# Justification for acoustic wave equation boundary conditions

In analyzing the standing acoustic waves produced by a wind instrument, one usually assumes that the openings of the instrument are antinodes of the acoustic wave (as depicted below). What is the justification for this boundary condition?

Before I answer the question, first some explanation about what the graph shows. There are generally two ways to graphically depict a one dimensional pressure wave.

One is by showing how the pressure is distributed over space. When a standing wave is depicted in this way, the antinodes indicate large pressure variations while the nodes indicate a constant pressure.

The second method is by showing the local displacement of the air from it's rest position. When a standing wave is depicted in this way, the antinodes indicate large displacement variations while the nodes indicate a spot where the air almost stands still.

At positions where the pressure has an antinode, the displacement has an node and the other way around.

The graph you give uses the second method and shows the displacement of the air. You can see that there is a node at the end of the tube, because the air there does not move.

Now about the hole. The (very rough) assumption that is made in this model is that the hole creates a connection to the environment, thereby reducing the pressure to zero at that point. This will thus lead to a node in a pressure graph and a antinode in a displacement graph.

This assumption is perfectly fine to get a feeling of how a flute works, but, as Alephzero's explains, is far from the complete story.

This is a very over-simplified approximation which doesn't explain the finger-hole positions on a simple instrument like a recorder or penny whistle.

The justification, so far as it goes, is that making a hole in the side of the tube is equivalent to cutting off the part of the tube below the hole. That is not really true, unless the "hole" was a slot that extended around the full circumference of the tube.

A relatively small hole, as in most real wind instruments, acts more like the opening of a Helmholtz resonator. The actual wave pattern is three-dimensional, not a simplified one-dimensional approximation to the waves in a pipe. For example, the resonant frequency of the instrument depends on the diameter of the hole, as well as on its position along the length of the pipe. One way to make fine adjustments to the intonation of an instrument is to change the diameter of the holes - for example by applying a thin coat of varnish to reduce the hole diameter slightly.

To get an accurate prediction of the acoustics of the instrument, you also need to take account for the finite impedance of the air outside the hole. That has an effect which is similar to the "end correction" to the resonant frequencies of a simple open or closed pipe.

A useful and fairly accurate acoustic model would therefore consist of a branching network of pipes, where each "hole" is modeled as a short pipe, including its diameter, length (i.e. the thickness of the instrument where the hole is drilled through it), and its own end correction factor - including the "shading" effects of a mechanical key mechanism that partial covers the hole when it is "open", or the same thing done by the player's fingers.

Note that even when the finger hole is "closed", there is still a closed-off cavity in the wall of the instrument, which modifies the pitch compared with a simple "one-dimensional wave in a pipe" model.