A THEORETICAL-STUDY OF END EFFECTS FOR RECTANGULAR RESONATORS ON NARROW CHANNELS

In order to understand the end effects of rectangular resonators on narrow channels a linearized potential theory has been applied to study the deformation of waves by rectangular resonators in narrow channels, when the width of the channels is less than one half the incident wavelength. The "exact" solution is obtained by matched eigenfunction expansions; the resulting matrix equation is solved numerically. The numerical solution of the boundary integral equation method (BIEM) has also been obtained for comparison. The present two-dimensional mathematical model complements the previous laboratory investigation of James (J. Fluid Mech. 1970, 44, 615-621). The numerical results of the present analytical and numerical models are compared to James' laboratory measurements. Results show that both analytical solutions and BIEM solutions for the fundamental resonance mode are in reasonably good agreement with the laboratory experimental data. Therefore the considerable effect of main channel width on optimum resonator width and resonator length is further confirmed by theoretical analysis. The usual resonance theory, which predicts complete reflection on the incident wave train when the length of the branch canal is one quarter of a wavelength, was found incorrect by James' laboratory study. James' finding is proved theoretically by rigorous analytical and numerical models in the present study. In addition, higher resonance modes of extreme shape as well as intermediate shape rectangular resonators, not discovered in either James' laboratory investigation or his one-dimensional approximate analysis, were also discovered in the present study.