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I'm trying to find a function in the following way:

$F = f(I, f, m, V, A, L) $

Where $I$ is sound intensity, $f$ frequency $m$ mass, $V$ volume, $A$ area, $L$ length.

So I wrote down:

$M^1 L^1 T^{-2} = [T^{-1}]^a [M T^-3]^b [M]^c [L^3]^d [L^2]^e [L]^f$

Then I get the equations:

$ M: 1 = b+c$

$ L: 1 = d+e+f$

$ T: -2 = a -3b$

There are too many unkowns to solve this and get the formula. Can someone help me please? Thanks in advance

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    $\begingroup$ The Pi theorem states that since you have 3 dimensions $(M, L, T)$ and 6 parameters, you can form $6-3=3$ dimensionless groups. Not all the parameters may be used in a group. From there it's a game of intuition and guessing until you get something that works. And even then, the group formed may or may not have physical relevance. This process is typically guided by empirical data and theory, not just ex nihlio. $\endgroup$
    – RC_23
    Jan 4, 2022 at 22:47
  • $\begingroup$ @RC_23 … looks to me like an excellent answer… $\endgroup$ Jan 4, 2022 at 22:53
  • $\begingroup$ THanks for the answer. How do you exactly proceed then? I'm looking at this example. How do you form the pi groups exactly? $\endgroup$ Jan 5, 2022 at 9:15
  • $\begingroup$ I responded as an answer $\endgroup$
    – RC_23
    Jan 7, 2022 at 2:07

1 Answer 1

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(Transferring comment into an answer)

The Pi theorem states that since you have 3 dimensions (M,L,T) and 6 parameters, you can form 6−3=3 dimensionless groups. Not all the parameters may be used in a group. From there it's a game of intuition and guessing until you get something that works. And even then, the group formed may or may not have physical relevance. This process is typically guided by empirical data and theory, not just ex nihlio.

Update responding to the follow on comment:

I'm not an expert on using Pi theorem, but in your example they wanted to isolate the effects of the 3 parameters diameter, density, and surface tension, and show how they might relate.

In your case, what is $F$ physically? What that is will help guide you in the relationships to expect. Based on your parameters, you can make a geometric parameter $V/(AL)$ that is dimensionless for example. $I$ is already dimensionless I believe.

It's pretty open ended. I might suggest looking at the existing literature on the field of study in question (acoustics?) and seeing what dimensionless groups are typically used

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