I've had a few people ask for an update on the sound testing so here it is. My sound engineer has been designing high end microphones for much higher paying clients, but now he is back on the job. I'm asking him to give me the raw sound amplitude vs. time signals that we collected in a form that I can create overlaid plots and also subtract one from another to show at what time before impact there is a discernible difference heard at the target location. The short answer is "it's complicated and is taking a while". I posted his actual update below for your entertainment:
Hi Bill - I completed a Mathcad script to enter 4 mic sound levels from recordings. It then cranks out x,y coordinates of where each arrow passed through the 2ft dia mic-array-circle, and an adjusted measured level as if the arrow had always flown a fixed 2ft from one specified reference mic. A scaling factor is also provided to adjust each recorded waveform's gain in my sound editor software, such that all 13 flights in a set can be fairly compared to each other. The raw (unadjusted) files I crudely prepared just after our testing, lacked such adjustment, thus they do not provide fair comparisons to each other (unless you happened to make 13 perfect shots always centered through the circle). It seemed intuitively obvious to me that amplitude info from the 4 mics would in theory allow a consistent adjusted level measurement, no matter where the arrow passes within bounds of the circle, but then I couldn't find the mathematics to support. So I proceeded to set up the problem and derive needed equations. Between Mathematica and the Maple feature of Mathcad, symbolic solving of systems of equations was possible, and barely manageable. Some equations in intermediate stages blew up into thousands of terms, so took much steering and strategizing. Resulting equation set fit on a single page. The script has been tested with a handful of theoretical data cases, including debug through boundary conditions and exception handlings. So it is now time to try real data!