**Nanomedicine,
Volume I: Basic Capabilities**

**©
1999 Robert A. Freitas Jr. All Rights
Reserved.**

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

**7.2.2.1 Acoustic Radiators**

The acoustic power generated by any vibrating source is P_{out}
= U^{2} R_{A} (RMS watts), where U is the rate of oscillatory
volume displacement of the fluid, measured in RMS m^{3}/sec, and R_{A}
is the acoustic radiation resistance seen by the source, measured in MKS acoustic
ohms.^{889} Two basic generators are
the vibrating piston and the pulsating sphere, both of radius r. From Stokes
Law (Eqn. 9.73) the input power required to
propel a circular piston through water at velocity v is P_{in} ~ 6 p
r h v^{2}, or P_{in} ~ 24 p
r h v^{2} for a radially oscillating sphere.
Hence U = p r^{2} v = (p
r^{3} P_{in} / 6 h)^{1/2}
for a vibrating piston to which P_{in} watts of mechanical input power
are delivered, and U = (2 p r^{3} P_{in}
/ 3 h)^{1/2} for the pulsating sphere.

The radiation resistance R_{A} = p
r n^{2} / k_{r}
v_{sound}, where k_{r} = 2 (piston) or 4 (sphere) and n
is acoustic frequency (Hz). This equation is valid only when r/l
<< 1 (e.g., radiator size is very small in comparison to acoustic wavelength
l = v_{sound} / n);
for r = 1 micron and n = 1 MHz, then r/l
~ 0.001 in water. Sound energy generated from a source with dimensions small
compared with the wavelength of the vibration in the medium produces an intensity
uniform in all angular directions; such a generator is considered to be a point
source. Formulas for radiators having r/l >>
1 are also provided by Massa.^{889}

Combining these results with Eqn.
4.53, transmitted acoustic pressure (A_{p}), output power (P_{out}),
and power intensity (I_{p}) at the radiator surface are given by

_{}
{Eqn. 7.7}

_{}
{Eqn. 7.8}

For example, a vibrating piston radiator of radius r = 1 micron
and input power P_{in} = 10 pW operating at n
= 1 MHz in vivo produces A_{p} = 0.0007 atm radiation pressure, P_{out}
= 0.005 pW (giving e% = P_{out} / P_{in} = 0.0005 (0.05%) and
an acoustic intensity of I_{p} = 0.002 watts/m^{2} at the surface
of the radiator, assuming r = 993.4 kg/m^{3},
h = 1.1 x 10^{-3 }kg/m-sec and v_{sound}
~ 1500 m/sec for human interstitial fluid at 310 K. Figure
7.1 summarizes Eqn. 7.6 for various parameter
choices.

Last updated on 19 February 2003