5.11 Vibrating air column: standing waves in a conical tube
The third configuration of air column that we shall consider is that enclosed by a conical tube. Figure 17 shows the normal modes of vibration for a conical tube plotted in terms of pressure. As you would expect, there is a pressure antinode at the closed tip of the cone and a pressure node at the open end of the cone. However, it is important to notice that the pressure amplitude decreases as the distance from the tip of the cone increases. This has the remarkable effect of making the resonance frequencies of a conical tube match those of a cylindrical tube open at both ends. That is to say, the resonance frequencies are given by:
where n is an integer number and L, in this case, is the slant length of the cone (the length along the outside of the tube, not the length down the middle). You don't need to know why a conical air column has the same resonance frequencies as a cylindrical tube open at each end. You just need to be aware that it does and that the resonance frequencies form a complete harmonic series.