On the adjoint problem in duct acoustics and its solution by the time domain wave packet method

Abstract

The Time Domain Wave Packet (TDWP) method has the advantage of obtaining radiated sound by a given duct mode at all frequencies in one computation. It also makes possible the separation of the acoustic and shear flow instability waves. In this paper, the TDWP method will be applied to the adjoint duct acoustics problem. As a microphone records the sound radiated by all duct modes combined, the adjoint approach has the advantage of obtaining the relative mode strengths of all the propagating modes at all frequencies in one computation. The novelty of the paper is on the formulation of adjoint equations in time domain and the numerical solution using the TDWP method. The theoretical formulation of the duct adjoint problem and derivation of various reciprocal relations are presented. The adjoint equations are then solved numerically in reversed time by the TDWP method, in which a point source in the far field is enforced with a Broadband Acoustic Test Pulse time function. PML absorbing boundary conditions for the adjoint equations are also discussed. By an FFT, the relative strengths of all duct modes at all frequencies to the far field point are computed at once.