In part one of this blog series, the benefits of having a small 1.4 mm (1/16”) vent drilled into a musicians’ earplug were discussed. Apparently this question was first addressed by Shakespeare himself, but later addressed by more modern Bards. In the original form, he wrote “To vent, or not to vent/That is the question…”, but of course his editor later changed it to the more boring existential question, “To be, or not to be…”
A musician, who was fit with a musicians’ earplug and a 1.4 mm diameter vent, felt that there was an unacceptable low frequency resonance, and he wondered if this was due to the presence of the vent. In my usual clinical genius, I simply said “I have no idea”, but did use some Kleenex to fully plug up the vent. I asked him to walk around, go back to work, and call me if this solved the issue. An hour later he called and said it was perfect. He returned the molds to me and I have sent them back to the lab to have something more “professional” and more permanent than Kleenex installed. (Toilet paper would also work, but I happened to have a box of Kleenex sitting right there).
So what was happening?
I immediately went to my acoustics dictionary and looked up the word “inertance”. (I do this whenever I can’t figure something out, and usually just say that “it’s probably related to an inertance”. You should try saying that- it will make you sound smart!)
In acoustics, an inertance is a mass of air that moves together as a single (or lumped element) mass. We see this in Helmholtz resonators all of the time, and an inertance is formed in a pop bottle by the mass of air vibrating in the neck of the bottle as you blow over it. Inertances are always low frequency since above 2500 Hz air tends to become compressible and it cannot move together as a single unit. That’s why in speech acoustics, Helmholtz resonances are only found for the lowest two formants, F1 and F2. Any higher formants need to be wavelength related.
But back to earmold acoustics or earplugs with vents. The mass of air in the vent acts like a single vibrating mass; in earmold acoustics we used to call this a “vent associated resonance”. This inertance actually can create a resonance which can be noticeable by some listeners, especially if they have good low frequency hearing thresholds.
Robyn Cox was the first to introduce me to this word in her 1979 monograph on earmold acoustics. This is an amazing monograph and there is nothing better than sitting down with your favorite wine or beer and spending an hour or so reading through it. Dr. Cox also came out with a simple formula to calculate the resonant frequency of the vent associated inertance.
Resonant Frequency = 5500 Hz ( cross sectional area/L0Ve)1/2
Actually, in the equation, it didn’t really say “LoVe” but its equivalent. The Ve is the equivalent volume of air that is trapped between the end of the earplug and the eardrum, and L0 is the Length of the vent…. I just put the subscript “o” in there because I am in a romantic mood, and it makes it easier to memorize.
In any event, this tells us that the wider the vent is (the top of the equation), the higher the resonant frequency or inertance. And, the longer the vent (bottom of the equation), the lower the resonant frequency or inertance. So, to make a very low frequency inertance, one needs a narrow, long vent. Believe it or not, this can be very useful to help a vocalist hear their own voice in a noisy musical background.
So, back to the musician that I talked about in part one of this blog series; the vent that I had drilled created a low frequency resonance that was just too much for him. Plugging the vent, in his case, resolved his problems and like all happy clients, he vowed to name his first born son after me.