Equation check: Dimensional analysis.

[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

 

[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.

 

[Beer-monster]

Thanks for the reply.

No mention of different units that I can see. The author also uses a similar formula for a cavity with a neck and includes a speed of sound term.

To be completely frank, I'm checking a wikipedia article. An error is therefore, not completely unexpected. Though I lack the confidence to be 100% confident in my argument.

 

[haruspex]

No mention of different units that I can see.

I didn't say different units, I said specific units. The article specifies metres, and the author may have felt it reasonable to assume that frequency is in cycles/sec. The next equation, where the speed of sound does appear, doesn't have a magic constant. This leads me to suspect the first equation is correct, just not ideally expressed.
I notice that if you write L=d and C=340m/s in the second equation you get something close to the first.

 

[Nickie]

for sharing this equation and your thoughts on it. I am always interested in checking the dimensional consistency of equations. After reviewing the equation, I agree with your analysis that there seems to be a missing dimension in the original equation. As you suggested, adding a term with units of speed squared to the numerator would balance out the dimensions on both sides of the equation. I also agree that this speed should be the speed of sound in air, as it is a fundamental property of the medium in which the resonant frequencies are being measured.

Therefore, I would also suggest that the correct equation should be:

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

Thank you for bringing this to our attention and for your insightful analysis. It is important to always double check equations and ensure that they are dimensionally consistent in order to avoid any errors or misunderstandings in their interpretation.