So far we have discussed the behaviour of acoustic waves in tubes assuming
that none of the acoustic energy is lost to heat. In reality there is a
boundary layer immediately beside the tube walls in which viscous and thermal
losses occur.
It is possible to use a lossy boundary condition to give lossy versions of
and but the
effect of losses will be noticeable in the direction only because
we will be considering objects which are significantly longer than they are
wide. The inclusion
or exclusion of the effect of losses will therefore be represented entirely
by the choice of direction wavenumber, .
Starting with a lossy boundary condition which allows a small
acoustic particle velocity flow into the wall of the tube, Bruneau et al
[44] have produced a complex direction wavenumber:

(2.54) |

The choice of signs is complicated by the fact that we are performing the
square root operation on a complex number. To split into real and
imaginary parts, it is helpful to first express (2.53) as follows

gives the imaginary part of the correction in :

(2.57) |

(2.58) |

Equating equations (2.55) and (2.59) we get

(2.60) |

(2.61) |

(2.62) |

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