Wave Impedance in a Cone
i.e., it can be expressed in terms of its own time derivative. This is a general property of any traveling wave.
Referring to Fig.C.46, the area function can be written for any cone in terms of the distance from its apex as
We can now solve for the wave impedance in each direction, where
the wave impedance may be defined (§7.1)
as the Laplace transform of the traveling pressure divided by
the Laplace transform of the corresponding traveling velocity wave:
We introduce the shorthand
Note that for a cylindrical tube, the wave impedance in both directions is , and there is no frequency dependence. A wavelength or more away from the conical tip, i.e., for , where is the spatial wavelength, the wave impedance again approaches that of a cylindrical bore. However, in conical musical instruments, the fundamental wavelength is typically twice the bore length, so the complex nature of the wave impedance is important throughout the bore and approaches being purely imaginary near the mouthpiece. This is especially relevant to conical-bore double-reeds, such as the bassoon.
Writing the wave impedance as
Up to now, we have been defining wave impedance as pressure divided by particle velocity. In acoustic tubes, volume velocity is what is conserved at a junction between two different acoustic tube types. Therefore, in acoustic tubes, we define the wave impedance as the ratio of pressure to volume velocity
This is the wave impedance we use to compute the generalized reflection and transmission coefficients at a change in cross-sectional area and/or taper angle in a conical acoustic tube. Note that it has a zero at and a pole at .
In this case, the equivalent mass is . It would perhaps be more satisfying if the equivalent mass in the conical wave impedance were instead which is the mass of air contained in a cylinder of radius projected back to the tip of the cone. However, the ``acoustic mass'' cannot be physically equivalent to mechanical mass. To see this, consider that the impedance of a mechanical mass is which is in physical units of mass per unit time, and by definition of mechanical impedance this equals force over velocity. The impedance in an acoustic tube, on the other hand, must be in units of pressure (force/area) divided by volume velocity (velocity area) and this reduces to
The real part of the wave impedance corresponds to transportation of wave energy, the imaginary part is a so-called ``reactance'' and does not correspond to power transfer. Instead, it corresponds to a ``standing wave'' which is created by equal and opposite power flow, or an ``evanescent wave'' (§C.8.2), which is a non-propagating, exponentially decaying, limiting form of a traveling wave in which the ``propagation constant'' is purely imaginary due to being at a frequency above or below a ``cut off'' frequency for the waveguide [295,122]. Driving an ideal mass at the end of a waveguide results in total reflection of all incident wave energy along with a quarter-cycle phase shift. Another interpretation is that the traveling wave becomes a standing wave at the tip of the cone. This is one way to see how the resonances of a cone can be the same as those of a cylinder the same length which is open on both ends. (One might first expect the cone to behave like a cylinder which is open on one end and closed on the other.) Because the impedance approaches a purely imaginary zero at the tip, it looks like a mass (with impedance ). The ``piston of air'' at the open end similarly looks like a mass .
More General One-Parameter Waves
Momentum Conservation in Nonuniform Tubes