Polar response
What's all that about then?
Here's a practical example of it. Turn a radio on, one of those little ones with a 2", 3" or 4" speaker in it. Do this in your bedroom, or another well furnished room, preferably carpeted, with the curtains closed.
Position yourself so that the speaker is pointing right at your ear. Note how it sounds. Now move yourself so your ear is 1" away and 6" off to the side and note down how the sound has changed - this is 30 degrees off-axis. Now move so that your ear is still 1' away but now 1' off to the side - this is 45 degree off-axis. And then so you're 1' away but and 2' off to the side - this 60 degrees off-axis. And then move right to the side so you're 90 degrees off-axis. And finally stand behind the speaker so you're a full 180 degrees off-axis.
In each position I bet you heard the sound change. The most obvious thing you'll have noticed is that the sound became less bright/clear and more dark/muddy the further you moved off-axis.
The simple (but incomplete) theory
Sound is a waveform, and when a wave is emitted from a source, the dispersion of that source is proportional to the ratio of wavelength to source size. The wavelength of low frequencies is very long whilst the wavelength of high frequencies is very short. The simplest way to consider a loudspeaker is as a rigid piston which produces all of its output at all frequencies evenly across the entire cone. Let's look at a pistonic 12" woofer - the 12" woofer actually has a radiating diameter of just over 10". When we stand directly in front of that woofer (on-axis) the sound emitted by from all points of the woofer all reaches us at the same time. However when we stand off to the side of the woofer then the sound emitted by the side which is closest to us reaches us sooner than the sound from the furthest side. If the different in time means that we hear the right hand side of the woofer moving forwards (compression) just as the left hand side of the woofer is moving backwards (rarefaction) then the sound cancels out. Clearly this is not a good thing!Simplistic beaming limits
With a completely rigid woofer which is acting in a totally pistonic manner, then the following frequencies are the points above which the dispersion becomes particularly narrow (assuming a typical surround width of 0.9"). Before you jump to any conclusions bear in mind that the diaphragms on woofers are surprisingly light and therefore rarely behave in this manner.
Nominal diameter Frequency
8" 2185Hz
10" 1650Hz
12" 1330Hz
15" 1025Hz
What matters more? Up and down or side to side dispersion?
Think about a typical venue - where are the audience's ears? Pretty much all within a 2' thick plane, positioned between 4'6" and 6'6" off the floor. Where are your bandmates ears? Within a similar plane but in this case raised above the stage rather than the main floor. What about horizontal positions? Your drummer is often directly to the side of your cab, your guitarist might be beyond the drummer along that line, you're probably off to the front and to the side, and the singer and audience are all over the place! The bigger the stage and the more players in the band the more varied the angles at which your bandmates' ears are to your cab. So what we want is for the cab to disperse really well horizontally.