© 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 = U² R_A (RMS watts), where U is the rate of oscillatory volume displacement of the fluid, measured in RMS m³/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 v², or P_in ~ 24 p r h v² for a radially oscillating sphere. Hence U = p r² v = (p r³ 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 r³ P_in / 3 h)^1/2 for the pulsating sphere.

The radiation resistance R_A = p r n² / 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. 7.6}

{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/m² at the surface of the radiator, assuming r = 993.4 kg/m³, 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