Comparisons of Sound Propagation in Rigid-walled and Soft-walled Elliptical Waveguides

Abstract

In a previous paper, sound propagation in soft-walled elliptical Wave-guides was investigated by employing the solutions of the wave equation expressed in terms of the Mathieu functions and modifie Mathieu functions. In the present paper, the cut-off wave numbers of the rigid-walled elliptical wave-guides are calculated in the same ways in the previous work and compared with the results of the soft-wall. To analyze exactly the sound propagation in the elliptical waveguide with a rigid-wall and a soft-wall the elliptical coordinate system is introduced as shown in Fig. 1. The velocity potentials which are solutions of the wave equation can be expressed in terms of the Mathieu functions Se_m(s, η), So_m(s, η) and the modified Mathieu functions Je_m(s, ξ), Jo_m(s, ξ) (see Eq. (1) for the rigid-wall and Eq. (2) for the soft-wall). Substituting Eq. (1) into the boundary condition Eq. (6) of the rigid-wall, equations of the dimensionless cut-off wave number (Eqs. (9) and (11)) were obtained. The numerical calculation of the dimensionless cut-off wave numbers by a digital computer was carried out for various values of the eccentricity (e), Figs. 6〜9 show the numerical results of k_cB represented as a function of e, where k_c is the cut-off wave number and 2B is the length of the minoraxis. From Fig. 6 for even-wave Φ^e_&ltmn&gt, it seems that the curves starting from k_cB of the circular wave-guide with no nodal circle, one nodal circle and two nodal circles approach 0, π, and 2π, respectively, as the value of e approaches 1. These values of k_cB correspond to those for a palallel waveguide with width 2B. On the other hand, from Fig. 7 (odd wave Φ^0_&ltmn&gt) the curves of k_cB starting from the values of the circular waveguide at e=0 approach π/2 at e=1 in the case of no nodal circle, and 3π/2 in the case of one nodal circle. Experiments were performed by the standing wave method to obtain the phase velocities in air contained in the elliptical waveguides made of aluminum (rigid-wall) and in water contained in the semi-elliptical waveguides made of foam-polystyrol (soft-wall). Examples of the measurement are shown in Fig. 13 for the semi-elliptical waveguides with soft-wall and in Fig. 14 for the elliptical waveguide with rigid-wall. It can be seen from Figs. 13 and 14 that the experimental results are in satisfactory agreement with the theoretical ones.