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

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.