Amplification and the Flaring of a Tube – Part 4

 

In the previous parts of this 5-part blog, we discussed quarter, half wavelength resonators, and impedance and damping. In this part, we will describe why some musical instruments have a flare or horn.

The formal name of this acoustic principle is called the acoustic transformer effect, but many just call it the “horn effect”. The horn not only makes some musical instruments look really neat, but also allows some neat things to occur in the output spectrum. Horns are no stranger to audiologists. For years, acoustic horns were used in behind the ear hearing aids to enhance the gain and output in the higher frequency region. This tends to improve the clarity of speech for those clients with a high-frequency hearing loss. Of late, because digital hearing aids can be programmed to generate more high frequency using the fitting software, the acoustic horn has fallen from grace.

Thomas Edison knew about the horn effect- he used it in his Victrolas at the turn of the last century- what he could not generate in the higher frequency region electronically, he was able to generate by using a gradually flared loudspeaker in the shape of a horn. Indeed, all commercially available loudspeakers use a horn configuration to enhance the level of the higher frequencies that are generated.

The horn effect refers to an enhancement in the amplitude in the higher frequencies, but what are “higher frequencies”? Let’s turn to an important equation- so important that it will guarantee to make you the life of your next party…. F = v/2L where, like the previous blogs, F is frequency, v is the speed of sound (34,000 cm/sec) and L is the length of the tube that is flared. For those who read part 2 of this series of blogs, you may recall that this looks suspiciously like the half wavelength resonator equation. For the purposes of this blog, let’s just call this a “coincidence”.

But what is L? The length of what? And why should the horn effect depend on any length? Well, it does, so unlike damping (see last blog part 3), the horn effect is discriminatory towards certain frequency regions.

In the formula, F = v/2L, the length L is a necessary element to see at which frequency the horn effect first begins to have an effect.   In the adult vocal tract, the length is 17 cm and when we open our mouth further for the vowel [a] as in ‘father’ the formula tells us that opening our mouths wide enhances the magnitude of all frequencies above F = 34,000/2 x 17 = 1000 Hz. If an adult opens their mouth wider when they articulate these low vowels, there is an increase in output above 1000 Hz. The output at 4000 Hz is greater than the increase at 2000 Hz, and this in turn, is greater than the increase at 1500 Hz. The horn effect continues to grow (but not indefinitely).

However if a young child with a vocal tract length of only 10 cm opens their mouth wide, the effect will only start to have an enhancement above F = 34,000/2 x 10 = 1700 Hz. That is, with a shorter vocal tract length, the horn effect will start at a higher frequency and have less of an overall effect.

The C trumpet has an overall length of 137 cm. Using the formula we find that the horn effect will be evident for all frequencies over F = 34,000/2 x 137 = 124 Hz which is an octave below middle C. For the trumpet, because the tubing is so long, the gradual flaring of the tubing serves to enhance the amplitude of almost all of the sounds of the trumpet.

Thomas Edison was very thankful that he knew about the horn effect. Assuming the length of the early Victrola was 1 meter (100 cm) the horn effect would begin to occur at F = 34,000/2 x 100 = 170 Hz, again a great boost for a large portion of the transduced frequency range. Next time you go to an antique store and are able to get a close up view of a Victrola, look down the neck of the flared horn. You will see what looks like a black tennis ball that is connected to a cable. The cable runs down the neck of the flare and disappears into the box of the Victrola. This was Thomas Edison’s attempt at a volume control. As a slider was adjusted, this caused the cable to be pulled back and forth into the neck of the flare. When the tennis ball was very close to the narrowing of the neck, the horn effect was destroyed. This had the effect of reducing the overall volume- actually it was a low pass filter that gradually cut the higher frequency amplitudes. The volume wasn’t reduced equally for all frequencies as it would be with today’s technology, but this was actually quite clever for over 100 years ago.

French horn players are pulling a “Thomas Edison” when they place their hands into the flare of their instrument. Trumpet and other brass players do this as well when they use a mute. These things reduce the effect of the horn effect thereby cutting the higher frequency energy from their playing.

Oh, and one more thing. The flare has to be at least 25-30% of the total tube length for any real effect to occur. In hearing aid acoustics, a flared tube, such as a Libby horn, is 22 mm of flare for a 75 mm earhook/earmold length. Anything shorter would have minimal effect. An earmold that has a short 10 mm flare at the end may be useful to avoid wax accumulation, but will have no acoustic effect. Another example is the clarinet. It has a very nice, but very short flare at the end- it is useful to allow it to stand up when placed on the ground, but has no acoustic value- it is purely ornamental.

In the fifth (and final) portion of this blog series on musical acoustics, we will be discussing what the pinna effect has to do with room acoustics in a musical venue. Stay tuned!

About Marshall Chasin

Marshall Chasin, AuD, is a clinical and research audiologist who has a special interest in the prevention of hearing loss for musicians, as well as the treatment of those who have hearing loss. I have other special interests such as clarinet and karate, but those may come out in the blog over time.