if the bottle had other holes in it (or multiple openings at the top let's say) wouldn't that affect the sound?
That's not really the acoustic equivalent of a slot resonator. It would be more like lining up several bottles of different sizes next to each other, and blowing across all of them. each will resonate at its own frequency.
But when you have different slat and width gaps, won't they all affect each other?
Not really. It is the volume of space directly behind the slot that acts as the resonator. Don't forget that air is very stiff, as far as sound is concerned: At a speed of 340 m/s, air is very solid. Ever drive at 100 km/h, and stick your hand out the car window? Imagine doing that at ten times the speed, and you get an idea of how solid air "feels" to a sound wave. So it only really "sees" the zone directly behind the slot itself, not the other zones above and below. It would be kind of like if you cut holes in the sides of adjacent bottles, and linked them together. It wouldn't change things much. In reality, a helmholtz resonator is a "mass/spring" resonator. The air in the gap between the slats acts as the mass, and the air in the cavity behind it acts as the spring.
The CORRECT equation for calculating the resonant frequency is:
F = 2160 * sqrt (r / ((d*1.2*D) * (r+w)) )
Where:
r = slot width
d = slat thickness
1.2 = mouth correction
D = cavity depth,
w = slat width
2160 = c/(2*PI) but rounded
c = speed of sound in inch/sec.
Notice that the only dimension regarding the cavity, is the depth of the cavity. It doesn't really matter how long the cavity (slot) is, and the effective volume is limited by the slot width plus the slat width. Since you normally use slats that are the same width for all of the slots, the only cavity dimension that changes the resonant frequency is the cavity depth: If you angle the slat away from the wall behind it, such that the depth changes along the slot (deeper cavity at one end of the slot, shallower at the other), then you make a broadband resonator. In effect, you can think of it as a series of several separate resonators, each with its own depth, as if you had actually made your slat into a kind of "sideways staircase", except on an infinite scale. So each tiny section of the resonator has it's own individual sharp frequency, governed only by the cavity depth immediately behind it, but when you consider all of the "steps" together, then you get a broadband resonant curve with a flat top, instead of a bell shape.
I'm not sure if I'm explaining this very well! But basically it is only the depth that governs the frequency at any give point on a slot resonator, since all of the other factors are fixed, and each portion of the sound wave only "sees" the slot and cavity right in front of it. It can't see, and is not affected by, adjacent "steps" on the resonator.
The calculator doesn't take the other slat and gap widths into consideration,
It does not need to. Each one acts on its own.
or how many there are for that matter.
Once again, It does not need to. Each slot acts on its own, independent of the others, since that portion of the sound wave that gets through the slot, only "sees" that specific width/depth of slot, and only that specific depth of cavity.
You are looking at the slot wall as an entire forest. You need to get closer, and see it as a large set of individual trees, each of which has its own characteristics. Al together they make up the forest, but each tree acts alone.
I guess I've always thought a Helmholz resonator worked with a cavity and a single slot/opening.
That's correct. But there is nothing to prevent you from putting a whole bunch of them next to each other! And you can tune them all to the same frequency if you want, or all to different frequencies, since each acts independently of the others.
I'm just wondering how useful it is to try and calculate the slat and gap widths.
Very useful! But John already did that for you with his design for a slot wall. If you stick to those dimensions, you will have a good broadband absorber that covers a wide range. If you angle it slightly, you get an even broader bandwidth, but at the expense of efficiency. You cover more frequencies, but get a lower amount of absorption for each individual frequency.
just trying to make the slats somewhat random and aesthetically appealing.
Probably not a good idea! You might end up duplicating or triplicating your coverage of certain ranges, and totally missing others.... Far better to just go with John's standard slot wall, since he has already done all the math for you, and oodles of those walls have already been built, very successfully. You might get luck with a random arrangement.. but then again, you might not. And your chances or randomly hitting on a BETTER design than John's, are about the same as your chances of winning all fifty State lotteries on the same day, then getting struck by lightning, shot, drowned and suffocated all at once, by accident!
- Stuart -
