Losses

So far the effect of propagating waves down each cylindrical section is simply represented by a delay of $T/2$, therefore ignoring losses. A way of including losses in the layer peeling algorithm has been presented by Amir et al. [18].

The frequency domain formula for losses associated with propagation of plane acoustic waves down a tube of length $L$, due to Keefe [54], forms the basis of the inclusion of losses in the layer peeling algorithm. The effect of losses are characterised by the complex wavenumber, $k$:

$$k = \chi + i\kappa$$ (5.25)

where $\kappa$ is the frequency dependent attenuation due to boundary layer effects, while $\chi = \omega / v_p$ is the ratio of the angular frequency and the phase velocity for propagation of sound along the tube. They are given by Keefe [54] as:

$$\chi = \frac{\omega}{c}\left[ 1 + \frac{A}{r_v} - \frac{C}{r... ... [ \frac{A}{r_v} + \frac{B}{r_v^2} + \frac{C}{r_v^3} \right ]$$ (5.26)

where the normalised boundary layer thickness is $r_v = R (\rho \omega / \eta)^{1/2}$ and depends on the tube radius, $R$. $\rho$ is the density and $\eta$ is the coefficient of viscosity of air. The coefficients $A$, $B$ and $C$ are also functions of the thermodynamic constants of air:

$$A = \frac{1 + b_1}{\sqrt{2}},\hspace{1cm} B = 1 + b_1 - \frac{b_1}{2\nu} - \frac{b_1^2}{2},\hspace{1cm}$$

$$C = \frac{1}{\sqrt{2}} \left(\frac{7}{8} + b_1 - \frac{b_1}{2\nu}... ...ac{b_1}{8\nu^2} - \frac{b_1^2}{2} + \frac{b_1^2}{2\nu} + \frac{b_1^3}{2}\right)$$ (5.27)

with $\nu = \frac{\eta C_p}{\kappa}$ where $C_p$ is the specific heat of air at constant pressure, $\kappa$ is the thermal conductivity of air and $b_1 = \frac{\gamma - 1}{\nu}$ where $\gamma$ is the ratio of the specific heats of air. Temperature dependent values of the thermodynamic constants of air due to Keefe [54] are provided in table 5.1. The imaginary part of the wavenumber responsible for attenuation is then $-2.92 \times 10^{-5} f^{1/2}/R$ for $T=20^{o}$, which agrees with the value quoted in Kinsler et al. [40] to within 1% and differs from the plane wave value from the multimodal losses theory due to Bruneau [44] by 3%.

Table 5.1: Thermodynamic constants

$\rho$	=	$1.1769 (1-0.00335 \Delta T)$kg m${}^{-3}$
$\eta$	=	$1.846 \times 10^{-5}(1+0.0025 \Delta T)$Pa s
$\gamma$	=	$1.4017 (1-0.00002 \Delta T)$
$\nu$	=	$0.8410 (1-0.0002 \Delta T)$
$c$	=	$347.23 (1+0.00166 \Delta T)$m s${}^{-1}$

This means that the resulting transmission coefficient is:

$$\Gamma(\omega) = \exp(-ikL)$$ (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. [18].