Impedance modelling of pipes

Abstract

Impedance models of pipes can be used to estimate resonant frequencies of standing waves and model acoustic pressure of closed and open ended pipes. Modelling a pipe with impedance methods allows additional variations to the pipe to be included in the overall model as a system. Therefore an actuator can be attached and used to drive the system and the impedance model is able to include the dynamics of the actuator. Exciting the pipe system with a chirp signal allows resonant frequencies to be measured in both the time and frequency domain. The measurements in the time domain are beneficial for introducing undergraduates to resonances without needing an understanding of fast Fourier transforms. This paper also discusses resonant frequencies in open ended pipes and how numerous texts incorrectly approximate the resonant frequencies for this specific pipe system.

Numerical simulation of sound propagation in and around ducts using thin boundary elements

“…The calculated impedance is compared with the values of the commonly used approximation formula for the radiation of an unflanged (circular) duct found in literature Re(Z) ≈ 0.25(ka) 2 and Im(Z) ≈ 0.6ka, cf. [3,9,30]. In this formula k denotes the wavenumber and a = w √ π is given by the equivalent radius of a circular tube with the same cross section as the benchmark duct.…”

“…In a second experiment, we look at the radiation impedance for a half open duct that is modeled using thin elements and compare it with an approximation formula for unflanged tubes commonly used in literature [3,30]. In a last example we calculate the resonance frequencies of a circular duct that is open at both ends (sound tube), compare the calculated resonance frequencies with measured spectra of the tube, and investigate the accuracy of the calculations with respect to the edge length of the elements used along the tube.…”

Adjustable pipes and adaptive passive damping

“…where f is the natural frequency, c is the speed of sound, and L is the length of the pipe. Impedance modelling (included in [2,6]) has also been used to derive equation (1) of the natural frequencies. The derivation uses the harmonic solution of the acoustic pressure of a plane wave, the definition of the particle speed from an acoustic wave, and the definition of mechanical impedance of the pipe.…”

“…The amplitude of a system excited by a forced oscillation at a natural frequency is a function of the damping in that system at that specific natural frequency [1]. The natural frequencies of a pipe is dependent on the geometry of the pipe and the boundary conditions of the pipe's ends where standing waves are created within the pipe [2][3][4][5][6]. The production of music using brass and woodwind instruments uses these principles to vary the frequency of a note by varying the effective length of the instrument through slides or openings that match the length of the standing wave of the desired note [3].…”

“…Pipes provide a platform for students to understand how changes in pipe geometry, pipe length, and pipe attachments change the amplitude of a standing wave and this platform can assist students in understanding the physics of standing waves. Multiple publications [6][7][8] have used labs to introduce students to this standing wave phenomenon associated with pipes because low cost equipment can be used to create these labs [8]. These labs use loud speakers to excite the natural frequencies of a pipe and compare measured results to different modelling schemes that predict the natural frequencies of the tested pipe.…”