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## Losses

So far the effect of propagating waves down each cylindrical section is simply represented by a delay of , therefore ignoring losses. A way of including losses in the layer peeling algorithm has been presented by Amir et al. .

The frequency domain formula for losses associated with propagation of plane acoustic waves down a tube of length , due to Keefe , forms the basis of the inclusion of losses in the layer peeling algorithm. The effect of losses are characterised by the complex wavenumber, : (5.25)

where is the frequency dependent attenuation due to boundary layer effects, while is the ratio of the angular frequency and the phase velocity for propagation of sound along the tube. They are given by Keefe  as: (5.26)

where the normalised boundary layer thickness is and depends on the tube radius, . is the density and is the coefficient of viscosity of air. The coefficients , and are also functions of the thermodynamic constants of air:  (5.27)

with where is the specific heat of air at constant pressure, is the thermal conductivity of air and where is the ratio of the specific heats of air. Temperature dependent values of the thermodynamic constants of air due to Keefe  are provided in table 5.1. The imaginary part of the wavenumber responsible for attenuation is then for , which agrees with the value quoted in Kinsler et al.  to within 1% and differs from the plane wave value from the multimodal losses theory due to Bruneau  by 3%.

Table 5.1: Thermodynamic constants = kg m  = Pa s =  =  = m s This means that the resulting transmission coefficient is: (5.28)

There are numerical difficulties which arise when trying to use this as a filter within the bore reconstruction algorithm. These issues are treated in detail by Amir et al. .

Back to Kemp Acoustics Home    Next: Apparatus Up: Layer peeling bore reconstruction Previous: Numerical implementation issues   Contents
Jonathan Kemp 2003-03-24