Sound Waves Legacy Problem #21 Guided Solution

Open-end air columns are simply long columns filled with air open to the surrounding atmosphere on both ends. If you were to blow through air somehow disturb the air inside of it with just the right frequency you would cause resonance to occur and the result would be a loud sound. Now in an open-end air column when it begins to resonate it forms a standing wave pattern where air along that column forms a collection of nodal and antinodal positions. Now the nodal positions, if we're speaking of air being disturbed or vibrating, the nodal positions are positions where air is not vibrating at all. You would say it's relatively undisturbed. Antinodal positions on the other hand are those positions where air is vibrating quite vigorously. Now for an open-end air column these displacement antinodes as we sometimes call them always occur at the ends of the open ends of the air column. So we get antinodes at the open ends of the air columns and if it's vibrating in the fifth harmonic what that would mean is that there's five different nodal positions within there and between the ends there's going to be four different antinodal positions. So the task of finding exactly where these nodal positions are demands that we first try to figure out where the antinodal positions are. And so when I think of this what I'm going to do is take the 80 centimeters length and I'm going to divide it up into five equal length sections. Each equal length section being 16 centimeters long. So I would find that there's displacement antinodes at zero centimeters, 16 centimeters, 32 centimeters, 48 centimeters, 64 centimeters, and 80 centimeters. Now that's four antinodes between the two ends, six antinodes in all. Now the nodes, the points of no displacement, are the points positioned midway between all those antinodes. So if you listed the antinodes down as 0, 16, 32, 48, 64, and 80, then you can quite easily find the nodal positions because they're halfway between 0 and 16 which puts one at 8 centimeters, between 16 and 32 which puts one at 24, between 32 and 48 putting one at 40 centimeters, between 48 and 64 putting one at 56, and between 64 and 80 putting one at 72 centimeters.