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

natural frequency Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all par ...

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 is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...

and

viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...

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

. 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 Ambient or Ambiance or Ambience may refer to: Music and sound * Ambience (sound recording), also known as atmospheres or backgrounds * Ambient music, a genre of music that puts an emphasis on tone and atmosphere * ''Ambient'' (album), by Moby * ...

, 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 easily 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 to ...

, or the gas

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

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 link

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)