How do cardioid loudspeakers work?

Here's my attempt to explain how cardioid (passive and active) loudspeakers work. I hope it makes sense!

An omni microphone

I find that the best place to start when it comes to understanding cardioid loudspeakers is with the omni mic. Sounds daft, but bear with me. Imagine a membrane stretched across a bucket - like a drum. Changes in air pressure (sound) can only act on the front (top) of this membrane no matter what direction the sound is coming from. For the purpose of this example our bucket is tiny, that means that sound hits the front of the diaphragm with equal force no matter what direction it comes from. It's pickup pattern is omni-directional.

Figure of eight microphone

Let's cut the end off the bucket. Now sound can get to the back of the diaphragm as well as the front of the diaphragm. So sound coming from the front of the diaphragm hits the front of the diaphragm and vice versa for the back.

Positive pressure from the top pushes the diaphragm down, and positive pressure from the back pushes the diaphragm up. This explains why sounds behind a figure of eight microphone are of the opposite polarity to sounds in front of the microphone.

Now imagine sound coming towards the edge of this diaphragm. An equal amount of pressure reaches either side of the diaphragm and so cancels itself out resulting in the distinctive figure of eight pickup pattern: sound from the front doesn't cancel it's self out so can be heard where as sound from the side gets cancelled out.

Cardioid microphone

So we can take the principals used above to create a cardioid microphone, we just need the situation that we had at the side of the figure of eight example to occur behind the diaphragm instead of to the side. To do this we need the sound from behind to arrive at the front of the diaphragm at the same time as the sound reaches the back of the diaphragm. This means that we need to slow the sound down that's going to hit the back of the diaphragm. This can be done in all sorts of fancy ways but we can just imagine an omnidirectional capsule with a little port in the back. Sound that enters this port has to travel the same distance to get to the diaphragm as sound that doesn't travel down the port.

From a microphone to a loudspeaker

You apply this to a loudspeaker by just turning it on it's head. Rather than air pressure moving the diaphragm, you use use the diaphragm to create the air pressure.

As a loudspeaker pushes air to create an area of high pressure an area of low pressure of equal amplitude is created behind it. If you were to add this area of low pressure to the area of high pressure they would cancel each other out (this is why drivers sound odd when they're not in a box).

So if you create the port and the hole in the back of the cabinet so that the pressure coming from the back of the loudspeaker meets the pressure coming from the front of the loudspeaker at the correct time they will cancel each other out.

Obviously if you want to do this in the real world it gets a whole lot more complicated! 

Examples of cardioid loudspeakers have been around for ages. Here's a patent for one in 1971!

Rather than building a cabinet and trying and do this with a single driver and amplifier channel it's much easier to fake the delay port with an extra loudspeaker. Unfortunately doing the passive version of this is not quite as easy as I have explained. As soon as you have to design a loudspeaker in a box it all starts to get very complicated. To design a passive cardioid loudspeaker not only do you have to do the electromechanical design work for a decent loudspeaker, you suddenly add a whole load more variables into your equation by needing to use the rear energy for fancy tricks.

A real world example

I did a quick experiment to demonstrate these principles in the real world. I used some small 4" loudspeakers. Delaying a rear facing loudspeaker to be in time with the forwards facing ones I flipped the polarity. Some measurements and a little EQ later (to switch off the rear facing tweeter!) I got the results shown below.

Figure 1 shows the response of three loudspeakers - two used for forwards pressure and the third for rear cancellation. The measurements show the SPL of the forwards addition (in green) and the rearwards cancellation (in orange). Note the LF boost caused by the coupling of three drivers. Also note the HF cancellation caused by interference from either the two HF tweeters (it's not the floor because the microphone was in contact with the floor). Figure two show the same principal but this time only using two loudspeakers.

Figure 1: Three loudspeakers. Orange = rearwards SPL. Green = forwards SPL.

Figure 2: Two loudspeakers. Orange = rearwards SPL. Green = forwards SPL.