Popping "vacuum bubbles" loudness
Stun-wizards specialize in blinding and deafening opponents at a distance. To do so, they can exert their magical power to manifest a spherical region of vacuum by essentially "telekinetically" pushing air out of the region. When the wizard releases their spell, air rushes back into the space, presumably with a loud "bang".
How do you calculate the acoustic loudness (in sound pressure, dB) of a sphere of vacuum, with arbitrary radius r, collapsing at a regular atmospheric pressure?
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1 answer
The collapse of bubbles on various scales has actually been an area of research for quite some time. Analyses are typically numerical, and rely on something known as the Rayleigh-Plesset equation, which tells you how the bubble's radius varies as it oscillates or collapses. If you know the pressure inside the bubble and the pressure far from the bubble, you can adequately predict its collapse by numerically solving the differential equation - you're not going to get an analytic solution in the general case.
Sound waves emitted by oscillating and collapsing bubbles - due to external perturbations - are discussed in Section 7 of Lauterborn & Kurz (2010). One of the earliest papers on solving the requisite equations is Hickling & Plesset (1964), which discusses the case of a bubble collapsing (and rebounding) in water. You may find that to be a good starting point. Regrettably, you'll have to carry out the numerical simulations on your own.
Some notes:
- The study of bubbles typically involves bubbles on the centimeter to nanometer scale; beware of assumptions that may become invalid at macroscopic scales.
- Solving the equations should give you the pressure of the shock waves; from there, you'll have to convert to decibels given the pressure of the external medium.
- This is a problem for which there is, unfortunately, no simple (i.e. analytic) solution.
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