Thursday, 18 May 2017

Low Frequency Fun With Organ Pipes

\/()43 |_|K19, Going Postal
Not \/()43 |_|K19's organ

What always throws me (and quite a few others judging by some of the diagrams on the web) when I try and calculate the frequency of the tone produced by an organ pipe, is that the arrangement at near the bottom, which is like a referee's whistle, counts as an *open* end to the pipe.
Therefore, for an ordinary organ pipe which is also open at the top, the fundamental resonance is a half-wave - that means, half the wavelength fits in the pipe.

\/()43 |_|K19, Going Postal

In fact the pipe "appears" to be slightly longer than it really is for these purposes because the air is still somewhat constrained immediately after it has left the pipe (imagine you're in the crowd pouring out of the theater after watching "Joseph and the Technicolour Dreamcoat" - you're outside the building but still constrained by the density of the of the crowd until you've traveled some distance away. (This "end correction" applies to reflex ports on loudspeakers too.)

Pipes can also be "stopped" by plugging the end. This creates a half-open pipe and the fundamental resonance is a quarter-wave. This will play about an octave lower.

Try it now - grab for example a Hoover attachments (other brands of domestic cleaning equipment are also OK). Blow over one end, leaving the other open. That tone is the result of a standing half-wave forming in the tube. Now cover the far end. About an octave lower? That's because only a quarter wave is inside the pipe, so the wavelength is twice as long as before.

\/()43 |_|K19, Going Postal

So, how low do they go? Well, the lowest frequency normally used is an unstopped 32 foot pipe. If the half-wave is 32 feet then the whole wavelength is 64 feet, which gives us a frequency of 16Hz. There is talk of stopped 32 foot pipes, 64-foot pipes and even 128-foot pipes, but I'll let you do your own googling on that.

Anyway, isn't the range of human hearing 20Hz to 20KHz? So 16Hz ought to be inaudible?

Erm, it's complicated. Very low frequencies are perceived by the ear as a flicking or fluctuation of pressure more than a "sound" as such. The basilar membrane is a long thing in your ear that has hairs on it that resonate at different frequencies.

\/()43 |_|K19, Going Postal

Recent research (ref the badly named "A ratchet mechanism for amplification in low-frequency mammalian hearing") shows that these resonances only happen down to about 1000Hz, below which a kind of "tuned amplifier" formed in neighboring tissues takes over down to about 40Hz.

However, all the resonances including the 40Hz one really respond to a range of frequencies, which could extend as low as 20, or 16Hz. The sensitivity would be lower, which seems to bear out, and in some sense the pitch would be less readily apparent, due to being "off the end" of the pitch scale. Does this bear out? Well, it's subjective.

Personally, I reckon that for any frequency, no matter how low, the ear detects it before it is "felt" by any other part of the body. But you could argue the ear is feeling it rather than hearing it if the oscillation is perceived while the tonality is not.

\/()43 |_|K19, Going Postal

Maybe you fancy these frequencies for your own home musical setup? For that you want a subwoofer. The bad news is most subwoofers for domestic use don't even reach down to 20Hz, let alone 16. The ones that work with small satellite speakers may be more accurately titled "freestanding woofer cabinets".

There are a couple of physical laws that make it hard to design the ideal subwoofer. One is Hoffman's law. It states that, for a given cabinet topology (I'll get back to that) the low frequency limit, the efficiency and the cabinet size are locked into a three-way trade-off.
Double the size and you can double the efficiency (efficiency is important because it's expensive to send thousands of watts of power into your speakers). To reduce the low-end by an octave, you have to either make the cabinet EIGHT TIMES bigger, or allow the efficiency to drop by a factor of eight. Low-end is expensive in this trade-off!

Clearly a "subwoofer" that only makes 32Hz (fairly typical) is much simpler to manufacture than one that gets an octave lower at 16Hz.

The topologies? Well, a sealed box is one, a ported reflex box is another, and there are a handful of bandpass topologies. Reflex is more efficient than sealed and bandpasses are the most efficient for producing low frequencies.

\/()43 |_|K19, Going Postal

The other problem is the amount of air a speaker has to move. Three dimensional air refuses to "accept" a source of sound that is smaller than about half a wavelength in diameter. So, when a subwoofer pushes out a blob of air, it has to spread according to thermodynamics (think gas pressure law) until it has grown to half a wavelength in size. Only then does it begin to propagate as a sound wave. For very low frequencies that might be bigger then the room you're using it in, and so the output of the sub never becomes "proper" sound at all.

The usual rule of thumb is that you have to displace not twice but FOUR TIMES as much air for each octave you go down in frequency because of this effect. So, a sub that will play 16Hz as loudly as your stereo can play, say, 100Hz would need a big and scary drive unit, and maybe multiple ones, just to shift that much air.

\/()43 |_|K19, Going Postal

Reflex and bandpass subs again have the advantage here: the ports are tuned to resonate and the resonance actually adds on more airflow than the driver alone produces. Designing a thing with resonators in it to provide a smooth frequency response and not *sound* resonant is an art and a science, but it can be done.

There is some respite if your listening room is quite small: you actually get an accumulation of pressure as the entire room pressurises and depressurises as a single unit. This is called cabin gain, and is most commonly applied to car interiors and car audio, but could be a factor in getting that 16Hz pipe organ sound.

And, of course, if you don't mind bringing out the kango hammer and making some adjustments, you can avoid the need for cabinets by mounting drivers directly in the wall. This can be a money saver, because drivers that don't need to be able to produce the extra force required to compress and rarefy the air in a cabinet are actually cheaper. Remember to seal around the edges properly.

Finally, there's this fun toy. It, again, goes in a hole between rooms but is a fan whose blades shift inclination with the music signal. So it starts off just spinning with no net airflow. Then, as a positive peak comes it becomes a fan that blows air into the room, and then as the negative part of the sound signal arrives it reconfigures to suck air out. This thing can get down to 1Hz, which would be a stopped 256 foot pipe or an open 512 foot pipe. L'eau!

\/()43 |_|K19, Going Postal

\/()43 |_|K19 ©

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