Viscosity Effects in Acoustic Inductances

The linear acoustic inductance and resistance of narrow circular tubes, short compared to a wavelength, have usually been represented by limiting formulas valid either for very low frequencies (viscous flow) or for relatively high frequencies (pistonlike displacement). Crandall's text gives a general mathematical development, with particular stress on resistance, but variation of inductance with frequency and viscosity has not been discussed generally, nor has numerical reduction of mathematical results been generally available. The present paper describes quantitatively the dependence of inductance and resistance upon the general parameter, radius times square root of the quantity, density times frequency divided by viscosity. With increase of this parameter resistance increases, while inductance diminishes from the “static” value to the limiting “high frequency” value. Experimental checks are made to ascertain whether resistance values are strongly affected by flow-transition end-effects found in hydraulics. Impedance-tube studies are made of five tube diameters from 0.0187 to 0.750 in., at 50 to 167 c/sec. Length/diameter is approximately 10, to render inductance end-corrections relatively small. Acoustic pressures are reduced until linear behavior appears. Measured resistance is generally within five percent of calculation as resistance increases to twice the “static” value. Measured inductance generally agrees with calculation within limits of the end-correction, over the range of significant variation.