Decomposing one-dimensional acoustic pressure response into propagating and standing waves

Abstract

Few actual sound fields are representative of ideal acoustic pressure responses and ideal boundary conditions, such as those nearly found in anechoic or reverberant rooms. Normally encountered enclosures have complicated responses that are difficult to relate to a boundary condition that is inbetween these two ideal extremes. Yet, the propagating-and standing-wave responses associated with absorptive and reflective boundary conditions seen in the ideal cases are fundamental bases to understand these more complicated problems. An analytical method is developed to decompose a one-dimensional acoustic pressure response associated with a specified partially absorptive boundary condition into an equivalent summation of propagating and standing waves usually.associated with absorptive and reflective boundary conditions, respectively. The propagating-and standing-wave responses are scaled and shifted in phase by factors that are dependent on the boundary absorptivity and frequency, but are independent of the spatial location. The complicated mixed response is decomposed into varying amounts of the ideal responses, which can be helpful during a design analysis.

Complex modes of vibration due to small-scale damping in a guitar topplate

“…In more general terms, Spiekermann and Radcliffe (5; 6) identify absorptive boundary conditions with a traveling-wave response and reflective boundary conditions with a standing-wave response, while noting that real conditions are necessarily associated to a mixed response. They then proceed to decompose both analytically (5) and experimentally (6) the total acoustic response associated with mixed boundary conditions into propagating and standing-wave components. A quantitative measure of the relationship between the two distinct ideal responses is given by scaling factors and phase angles obtained from the total mixed response.…”

A Comparison of Acoustic Impedance Measurement Techniques

“…The excitation speaker at the duct end at x = 0 is modeled as a harmonic pressure excitation (or source). This boundary condition is [ 16]…”

“…Equations (1) -(4) can be solved by using separation of variables and then equating the ordinary differential equations to the boundary conditions. This technique results in the following response of the system [61, [16]:…”

Scattering investigation based on acoustical holography

“…C. Spiekermann [10] considers the simultaneous presence of propagating and standing wave fields which add together to form the total sound field in a room (this result was reported by van Zyl et al shifted by constants that depend on frequency, so that they sum to form the mixed response. The scaling factors and phase angles indicate the portions of purely prop agating and standing wave components necessary to form the total mixed response.…”

“…Several methods have been developed for field separation [9,10,12,14]. The basic idea is that if the total field is composed of both the forward (incident) and backward (scattered) propagating waves, the direction of wave propagation for the forward and backward waves are in opposite directions and the pressure is the sum mation of both.…”