Inverse potential scattering in duct acoustics

Abstract

The inverse problem of the noninvasive measurement of the shape of an acoustical duct in which one-dimensional wave propagation can be assumed is examined within the theoretical framework of the governing Klein-Gordon equation. Previous deterministic methods developed over the last 40 years have all required direct measurement of the reflectance or input impedance but now, by application of the methods of inverse quantum scattering to the acoustical system, it is shown that the reflectance can be algorithmically derived from the radiated wave. The potential and area functions of the duct can subsequently be reconstructed. The results are discussed with particular reference to acoustic pulse reflectometry.

Optimal Control Theory: A Method for the Design of Wind Instruments

“…, N, ϕ 2n (L) = X 2n (L) = 0. These conditions lead to a Two-Point Boundary Value Problem (TPBVP) for the differential system (14), (16) : one part of the state is fixed at one end and another part is fixed at the other end. When some state component X j is left unspecified at one end, the corresponding costate component µ j must vanish there [23].…”

“…The few simulations presented here have been done while fixing, respectively, two, five and ten components for the overall potential inside the duct, i.e. N = 2, 5, 10 in equation (14). In each case, the duct shape is shown first, followed by the corresponding modes shapes that have been normalized to their maximum value, at x = 0.…”

“…Through the use of an appropriate choice of design variables, the finite dimensional optimization finds smooth horn profiles, that can also help in correcting existing instruments, the shape having been designed from an a priori choice of a succession of cones. In [14], a frequency-domain method, using inverse quantum scattering for the one-dimensional Klein-Gordon equation, allows to recover the area function of a given acoustical duct in a noninvasive way, without measuring directly neither the input impedance nor the reflectance : this last one is mathematically derived from the wave radiated in response to a high-impedance source. In [16], parameter optimization techniques have been used to design shapes for brass-trapping Helmholtz resonators that resonate at a design set of acoustic eigenvalues, taking into account physical and geometrical constraints.…”

Acoustic reflectometry for anin vitromodel of a human upper airway using sinusoidal wave packets and the Ware–Aki algorithm

“…Then it is eliminated The first reconstruction algorithm used in acoustic from the entire impulse response. In this latter case, it is reflectometry was developed (Ware and Aki 1969) under also necessary to estimate a compensating factor due to a the ideal conditions of no losses in the propagating wave low signal-to-noise ratio (SNR) at high frequencies in and that all frequencies were covered by the acoustic order to avoid a singular solution (Forbes et al 2003(Forbes et al , 2006; pulse. As a follow-up, a second reconstruction algorithm Aijun et al 2005a).…”

On the accuracy of bore reconstruction from input impedance measurements: Application to bassoon crook measurements

“…The measured quantities for the bore reconstruction are mainly the input impedance in the frequency domain and the reflection function in the time domain, but other quantities can be used. 1 Historically the first reconstructions performed in musical acoustics have been done using the so called "layer pealing" method with an algorithm developed by Ware and Aki 2 applied to the measured reflection function. 3,4 These works followed the pioneering works by various authors 5,6 for the determination of the vocal tract geometry.…”