meat,
Apparently, I cannot edit the previous message anymore? (Don't know if there's a time limit on that...)
Anyway, I've obviously taken a while to get back to this and for good reason: I'm hopefully going to have quite a useful spreadsheet tool that anyone can use to calculate the resonant properties of a riser or riser cavity.
Since that is, naturally, taking more time than I'd anticipated, I will at least post the basic formula for calculating the resonant frequency,
fr, of a Helmholtz resonator, which is what is created by punching a hole in the end of a joist cavity in a riser:
fr = (
c/(2*pi))*[sqrt(
S/(
V*
L'))] ................... Eq. 1
where:
c = speed of sound in air
S = area of opening
V = volume of cavity
L' = effective length of opening
There are two tricky parts for Eq. 1:
1. Mind your units. E.g., if you calculate
V in ft³, use
c in ft/s,
S in ft², and so on.
2. The effective length of the opening is calculated by:
L' =
L + (
x*
d) ................... Eq. 2
where:
L = actual length of the opening
x = mass correction term (unitless)
d = diameter of the opening
I have seen values of
x between 0.70 and 0.85. I like
x = 0.80, which should get the resulting
fr in the ballpark.
For the example case of a riser joist cavity, where a hole is being cut or drilled into a "2x" piece of lumber,
L would equal 1.5 inches and
d would be the diameter of the hole. (Think of the hole in the joist as a cylinder with diameter
d and "height"
L; I'm calling the "height" the length here, just to confuse.
)
Calculating
fr, it will become apparent that it can change significantly for small changes in opening or cavity properties, especially since
S and
L' are interrelated. I hope to have (
much later
) a spreadsheet with some macros that will optimize the hole diameter, even if the plan is for more than one hole to open up the cavity. If you (or your math friend) come up with anything in the meantime, let me know.