Passive End Correction of Pipes and Resonators

Abstract

The usual assumption of an “inside” as well as “outside” end correction is shown to be incorrect. The size of the outside end correction at the mouth of an organ pipe depends upon whether it is referred to the area of the mouth or to the internal pipe cross section. In the former case, it is identical with the so-called “Helmholtz resonator” end correction. For large pipe-to-mouth area ratios, the end correction referred to the pipe must approach λ/4. The natural frequencies of an ideal closed pipe are given by solutions of cot kL=tan kΔL, where k=2π/λ, L is the length of the pipe, and ΔL is the end correction, =k−1 tan 1IS/σ[ε+0.6(σ/π)12], where I is image factor, S is pipe cross section; σ, mouth area; and ε, effective mouth depth. Complications due to annular effect, nonlinear reactance, jet-induced reactance, and orifice shape may explain why the present considerations were overlooked in the past. Experimental data are presented. [Work supported by the Office of Naval Research.]