next up previous
Next: Rayleigh's Rule Applied to Up: F1:F2:F3=1:3:5 in Uniform Tube Previous: Rayleigh's Rule

Explaining the Node-Antinode Rule.

Consider a velocity node in a standing wave somewhere in the middle of a uniform tube. On either side of the node is a velocity antinode; you can think of two plugs of air around the velocity antinodes, moving in towards the node, and out again. As the plugs move closer together and farther apart, the air pressure between them rises and falls. In the middle, at the velocity node, the motion of the two plugs towards each other cancels out, and pressure varies rather than particle velocity. Consider two plugs of air (symbolized | |) moving in and out relative to a velocity node, labelled N, in Figure [*].

Figure: Movement of air in a standing wave cyclically towards and away from around a velocity node.

The pressure in the middle is inversely proportional to the space between the plugs. The closer they get to the middle, the more they squeeze the air between them, the less the space between them, and the higher the pressure at the node in between them. At a certain point dependent on the amplitude of the standing wave, the pressure between them gets so high that it slows down the movement of the plugs towards the middle, eventually stops them, and then pushes them out again.

This is how a standing wave works: the plugs move towards each other, building up pressure between them and reducing pressure outside of them; as pressure builds between them, the resistance to their inward movement increases, so that they slow down, and eventually stop. At this point the pressure between them is raised and the pressure outside them is lowered, so that the plugs are pushed outwards towards the low-pressure region. As they travel outward, the pressure in the middle falls, and the pressure outside rises; when the pressure outside is greater than the pressure inside, it starts to push them back towards the middle again. It is this process of alternation of pressures as air plugs at the velocity antinodes move alternately towards each other and away, which constitutes the standing wave.

Now suppose that the tube containing the standing wave becomes constricted at the velocity node. This means that there is a smaller volume of space between the two velocity-antinode plugs. What happens now when the two plugs move towards each other? Because they are pushing into a smaller space, the pressure between them will rise more quickly, and in response to the heightened pressure between them, they will slow down and begin to move in the opposite direction sooner. Similarly, as the plugs move outward there is less of a space to draw on, so the pressure between them falls more quickly. Thus the standing wave will vibrate in and out more quickly when there is less volume in the tube at the velocity node; that is, when the tube is constricted there.

Suppose conversely that the tube is widened at the node. Then the plugs at the velocity antinodes will be able to move farther towards each other before raising the pressure between them a given amount. The larger volume between them allows them to push farther inward, and the pressure between them builds up more slowly, given a fixed initial inward velocity. Similarly as the plugs move outward, the greater volume between them means that the pressure will not fall as quickly for a given amount of outward movement. Thus the vibration of the plugs will occur more slowly, which means that the standing wave will have a lower frequency.

What about constriction at the velocity antinodes? This is an even simpler problem. The situation is just like letting pressurized air out of any cavity. Getting a flat tire is a good example: if the hole is large, the tire will go flat right away; if the hole is tiny, the pressure will leak out more slowly. The larger the escape route for the pressurized air, the more rapidly the pressure will fall. If the tube is constricted outside the velocity node, then high pressure will take longer to fall, because less air will move out past the constriction in a given unit of time; conversely, low pressure will take longer to rise, because air can't move in as quickly to equalize the pressure when it has to move through a smaller opening as through a larger opening.

If the tube is widened at the velocity antinode, then high pressure can drop more quickly, and low pressure can be equalized more quickly too. Thus widening at the velocity antinode results in more rapid pressure fluctuation; i.e., the standing wave vibrates at a higher frequency.

We have thus explained Rayleigh's node-antinode principles, by which constriction or widening at a node or antinode of a standing wave will raise or lower, or lower or raise, respectively, the frequency of the standing wave.

It is important to note that this was called by Fant a ``rule of thumb'', which accurately describes the effects of relatively small changes in the shape of an acoustic tube. The rule is not as clearly applicable when the constrictions in the vocal tract are not symmetrical with respect to the nodes and antinodes. Consider a velocity node between two velocity antinodes, at the instant that pressure is maximum at the node, as in Figure [*], time t5. If equal constrictions are made at the antinodes on both sides of the node, then the air that escapes from the cavity between the constrictions during the outward-moving phase of the standing wave will escape symmetrically out the sides. Half of the escaping air will leave on each side. Thus the symmetry of the constrictions around the node ensures that the air on both sides moves away from the node symmetrically. Similarly, if there is relatively low pressure at the location labelled N in Figure [*], as at time t11, the incoming air will come in symmetrically if the constrictions are symmetric around the node.

Now consider what happens if a severe constriction occurs on the left of the location labelled N, but only a mild constriction occurs on the right: then, when the pressure is maximum, the air can more easily escape through the relatively unconstricted side. Because the escape routes for the release of the relatively high pressure in the cavity are not symmetrical, more than half of the air will move in the direction of the open side. Therefore, even the air at the location labelled N will move away from the greater constriction. Only the air closest to the severely constricted end moves toward that side. Thus the node, which was formerly at the location labelled N, is not there any more, because the air at that location moves out (and in) towards the more open side. The node itself has shifted towards the constriction.

In this way, when constrictions in the vocal tract are not symmetrical with respect to the nodes and antinodes, the nodes and antinodes move away from the positions that they occupy in the uniform tube. The important point to remember is that if the constrictions are symmetrical with respect to the nodes and antinodes, then they will remain in the same location as the tube shape changes.

The main reason, in fact, that the rule applies so well to large as well as small changes in the shape of the acoustic tube formed by the vocal tract is, as we will see below, the striking fact that constrictions are typically symmetric with respect to nodes and antinodes in the shaping of the vowel-like sounds of language. In particular, a number of vocalic sounds have been found to be distinguished by differences in the higher formants (particularly F2 and also F3). These are just standing waves with multiple nodes and antinodes, and these sounds generally do tend to have multiple constrictions.

next up previous
Next: Rayleigh's Rule Applied to Up: F1:F2:F3=1:3:5 in Uniform Tube Previous: Rayleigh's Rule
Thomas Veatch 2005-01-25