# Bubble Resonance Frequencies

#### Regger

This is very important. Can someone tell the dependency for the resonance frequencies of bubbles? i.e if know a bubble's diameter, the surface tension, the liquid and the gas' densities how do I calculate the sound wave frequencies that causes a resonance in a bubble. I have Landau's formula for droplets. But since gas is compressible and liquids aren't I guess there sould be some differences.
I have also found another formula in the net which is an aproximation for vey small bubbles, but gives radical deviation for bigger quantities. In this formula the raduis of the bubbles raised to the power of -1. While in Landau's formula it in -3/2 (i.e. the resonance frequency is reversly proportional to square root of the 3-rd power of the raduis).
If I'ld be very gratefull if someone gave me the exact formula.

#### FredGarvin

Admittedly this has been on my to-read list. I haven't had a chance to digest this at all. Hopefully it helps you out. It does appear that resonance does depend on radius.

http://caltechbook.library.caltech.edu/1/04/chap4.htm#L2 [Broken]

This is taken from "Cavitation and Bubble Dynamics" by Christopher Brennen.
http://caltechbook.library.caltech.edu/1/04/content.htm [Broken]

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#### Regger

Thank you very much.
I would have tor read through the article some time soon. But I think this wouldn't be very usefull for me. See what I need is the frequency of a sound wave that causes a resonance. The article talks about pressure for instance, which I can use because I can't measure it. If could tell me what how does a sound frequency causing resonace in the bubble depend only on the bubble's characteristics. Because you know that bubbles soak up the intensity of a sound wave with a frequency matching their own resonating frequency.

Here is the set up. I have made two experiments. I the first I propagated white noise trough foam recorded it and compared it to one the same noise not passing through foam. The idea is that by the lowering in sound intensity for each frequency I could make a judgement for size distribution of the bubbles (because the resonance frequency depends on the size of the bubble).

In the second experiment I produced foam consisting of same-sized bubbles. Again I propagated white sounf through the foam and recorded it. I noticed that for the intensity for two frequencies was notebly lower than for the rest. Of cource those two frequencies should be the first two resonance of the bubbles.
Using Landau formula for droplets, I mentioned, I calculted what resonance frequencies should the bubbles I produced have. But it deviated severly from the actual frequencies.

If someone could help me.

#### Regger

I am going to ask again 'cause I really, really need it - and need it soon. I'll try putting it this way. How do the characteristics of a bubble affect its resanating frequency. And I'm talking about sound - resonance aroused by sound. I'm more specifincally interested in the dependency between a bubbles' size - its diameter (assuming it's spherical) and the sound frequency that causes it to resonate.

If someone please knows somethink abou it, if he could share it. My ass is on the line.

#### Clausius2

##### Mechanical Engineer
Gold Member
Regger said:
(i.e. the resonance frequency is reversly proportional to square root of the 3-rd power of the raduis).
That's curious, because doing only dimensional analysis:

$$\omega\sim \phi\left(\sqrt{\frac{\rho R^3}{\sigma}}\right)$$

where $$\sigma$$, $$\rho$$ and $$R$$ are the surface tension, density, and bubble radius. It seems to me this is only for asymptotically large Reynolds Numbers.

Bubbles are not my area of expertise, but you should take a look at on line journals such as Journal of Fluid Mechanics or Physics of Fluids about this subject.

#### Fizzywater

Article on "forced oscillations"

Regger,
Did you find information on induced bubble frequencies? I found you posting while Googling the subject. Not a lot out there. I also found this article, which may be close to what you were looking for(?). Good luck. This is a new topic for me.

Nigmatulin, Akhatov, and Vakhitova (1999). "FORCED OSCILLATIONS OF A GAS BUBBLE IN A SPHERICAL VOLUME OF A COMPRESSIBLE LIQUID," Journal of Applied Mechanics and Technical Physics, 40(2):285-291
ABSTRACT: A spherically symmetric problem of oscillations of a single gas bubble at the center of a spherical flask filled with a compressible liquid under the action of pressure oscillations on the flask wall is considered. A system of differential-difference equations is obtained that extends the Rayleigh-Plesset equation to the case of a compressible liquid and takes into account the pressure-wave reflection from the bubble and the flask wall. A linear analysis of solutions of this system of equations is performed for the case of harmonic oscillations of the bubble. Nonlinear resonance oscillations and nearly resonance nonharmonic oscillations of the bubble caused by harmonic pressure oscillations on the flask wall are analyzed."

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