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Solutions for a cylinder
In this section we will show the solutions of the wave equation for
pipes of uniform crosssection.
Consider a cylinder of radius (crosssectional area is )
as shown in figure 2.3.
Figure 2.3:
Cylindrical waveguide with radius

First we will examine the case of lossless
propagation along a duct of infinitely hard walls. No flow is allowed through
the walls so the gradient of the pressure along the radial direction must
equal zero at the wall. This boundary condition will be used to derive
expressions for the mode profiles on the xy plane, and the corresponding
wavenumbers along the different dimensions. The modes in a uniform cylindrical
duct will then be illustrated in a colour pressure map. We then go on to
show the solutions for lossy propagation.
The profile of the various modes are the eigenfunctions defined in
equation (2.31). In general, the solution need not be
axially symmetric. Musical wind instruments are, however, generally designed
so that the internal profile has no sharp sideways steps and only feature
relatively gradual bends. This means that the behaviour may be modelled
accurately
by approximating the profile by a series of concentric cylinders.
Only axially symmetric modes will have an effect on the response of
an instrument assuming the excitation is also axisymmetric. The present
discussion will therefore be limited to axisymmetric modes only.
This approximation ignores the fact that the excitation from
a brass instrument player's lips is not perfectly axially symmetric.
Subsections
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Jonathan Kemp
20030324