I Equations for Spherical Resonators

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The web app hosts calculations for determining the diameter of a sphere to resonate at a specific frequency, but issues have been identified with the equations used, particularly regarding material thickness and its impact on the sphere's neck. The equations provided are only accurate to three significant digits, which is comparable to the temperature variation affecting the speed of sound. Users are questioning the accuracy of these equations, specifically whether the three-digit accuracy refers to total digits or digits after the decimal point. To improve accuracy beyond three digits, refinements in constants and adjustments for temperature variations in the speed of sound are necessary. Overall, the discussion emphasizes the need for enhanced precision in the app's calculations.
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I'm trying to determine the accuracy of a couple of equations to determine the diameter of a sphere given the frequency and the diameter and length of a sound hole and ask related questions.
I host freely for the public a web app for determining the diameter of a sphere to resonate a given frequency and sound hole diameter and length, and then download a stl file for 3D printing. I've realized it has some issues and part of it is the equations i use to determine the sphere's diameter. I offer two styles in the app using equations given to me by a physics professor in 1980's. They are...

sphere no neck1.jpg

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sphere with neck1.jpg
At this point, i believe i should be most interested in the accuracy of the equation next to the sphere with a neck in the second picture. I've realized that the equation for a sphere with no neck in the first picture does not consider the thickness of material used since it will in essence create a neck of some length. The expression below each equation, i'm hoping, is an accurate representation of the equation above it.
 
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Baluncore said:
Welcome to PF.

Those equations are only three digits accurate, which is about the same accuracy as the temperature variation of the speed of sound.

What is your actual question ?

https://en.wikipedia.org/wiki/Helmholtz_resonance#Quantitative_explanation
https://en.wikipedia.org/wiki/Acoustic_resonance#Resonance_of_a_sphere_of_air_(vented)
Thank you.
My question is, are the equations accurate?
Do you mean three digits accurate to the right of the decimal point?
 
DrewPear said:
Do you mean three digits accurate to the right of the decimal point?
No, I mean 3 digits in total, wherever the decimal point may be.
When Pi = 3.14 is used, there can be only three valid digits in the result.
The neck end correction has only two digits, but is inside the root computation, so all is not lost.
To get more than 3 digits, you will need to refine all constants, and correct the speed of sound in air for temperature.
 
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