Resonance mode analysis for volume estimation of asymmetric branching structures

The resonance conditions associated with the propagation of a harmonic wave within a rigid, lossless branching structure can be explicitly derived. In this study, exact resonance conditions are derived for multi-order, rigid, asymmetric branching structures. These are compared with resonance conditions for rigid, multi-order, symmetric branching structures which we reported previously. The effect of asymmetry on the form of the higher-order resonance condition is discussed. In the low-frequency range, the resonance condition can be modified into simpler forms which facilitate volume estimation of the branching structure. Two such volume approximation techniques are presented: (a) a fundamental frequency method, in which the lowest resonance frequency is inversely proportional to the structure volume, and (b) an effective-length method, in which an effective length is calculated for all branches distal to the first bifurcation. Equivalence of the two methods is demonstrated. An experimental study was performed to measure the resonance modes of several second-order glass models with asymmetric branching structures similar to those of mammalian lungs. The resulting volume estimates were in close agreement with the true volumes.