For a very rough classification of the different chassis sizes, such a loudspeaker cabinet-independent formula is useful, especially if you have no other option.
So let's apply all formulas for a 6.5'' woofer. Specifically, it should be the 6.5'' Purifi driver, with about 0.131m effective diameter.
Besides this formula, other formulas were mentioned. Here is an overview of all formulas:
c = speed of sound
dia = effective driver diameter
f-beam < c/N*dia = 2600Hz / 1700Hz / 1300Hz (N = 1, 1.5, 2)
f-beam = 2*c/pi*dia = 1700Hz
f-beam = 2*c/dia = 1300Hz
f-beam = c/dia = 2600Hz
Exactly, nowadays, unlike 40 years ago, every developer has tools at his disposal that more or less accurately predict the radiation behavior.
For very accurate predictions the effort is quite high (because the behavior of the whole driver has to be simulated), for good predictions (around +-1 or +-1.5dB) simulations are relatively easy to set up.
In addition to the obvious influences such as cone shape and driver surround, the shape and dimensions of the loudspeaker cabinet also have a major impact on the radiation pattern of the driver (as already mentioned by others) - the depth of the cabinet also plays a role, which is often not taken into account during preliminary considerations.
So let's take our driver from above and put it into different speaker enclosures.
1) a very narrow cabinet with 17cm width
2) a wide cabinet with 30cm width
3) a very narrow cabinet of 17cm width and shallow depth
4) an asymmetrical cabinet with a width of 8cm at the top and 33cm at the bottom
View attachment 133291 View attachment 133292 View attachment 133293 View attachment 133294
As a measure of beaming, we consider the -6dB point (sound pressure level is lowered by 6dB) relative to the normalized axis frequency response, for each enclosure shape:
View attachment 133302
After the transition from full-space to half-space, the loudspeaker cabinet significantly determines the radiation pattern.
- Wide cabinets show a significantly different radiation pattern than very narrow ones (green and cyan curves).
- The cabinet depth also has a significant influence on the radiation pattern (cyan and red curves).
Only above 2.2 - 2.6 kHz the driver itself determines the radiation.
For all those who think it would change fundamentally, if one does not normalize to the axial frequency response, here the same analysis normalized to the 30° frequency response:
View attachment 133303
For all those who can't do anything with the upper diagram, here are the simulations of the frequency responses 0-180deg (normalized on-axis):
View attachment 133307 View attachment 133308 View attachment 133309 View attachment 133310
How well the above formulas apply, everyone must decide for themselves.