Ideal open end condition

A first approximation to the behaviour at the end of a duct may be obtained using the plane wave approximation. Recalling equation (2.20), the plane wave approximation reflection coefficient at a change in cross-section from $S_1$ to $S_2$ is given by:

$$\frac{B}{A} = \frac{S_1/S_2 - 1}{S_1/S_2 + 1}.$$ (3.1)

An open end corresponds to $S_2$ tending to infinity, so $B/A$ tends to -1. The acoustic wave is reflected back down the duct $180^o$ out of phase implying standing waves with zero pressure amplitude at the open end as mentioned in chapter 1. While this is useful for obtaining a first approximation for the behaviour of musical instruments it is obvious that, even ignoring mode conversion at the opening for the moment, a more accurate analysis of the end condition should account for the sound radiated from the end of the instrument.