On the acoustic modes in a cylindrical duct with an arbitrary wall impedance distribution

The present paper considers the propagation of sound in a cylindrical duct, with a wall section of finite length covered by an acoustic liner whose impedance is an arbitrary function of position. The cases of (i) uniform wall impedance, and wall impedance varying along the (ii) circumference or (iii) axis of the duct, or (iv) both simultaneously, are explicitly considered. It is shown that a nonuniform wall impedance couples modes with distinct azimuthal l or axial m wave numbers, so that their radial wave numbers k can no longer be calculated separately for each pair (m,l). The radial wave numbers are the roots of an infinite determinant, in the case when the wall impedance varies either (i) circumferentially or (ii) radially. If the wall impedance varies (iv) both radially and circumferentially, then the radial wave numbers are the roots of a doubly infinite determinant, i.e., an infinite determinant in which each term is an infinite determinant. The infinite determinants specifying the radial wave numbers are written explicitly for sound in a cylindrical nozzle with a uniform axial flow, in which case the radial eigenfunctions are Bessel functions; the method of calculation of the radial wave numbers applies equally well to a cylindrical nozzle with shear flow and/or swirling flows, with the Bessel functions replaced by other eigenfunctions. The radial wave numbers are calculated by truncation of the infinite determinants, for several values of the aspect ratio, defined as the ratio of length to diameter. It is shown that a nonuniform wall impedance will give rise to additional modes compared with a uniform wall impedance. The radial wave numbers specify the eigenfrequencies for the acoustic modes in the duct; the imaginary parts of the eigenfrequencies specify the decay of the sound field with time, and thus the effectiveness of the acoustic liner.