Wall Loss and Pipe Radius

• The walls contribute a viscous drag dependent on the ratio of the pipe radius $a$, given by the parameter $r_v$.

• Losses due to thermal conduction are also dependent on the pipe radius and are given by the parameter $r_t$.

• The effects of the viscous and thermal losses lead to an attenuation in the waves propagating down the length of the pipe. The propagation constant is given by
  $$\Gamma(\omega) = \alpha(\omega) + j\frac{\omega}{v(\omega)},$$
  where $\omega$ is the radian frequency and
  

  $$\alpha(\omega) \triangleq ,$$

  $$v(\omega) \triangleq$$ the phase velocity

  
  

• The attenuation coefficient $\alpha(\omega)$ is a function of $r_v$ and the phase velocity $v(\omega)$ is a function of $r_t$. Both therefore, are dependent on the tube radius.³