopping 0.0033 cm, about 0.033 mm or about   
   1/1000 of an inch. At that point, to produce a sound   
   level that is ooud enough to seriously impair conversation   
   between two people standing 2' apart would result in an   
   acceleration on the order of 325 m/s^2. Hardly "infinite",   
   hardly "thereabout infinite", and, indeed, not even in   
   the same zip code.   
      
   And the amplifier at that point would be putting out a   
   blazing 15-20 watts or so.   
      
   But the fact that some hypothetical woofer COULD have an   
   excursion of 1" and COULD go to 500 Hz, does not mean that   
   in any realizable universe it will do both at the same time.   
    From the above, just on the issues sound pressure level and   
   electrical power requirements, it's a technically absurd   
   assumption that it even might.   
      
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