Wave-Envelope Technique for Wave Propagation in Nonuniform Ducts

Abstract

The method of variation of parameters is used to analyze wave propagation in variable area, plane ducts with no mean flow. The method has an explicit representation of the fast axial variation of the acoustic modes, and numerical integration is required only for the slower axial variations of the mode amplitudes and phases. Results are presented which demonstrate the numerical advantages of the method. Comparison of the results with those of a small perturbation theory are given. The relationship between this method and the method of weighted- residuals is discussed. 7 among others. Each approach has unique characteristics and advantages, as well as obvious limitations, either of a numerical or physical nature. For example, no tests have been made to establish the range of geometry variations for which the multiple-scale solutions remain valid; direct numerical analyses require small step sizes and large computation times for high frequencies and high-order modes because of the rapid axial and transverse oscillations; and weighted-residual analysis7 also requires a numerical integration over each axial oscillation of the signal although the transverse variations of high-order modes present no difficulty. The computation of these axial variations has been sim- plified in a direct numerical analysis of wave propagation in constant-area ducts by using an estimate of the harmonic axial variations of the fundamental mode.8 This procedure was reportedly advantageous even with estimates that were only moderately accurate. In the work reported here, acoustic propagation in variable- area ducts without mean flow is analyzed by the method of variation of parameters 9 in a manner that incorporates features of several previous investigations. To facilitate the study of high-order modes and multimodal interactions, the acoustic disturbance is represented as a superposition of parallel-duct eigenf unctions. Moreover, the fast axial variation is given explicitly, and numerical integration is required only for the slower axial variations of the amplitudes and the phases of the modes. We have borrowed the term 'wave-envelope method' as a description of this aspect of the method, although in most respects it bears little resemblance to the procedure used by Baumeister.8 Finally, the representation of the acoustic wave is required to satisfy an integrability constraint derived from the wave equation; this feature is similar to that used in the multiple-scale analysis by Nayfeh and Telionis,3 but does not have the small per- turbation limitation.