Linear Acoustic Field in a Closed Pipe with an Abrupt Change in Cross-Sectional Area Lineares Schallfeld in einem einen Querschnittssprung habenden geschlossenen Rohr

Abstract

Culick and Yang studied an acoustic field in a closed pipe with an abrupt change in cross-sectional area [vol. 143, Progress in Astronautics and Aeronautics, 1992, p. 759]. They presented only fundamental results; here the field is fully analyzed. The boundary conditions at the area discontinuity are equality of acoustic pressures and that of acoustic mass fluxes. For nonzero, finite S 1 /S 2 (S 1 , the cross-sectional area of one part of the pipe; S 2 , that of the other part), the characteristic wave number multiplied by the total pipe length (simply, the wave number) of the m-th mode lies in ((m- ½)π, (m+ ½)π); for zero S 1 /S 2 , it lies in [(m- ½)π, (m+ ½)π] and can analytically be calculated. If the discontinuity is located at 2n/2m (n = 1,..., m - I) or (2n — 1)/2m (n = 1,..., m) of the pipe length, the wave number equals mπ for all S 1 /S 2 . In the former configurations, the field is identical with that in a straight pipe. In the latter, the ratio of the pressure amplitude in one part of the pipe to that in the other depends on S 1 /S 2 . For zero S 1 /S 2 , if the discontinuity is located at n/(2m + 1) (n = 1, 3,..., 2m - 1) or n/(2m ― 1) (n = 1, 3,..., 2m - 3) of the pipe length, the wave number equals (m + ½)π or (m - ½)π and coincides with that of the (m + 1)-th or (m - 1)-th mode, respectively; the wave numbers are degenerate.