In plane wave propagation the continuity condition was that the volume
velocity must be equal on and meaning that the
mass of air flowing out of equals the flow of mass into at any
given time. This implies that the velocity, which is assumed to be constant
over the cross-section, is different on either side because of the
difference of cross-sectional area. When we are treating the
velocity field accurately in three dimensions it is clear that the
velocity on the two surfaces should match and that the velocity into the
- plane wall on the larger cross-section is zero. For the
case where we have

(B.10) |

where is a shorthand for the - plane wall resulting from the part of surface 2 which is not shared with surface 1. In terms of volume velocities this means that on and on . Now we will use equation (2.27) (again ignoring the time factor) and use the orthogonality of the modes. This time in order to include the fact that the volume velocity is zero on we must perform the integration over surface 2:

(B.11) |

where is given in equation (B.6). It should be noted that the integration in is this time over the on surface 1 and over on surface 2, hence and in the subscript to are swapped for equation (B.12). In matrix notation the result is

(B.13) |

As with the pressure, when the volume velocity calculation
can be performed simply be interchanging the labels 1 and 2 giving
equation (2.84):

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