# Equation check: Dimensional analysis.

1. Feb 10, 2013

### Beer-monster

I came across this equation, said to describe the relation between the resonant frequencies of air in a spherical cavity open at the top.

$$D = 17.87 \sqrt[3]{\frac{d}{f^{2}}}$$

Where D is the sphere diameter, d is the diameter of a small circular cavity at the top of the sphere and f is the resonant frequency.

Is it me or is this equation wrong?

The dimensions do not seem to check out. The frequency term introduces a dimension of $T^{2/3}$ to the RHS which is not balanced on the LHS.

I would guess that a term with units of speed squared should be added to the numerator inside the cube-root. That would add dimensions of $L^{2/3} T^{-2/3}$. I would also suspect that this speed of be the speed of sound in the air (C).

i.e. I think the equation should be:

$$D = 17.87 \sqrt[3]{\frac{dC^{2}}{f^{2}}}$$

Can anyone tell me if I'm right?

Thanks

2. Feb 10, 2013

### haruspex

Your argument makes sense, but it is possible that the author presumed/specified certain units to be used and has incorporated a standard value for the speed of sound in air, based on that assumption of units, into the constant.

3. Feb 10, 2013