Sound Pressure Generated by a Bubble
Adrian Secord
Dept. of Computer Science
University of British Columbia
ajsecord at cs ubc ca
This report summarises the analytical expression for the sound pressure
generated by a gas bubble in a fluid. The main reference is T.G. Leighton's
excellent The Acoustic Bubble [1].
Assume a Newtonian fluid of constant density and sound speed
. A spherical body of rest radius vibrates radially
with wall amplitude , wave number and angular frequency .
We have the following analytic expression for the pressure in a fluid
at a radius and time :

(1) 
The parameter relates the wave number of the vibration to the
size of the body and is defined by
In the long wavelength limit
we have
and
. The expression for
pressure simplifies considerably:

(2) 
In section 3.4, the resonant frequency , and
hence
the wave number , is related to the bubble radius. Figure 1
shows the dimensionless parameter for a large range of bubble radii
vibrating at their resonant frequency. Clearly for bubbles larger than
1 m the long wavelength approximation is valid.
If we are concerned with nonresonant frequencies, then in water we need
, where
is the linear frequency. At the highest end of human hearing,
Hz and so we need m, which
implies again that the long wavelength approximation is valid.
Figure 1:
Parameter

Acoustic vibrations are typically small in amplitude, which leads to
linearisation of the standard wave equation and other simplifications.
The radial vibration of an gas bubble in a fluid is modeled well as a
simple harmonic oscillator whose stiffness and mass parameters relate
to the properties of the gas and fluid. In particular the inertial mass
of the bubble will relate directly to the work required to move the
surrounding fluid. We will take the
As a harmonic oscillator, the bubble will have a resonant frequency at
which it will vibrate after being subjected to an impulsive force. The
resonant frequency is

(3) 
or

(4) 
where is the linear resonant frequency and is the static
pressure in the fluid. The effects of heat conduction are represented
by , where
and is the ratio of specific heats for the gas. For the
isothermal case, , and for the adiabatic case,
. In most practical cases will take on some
midrange value. The effects of surface tension are assumed to be negligible.
The classical Minnaert resonant frequency assumes that the gas in
the bubble is compressed adiabatically, so
.
We can extend equation 2 to include the effects of
damping at the bubble wall. The bubble will lose energy from the radiation
of sound, thermal conduction and viscosity. In holding to our harmonic
oscillator model, we rewrite the wall amplitude to be
where is the time constant of
decay of amplitude. In addition, classical harmonic oscillator theory shows
that the resonant frequency will be shifted to
. An examination of the quality
factor of the system shows that we can relate to , the
resonant frequency, by way of a dimensionless damping constant
Further, it is assumed that radiation, thermal and viscous losses are the
only relevant factors in the damping, which contribute linearly:
Figure 2 shows all three damping coefficients and their sum.
The radiation damping coefficient can be derived by examining the acoustic
impedance for spherical waves, which includes the radiation resistance. The
radiation damping term is

(5) 
The viscous damping coefficient can be derived by examining the NavierStokes
equations at the bubble wall. While there are no viscous losses inside the
bubble, at the wall the bubble loses energy on compression. The viscous
damping term is

(6) 
where is the coefficient of shear stress in the fluid.
The thermal damping coefficient can be derived using a painful examination
of the thermodynamics of the gas inside the bubble. The net result is that
if the thermal boundary layer of the bubble is large compared to the radius,
then on compression the bubble behaves isothermally and loses energy to
the surrounding fluid. If the thermal boundary layer is small compared to the
radius, then the bubble behaves adiabatically and no energy is lost through
thermal conduction.
The thermal boundary layer thickness is given by
where
is the thermal diffusivity of the gas,
is the thermal conductivity of the gas, is the density of the gas
and is specific heat of the gas.
The thermal damping coefficient is then given by

(7) 
where
is the ratio between the bubble radius and the
thermal boundary layer thickness.
Figure 2:
Resonant bubble damping coefficient

Resonant Frequency with Damping
The effects of damping change the resonant frequency at which the bubble will
vibrate. The new frequency is given by

(8) 
where is the classical Minneart frequency.
The effects of thermal conductivity are incorporated into , a
dimensionless constant:
The effects of the surface tension are incorporated into , a
dimensionless constant:
where
.
Figure 3 shows
for various bubble radii
and figure 4 compares the damped resonant frequency to the
classical Minneart frequency.
Figure 3:
Damped resonant frequency scaling factor

Figure 4:
Resonant bubble frequency

The equation for the damped resonant frequency relies on
, which uses the thermal boundary layer , which itself requires
the frequency. More explicitly, we have
where we have dropped the subscripts on . This is a nonlinear
equation in to solve and the solution cannot be written simply. We
used Matlab's fzero command to solve the equation. For a starting
guess we used either the Minneart frequency or if we were calculating the
frequencies for a range of radii, the frequency of a nearby radius.
 1

T.G. Leighton.
The Acoustic Bubble.
Academic Press, London, 1994.
Adrian Secord
20011022