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.