The Minnaert resonance is a phenomenon associated with a gas bubble pulsating at its

natural frequency Natural frequency, measured in terms of '' eigenfrequency'', is the rate at which an oscillatory system tends to oscillate in the absence of disturbance. A foundational example pertains to simple harmonic oscillators, such as an idealized spring ...

in a liquid, neglecting the effects of

surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...

and

viscous Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...

attenuation In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a Transmission medium, medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and ...

. It is the frequency of the sound made by a drop of water from a tap falling in water underneath, trapping a bubble of air as it falls. The natural frequency of the entrapped air bubble in the water is given by :

f = \cfrac\left(\cfrac\right)^

where

a

is the radius of the bubble,

\gamma

is the

polytropic A polytropic process is a thermodynamic process that obeys the relation: p V^ = C where ''p'' is the pressure, ''V'' is volume, ''n'' is the polytropic index, and ''C'' is a constant. The polytropic process equation describes expansion and com ...

coefficient,

p_A

is the

ambient pressure The ambient pressure on an object is the pressure of the surrounding medium, such as a gas or liquid, in contact with the object. Atmosphere Within the atmosphere, the ambient pressure decreases as elevation increases. By measuring ambient atmosp ...

, and

\rho

is the density of water. This formula can also be used to find the natural frequency of a bubble cloud with

a

as the radius of the cloud and

\rho

the difference between the density of water and the bulk density of the cloud. For a single bubble in water at standard pressure

(p_A=100~ , ~ \rho=1000~ )

, this equation reduces to

f a \approx 3.26~m/s

, where

f~

is the natural frequency of the bubble. The Minnaert formula assumes an ideal gas. However, it can be modified to account for deviations from real gas behavior by accounting for the gas

compressibility factor In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. It is simply defined as the ratio of the molar volume of a gas ...

, or the gas

bulk modulus The bulk modulus (K or B or k) of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other mo ...

K = \rho_g c_g^2

f = \cfrac\left(\cfrac\right)^{1/2}

\rho_g

and

c_g^2

being respectively the density and the speed of sound in the bubble.

References

External links

Low-Frequency Resonant Scattering of Bubble Clouds
by Paul A. Hwang and William J. Teague, 2000, Journal of Atmospheric and Oceanic Technology, vol. 17, no. 6, pp. 847–853. journals.ametsoc.org Sound Bubbles (physics)