Axial wavenumber-merging design method for finite-length liners

Abstract

The Cremer concept via the classical entire-mode merging fails in most cases to achieve the optimal design for finite liners, whose efficient design remains unsolved. To this end, based on not only the simple- but also the cross-modal power analyses, the axial wavenumber(kx*)-merging design method (AWMDM) is proposed herein, merging the imaginary part Im(kx*) and modulating the real part Re(▵kx*)=2πq/L*. It is found for a not-too-short finite liner of length L* that, the AWMDM of q = 1 and 2 can almost capture the globally optimal and suboptimal solutions, respectively, while the Cremer concept is a particular case of the AWMDM at q = 0. The physical mechanism to optimize a finite liner is further revealed: 1) the simple-modal power shows purely exponential attenuation and can be minimized directly by merging Im(kx*); 2) under the premise of merging Im(kx*), the total cross-modal power exhibits a periodic oscillation, and just reaches the first/second trough at the liner exit when one/two (q = 1 or 2) complete oscillation(s) occur(s) within L* by modulating Re(kx*), and thus can be minimized. However, the Cremer concept is based on the infinite-length assumption and considers only a single simple-modal power, so that 1) it cannot capture the inlet effect of the total power in a finite liner; 2) the unexpected cross-modal power has an oscillating period tending to be infinite and thus cannot reach the trough at the exit, which result in its limitation. Generally, the AWMDM of q = 1 and q = 0 exhibit the optimality at lower and higher Helmholtz number scopes respectively, and the valid scope of q = 1 is significantly expanded with the increases in both the length L* and the circumferential mode.