Acoustic resonance inside an enclosed sphere

Question

I apologize if this question is too basic.

Assume an acoustic source suspended at the center of a hollow spherical shell. (The reason why the central acoustic source is suspended inside the sphere can be whatever's most helpful: a zero-G environment; electromagnetic levitation; or anything else.)

Other than the solid acoustic source, the shell is otherwise filled with air at standard ambient temperature and pressure. The size and shape of the acoustic source, its density, the radius of the enclosing spherical shell, its thickness, density, and other properties, and the frequencies and amplitudes of the sounds produced, are up for grabs.

Crucially, the sphere does not have a hole or "neck" or similar topological features.

Under approximately what conditions/values would the sound(s) produced by the central acoustic source cause resonance in the spherical shell? Ideally, resonance that would lead to visible vibration of the shell but not breakage.

This isn't Helmholtz resonance because the sphere isn't vented.

Any insights appreciated. Thanks!

Answer 1

You are asking about the resonant modes of a hollow sphere. I am assuming the sphere is stiff, so we are looking at standing waves in air pressure, not a flexing of the sphere.

You have probably looked at the resonant modes of a drum head, but if not that’s a great place to start. You will see a whole set of standing waves with both radial nodes and angular nodes. These become more and more complex as the harmonic frequency goes up.

Same with the sphere. Starting with the fundamental tone, the higher-order modes become more complex. The math is not simple, but part of the solutions use a function called “spherical harmonics”. Look it up, it’s interesting.

This topic is very closely related to the quantum structure of atoms. An atomic nucleus is a kind of spherical cavity. A strange one, hard in the middle and soft at the edges. This odd sphere has a very similar family of fundamentals and overtones, but in this case instead of pressure waves they are electromagnetic waves.The shapes of these vibrational modes are the electron orbitals.

Answer 2

I found your question to be very interesting, although I am unable to give a complete response on the conditions that would cause a spherical shell to vibrate visibly without breaking. (I imagined that all frequencies higher than a certain limit would generate vibrations since the interior has a variable pressure "P(t)" while the outside has a uniform pressure "P0"). However, it is fascinating to study modes of vibration. For example, if we consider the acoustic wave trapped inside the shell, similar to a waveguide, then the permitted frequencies would be determined by the shell's characteristics. We could represent the shell locally as a piston with the surface "s" attached to a spring with rigidity "r" and define the characteristic pulsation "ωc" (where wc=(r/μs)^0.5) and "μ" represents the surface mass of the shell. If "𝛒" is the density of the gas, "a" is the radius of the sphere, and k=ω/c (where c is the speed of sound), you would be able to find that permitted waves are given by ω = c/a * z (where z is a positive solution of tan(z) = z / [1 + 𝛒aω^2/(μ(ω^2 - ωc^2)]). It took many steps to come to this conclusion.