With you so far!
Is there a quick and dirty way to determine at what sort of frequency a cone will start to become omni directional?
Keith
I am glad you asked for quick and dirty as the exact computation is quite complex. The simple concept goes like this:
Here we have a circular transducer. If it were 100% beaming then there would be no spreading and the waveform goes out with the same diameter of the transducer as shown with the dashed horizontal lines ("BEAM AXIS"). The degree the beam spreads is said to be the angle/2 or alpha/2. Computation of that is as follows, assuming we stop when the beam strength is at -20 db
S (alpha/2) = .870 * c / fD
c = speed of sound in the medium
f = frequency of the sound
D = Diameter of the transducer
It is super important to stay consistent with units of f, c, and D. Don't mix inches, meters and millimeters
. I have a spreadsheet for this although my application was computing this for transducers underwater (fish finders) so uses different constant for "c".
Without going through calculation, we see that as frequency gets higher, S becomes smaller. When S becomes smaller, then the beam axis is getting narrower and closer to the transducer width.
When diameter gets larger, the same thing happens. So all else being equal, a larger woofer will be more directional than a smaller one at the same frequency.
A typical bookshelf speaker with an 8 inch woofer and 1 inch tweeter falls victim to this. The crossover frequency will be too high forcing the 8 inch woofer to play higher frequencies. Higher frequencies means our beam angle gets smaller and hence the woofer will become directional. A directional woofer then means there is less sound coming out the sides. Translated, we have a dip in mid frequencies off-axis. That dipped response combines with the direct sound and screws up the overall response.
A better speaker would use a 5.25 inch woofer which would become less directional per above formula. But it won't sell as well because it won't play as low and look cheaper.