Small acoustic tubes: New approximations including isothermal and viscous effects

Abstract

Algebraic expressions are given (in fortran format for convenient use) that accurately approximate the value of the per unit-length acoustic impedance of small tubes (radii less than (1)/(16) acoustic wavelength) including isothermal and viscous effects. Their accuracy of typically 3% or better was obtained as a result of adding a shunt conductance term to the traditional approximations and including frequency-dependent multiplicative factors. The multiplicative factors are simple expressions for the real or imaginary terms, providing a factor of 40 increase in calculational speed over the full Bessel-function solution. Experimental verification is presented for various combinations of small tubes and cavities relevant to hearing aid applications, and the differences between the present approximations and previous approximations are illustrated using a 1-mm-diam tube as an example. Normalized complex series impedance and shunt admittance, characteristic impedance, time delay, and attenuation constant plots are provided as a reference utilizing the more nearly exact Bessel solution.