Wednesday, December 21, 2011

A Parabolic Reflector

So, I want to play with a parabolic reflector to make a directional microphone, bit I don't want to spend a lot of money.  I can buy 1/32" polycarbonate sheets from TAP plastics, and maybe I could try forming that into a parabola.  I thought about just heating it and laying on top of a round trash can, letting gravity and maybe a small weight shape a curve.  Then I researched vacuum forming on the internet.  I think I could cobble something together...

There are instructions on the net for making parabolic microphones out of dollar store umbrellas, mixing bowls, and woks, but those aren't likely the right shape or size to really do the trick.  A parabola is a specific shape that reflects a set of incoming waves onto a point in the middle, so random bowl shapes won't necessarily give the concentration of sound energy I'm after.  The other issue is the size.  It takes a large reflector to deal with low frequency sound - an efficient reflector apparently needs to be twice as large as the sound's wavelength.  The speed of sound is about 1100 ft / sec, so a 2 ft diameter reflector is efficient to about 550 Hz, which is low enough for many wildlife sounds, and acceptable for human speech.  I also know that the $1200 Telinga parabolic mic that a lot of pros use is about 23", so mine will be one bigger.  Yeah.

For this purpose, a parabola is expressed mathematically by the equation:

y = x2 / 4f

where f is the height of the focus.  I wanted the focal point to be within the bowl of the parabola, so that sound from the sides would be blocked from the microphone element, and I wanted the diameter to be 24".  After testing a few values for f, I settled on 4.5.  That creates a parabola 8" deep, with a focus 4.5" from the bottom.

I calculated a y value for every half-inch from -12 to 12 (to make a 24" diameter dish), and plotted out the shape on a piece of construction paper, using a straightedge for the x axis and sliding a carpenter's square along it to measure up for the y axis.

I connected the dots, cut out the paper, and transferred the shape carefully to a piece of melamine-coated hardboard.  Then I cut the hardboard on the band saw and gave it a light hand sanding to make a smooth curve.  I might have been slightly more accurate to draw it directly on the hardboard, but I think my error was less than 1/32".

Next I cut a 26" diameter circle of 1/2" OSB sheathing with a router jig, and mounted a 8" long scrap of 4x4 to the center of it.  I carefully located the exact center, drilled a small hole in the top of the 4x4, and stuck a small finish nail in.  I cut the head of the nail off, leaving a small pin as a pivot point.  I drilled a mating hole in the center point of my hardboard pattern, and cut a notch in the outer part to ride on the edge of the circle.  Now sweeping it around on the pivot will form a 3D parabola.

Next I cut some rings out of a chunk of 1-1/2" foam insulation scrap.  I calculated the outer diameter at 1-1/2", 3", 4-1/2" and 6" high, and cut circles at that size.  I stacked them up, glued them together with some spray adhesive, and made a base for a plaster layer.  I trimmed the corners off to start rounding the shape.

I coated the foam layers with strips of fiberglass drywall reinforcement mesh tape.  It's a little sticky on one side, but I stapled the longer pieces to the wood block at the top and / or the wood base.  I used the pattern to make sure there was around 1/2" gap everywhere.  I also put a big lazy susan bearing under the wood base, which makes it much easier to work on.

Then I applied a layer of setting-type drywall compound to the form.  This stuff is basically plaster of paris with some fillers and retarders to slow the setting rate.  The type I used solidifies in 40 minutes after you add water.  Again, I was sure to leave this rough surface 1/8" - 1/4" low of the final finished height.  I let the whole thing dry out for a couple of days at this point.

Finally, I wiped a couple coats of Zinnser shellac on the hardboard pattern to waterproof it, and then used it as a screed to apply another coat of the setting compound.  The resulting surface has a few holes and defects, but is overall a very accurate shape.

After some filling and sanding, it's pretty smooth and I hope ready to use as a vacuum forming mold.

Update:  to my chagrin, I never got to test this mold out.  It turns out I'd have to build a big insulated box to make an oven to heat the polycarbonate sheet up, if I want to form it.  My original quick and dirty forming oven didn't have enough capacity.  I might do another one of these, in a size that I can just use our household oven to heat the plastic.  Don't tell my wife.