Equations for Spherical Resonators

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In summary, the conversation discusses a web app that calculates the diameter of a sphere for a given frequency and sound hole dimensions, and allows for downloading an stl file for 3D printing. The speaker mentions issues with the equations used in the app and offers two styles based on equations from a physics professor in the 1980s. They also mention concerns about the accuracy of the equations and discuss the limitations of using only three digits in calculations. The speaker's question is whether the equations are accurate.
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DrewPear
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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?
 
  • #4
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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Related to Equations for Spherical Resonators

1. What is a spherical resonator?

A spherical resonator is a type of cavity resonator where the cavity is in the shape of a sphere. It is used to confine electromagnetic waves or acoustic waves within a spherical volume, allowing for the study of resonant modes and frequencies specific to the spherical geometry.

2. What are the fundamental equations governing spherical resonators?

The fundamental equations governing spherical resonators are derived from Maxwell's equations for electromagnetic resonators or the wave equation for acoustic resonators. For electromagnetic resonators, the key equations include the Helmholtz equation in spherical coordinates. For acoustic resonators, the wave equation in spherical coordinates is used. These equations describe how the fields or waves behave within the spherical cavity.

3. How are the resonant frequencies of a spherical resonator determined?

The resonant frequencies of a spherical resonator are determined by solving the Helmholtz equation or the wave equation with appropriate boundary conditions. For a spherical resonator with radius \( R \), the resonant frequencies are given by specific values of the spherical Bessel functions and their derivatives, which satisfy the boundary conditions at the surface of the sphere.

4. What are the modes of vibration in a spherical resonator?

The modes of vibration in a spherical resonator are characterized by three quantum numbers: the radial mode number \( n \), the angular mode number \( l \), and the azimuthal mode number \( m \). These modes describe the spatial distribution of the fields or waves within the resonator. The solutions to the wave equation in spherical coordinates yield spherical harmonics for the angular part and spherical Bessel functions for the radial part.

5. How does the size of the spherical resonator affect its resonant frequencies?

The size of the spherical resonator, specifically its radius \( R \), has a direct impact on its resonant frequencies. Generally, larger resonators have lower resonant frequencies, as the wavelengths of the resonant modes are longer. The specific relationship between the radius and the resonant frequencies is determined by the solutions to the wave equation or Helmholtz equation, where the resonant frequencies are inversely proportional to the radius of the sphere.

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