The inverse square law. Sound and radio waves spread out as they leave their source. The intensity of the waves is inversely proportional to the square of the distance of the wave from it's source. So if we require 1 watt of energy to achieve a range of one meter, we will need 4 watts to get the same signal level at two meters. Nine watts at three meters, sixteen watts at four meters, 25 watts at five meters and 36 watts at six meters etc.. If we require long ranges then we either need very high power levels .... or we need to stop the dispersion in the first place. One means of reducing beam dispersion is to use a parabolic reflector to produce a very narrow beam.
A similar problem exists with receiving systems in that the apparent area of the radiator decreases by a quarter each time the range is doubled. By using a parabolic reflector for reception we are focusing on the radiator and effectively pick up the radiated signal before dispersion occurs.
Parabolic reflector size
As always ... size matters !. In practice this means that we cannot expect a surface to reflect if it is narrower than the wavelength of the signal. The usual example is that a one kilohertz signal has a wavelength of about one foot. This means we can expect a one foot diameter parabolic reflector to perform badly at 1 kHz. For example a 36" diameter reflector with a 12" focus only starts to become reflective at about 1 kHz. To cover the normal speech range of 300 - 3300 Hz, we need very large reflectors. However at ultrasonic frequencies and above parabolic reflectors work extremely well and are relatively simple to construct.
Focal length considerations.
By varying the curve of a reflector we can vary it's focal length. Short focal length reflectors are deeply dished. For audio use it might be thought that by using a deeply dished reflector would mean that the microphone would be situated within the rim of the dish, thus improving the directional response. Unfortunately this causes a resonant cavity effect and a large peak on the response curve, which is omnidirectional. By using a longer focal length the microphone is now outside of the 'flatter' reflector, which avoids the resonant cavity effect ..... but decreases protection against side noise. The trick is to make the microphone element directional too.
Construction of a parabola template
I have adopted a non-mathematical approach to this introduction, which means that it does not tell the complete story. If you are interested in looking deeper into the subject you will find lot's of information on the internet. For a non-mathematical method for constructing a parabolic curve template of any focal length ..... you need two nails and a bit of string !. From the template we will then create a male master of the reflector and from that you can make a fibre glass reflector to actually use.
Not a pretty picture but ..... tie the sting into a loop and hook it over the two nails. Now place the pencil inside the string loop and pull make the string taut. By swinging the pencil around in an arc .... it will trace a perfect parabolic curve. The focal length of the curve is measured from this line to the right hand nail. If you don't like what you get, then move the position of one nail or get a different length loop of string !. When you are happy with the result, draw a line from both nails through the traced curve. This will tell you the exact center of the curve. You now have your paper template.
Construction of a plaster lathe.
In order to make a glass fiber reflector we first need to make a master to lay the GRP up on. If we had the means and the skill we could turn a master up out of wood or plastic, but we can also make one out of plaster using a plaster lathe that you can easily make yourself. First glue your paper template onto a piece of stiff sheet such as aluminum, plastic or even hardboard (with a coat of varnish to make it waterproof). Then carefully cut out the surplus from the inside of the curve. Next you need to slot a length of wooden dowel and glue the exact center of the curve into the slot. We next get a piece of board and drill a hole in it for the wooden dowel. This will make a pivot point around which we can rotate our sheet template.
The black is the template ... that can rotate on the grey board .... by using the red dowel as a pivot point. This, believe it of not is a plaster lathe !. First apply a thin coating of Vaseline onto the dowel, which will assist in removing it later. To make smooth strong plaster master we will use plaster of Paris, which we need to mix with water to get a creamy mix. We then dribble it onto the board to build up the male parabola. If you occasionally rotate the template it will scrape away any surplus plaster. Keep dribbling and scraping until you have built up a smooth plaster master that fits the template perfectly. The plaster will set hard within 20 minutes. Plaster is porous so the next task is to seal it and the board ... with two coats of varnish. Your master is now ready to use to make the GRP reflector.
Construction of a GRP reflector using the plaster master.
First apply mould release to the master. This could be two coats of PVA of five coats of wax polished off between applications. For reflector up to three feet in diameter you will probably need to lay up two layers of 400 chopped strand mat, four to five feet diameters three layers with perhaps a two inch stiffening strip around the rim. It is normal to apply a gel coat to the master before laying up the chopped strand mat. This will ensure that the reflective surface of the finished GRP reflector is perfect. In short wet the master and mat with resin and apply to the master. Consolidate with either a roller or a stippling action with a brush. You should be able to lay up two layers in one session and the pot time of the resin will be between 15 -30 minutes depending upon how much catalyst (MEKP) is mixed with the resin and also the ambient temperature. The cured resin laminate will need to be forced off the master. The usual way of doing this is to go around the casting knocking thin wooden wedged between the laminate and the master, until it pops off.
We normally think of parabolic reflector surfaces as being a smooth continuous curve, but it is possible to have a facetted surface made up of many small reflective area's. I mention this because I have seen an Ultrasonic Audio spotlight that appeared to use a facetted surface. In practice a good faceted surface can be almost as efficient as a conventional continuous surface.
At any wavelength unwanted vibration to the transducer can be a problem. The solution is to mechanically decouple it from the reflector and mounting assembly using anti vibration mounts. As I mentioned earlier tranducer's can also be made to be directional by encasing them with a sound proof material and only leave the intended active surface of the transducer open. Plasticene work well with microphones !. Have fun .....
John Kent 2007