In the case of rectangular cross-section, the modes of the duct will have an integer number of nodal lines parallel to the axis and an integer number of nodal lines parallel to the axis. We will only treat systems that preserve symmetry about the central axis, so only modes with an even number of nodal lines in both dimensions need to be considered and the subscript will be used where there are nodal lines parallel to the axis and nodal lines parallel to the axis.

The pressure for each mode obeys the 3
dimensional wave equation:

(2.23) |

(2.24) |

(2.25) |

We can then solve the problem by separation of
variables ([43] pp.540-556).
From Pagneux et al. [32] the pressure and axial velocity are:

where is the pressure profile on the - plane and is the pressure profile of the th mode along the length of the tube. Similarly, is the axial volume velocity profile of the th mode along the length of the tube. and are in general complex numbers to take phase into account. Note that although it is convenient to refer to as the volume velocity, the net volume velocity is ; the other entries are the amplitudes of the axial velocity distributions multiplied by the surface area but have no net contribution to the volume velocity [32].

Equation (2.26) gives the pressure as a series of terms,
being unity, and so the contribution represents plane wave propagation
while the other modes have a non-uniform pressure profile. Substituting
from equation (2.26) into the wave
equation (2.22) and dividing through by gives:

with the eigenvalue being the free space wavenumber. Defining to be the wavenumber along (commonly referred to as the propagation factor):

The transverse term gives

with being the eigenvalue of the th mode. Physically is the wavenumber in the - plane and is zero for plane wave propagation and is positive and real for the modes which feature nodal lines or circles.

It then follows from equation (2.28) that the
wavenumbers follow the relation

Note that in a pipe of uniform cross-section the solution of equation
(2.30) gives

giving the corresponding axial volume velocity as

We can note from this that the characteristic impedance, defined to be the ratio of pressure and volume velocity of forward travelling waves, also depends on the mode number, . For plane waves it was but for the th mode this becomes:

Assuming loss-less propagation is positive and real above cut-off point ( ) so the pressure varies sinusoidally along the axis with a wavelength of where is the free-space wavelength. Below the cut-off point is negative and imaginary [42] so the pressure will be exponentially damped. Calculation of first requires calculation of which in turn depends on the boundary conditions in equation (2.31) (and therefore on the geometry of the duct). It will be treated for both lossy and non-lossy propagation in section 2.4.1 for ducts of circular cross-section and in section 2.4.2 for rectangular ducts.

We have seen that plane waves travelling across a section of tube are only
reflected when the cross-section changes.
The plane wave pressure amplitude at any point in a uniform section of tube
may be obtained from a known plane wave pressure amplitude at a single point
simply by projection using equation (2.10). Each of the other modes of
the pipe will also
only be reflected where the cross-section changes and so may be treated
independently in a section of uniform cross-section. The wavenumber along
the axis becomes for the th mode and, as discussed earlier in the
section, the characteristic
impedance of the th mode is
.
Projection of pressure and volume velocity complex amplitudes
of the th mode then follow from equations (2.10) and (2.11).

Here and are the complex amplitudes of the pressure and volume velocity of the th mode on plane 0 and and are the complex amplitudes of the pressure and volume velocity of the th mode on plane 1 with the planes separated by a distance (see figure 2.1).

We define the pressure vector, , as a column vector consisting of the modal pressure amplitudes and as a column vector of the corresponding values. A method of projecting the pressure and velocity vectors along a uniform section of tube will now be discussed.

In matrix notation the pressure vector on plane 0 is given in terms of the
vectors on plane 1 by

(2.40) |

Similarly the volume velocity on plane 0 is given in terms of the vectors on plane 1 as

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