A cone’s apex turns what would be a wave node into an antinode

Evidence

Starting with the notion that the apex of a cone acts like an open end, let’s see whether we can justify the shape of the modes of the cone, and in particular their apparently increasing wavelength near the apex. The pressure variation goes to zero at the end of an open cylinder, but we need to collect that variation into a small volume for the analogous cone. If we think of pressure as the amount of color, orange or blue, the thinning “pigment” near the end of the analogous cylinder is collected finally to a point at the apex of the cone. This compression keeps the color density (the pressure) from dropping toward the end of the cone, remaining finite instead. Instead of finishing off as a node, the wave fails to go through that last quarter-wave from antinode to node, but takes up the same length as the open-ended tube. Thus the apparent increase in wavelength. The color in the conical sector in figure 16.9, representing pressure above ambient, is shown blurred over the whole rectangle on the right. Note that the effective pressure variation goes to 0 at the end. Mathematically, a standing wave inside the cone can be shown to behave as sin(kr)/r, where r is the distance from the apex of the cone. The numerator and the denominator both go to 0 as r → 0, but they do so in such a way that the ratio is finite and nonzero. Thus what would have been a node of sin(kr) is turned into an antinode by the vanishing denominator. — Eric J. Heller, Why You Hear What You Hear- An…, 1471

Categorization

• DEFINE: Explains the sin(kr)/r standing wave solution that underlies conical instrument acoustics—why conical bores behave differently from cylinders.
• CANDY: The image of wave phase collected ‘like thinning pigment’ to a point, turning a node into an antinode, is a vivid and tweetable physics gem.