Title:

Wave propagation in pipes of slowlyvarying radius with compressible flow

The work presented in this thesis studies acoustic perturbations in slowly varying pipes. The slow variation is introduced in the form of a small parameter ${\epsilon}$ and through this in turn gives rise to a slow axial scale $X$ such that $X = {\epsilon}x$ where $x$ is the normal axial coordinate. This allows an asymptotic approach and the WKB method is used to solve the subsequent mathematical problems. The first deals with the existence of a trapped mode in a hardwalled pipe of varying radius conveying fluid. For the derived leading order propagating mode solution, its amplitude becomes singular at transition points $X_{t}$ and $X_{t'}$ where $X_{t} > 0$ and $X_{t'} < 0$ and thus is unable to propagate past these points. Because of the break down in the solution, this leads to the theory that in the neighbourhood of these points there exists a boundary layer in which the original assumption about having slow variation does not hold. By first seeking the thickness of the layer, valid solutions can then be derived and then matched to the outer solutions in order to produce a uniform solution which holds for the entire axial domain. Once this is achieved, it is then used to derive trapped mode solutions. In this case, the theory used is that of two single turning points which are then combined to obtain the full solution. It is illustrated through consideration of examples and the dependence on ${\epsilon}$ is also shown through various plots. This problem will be considered for a symmetric and asymmetric duct and for differing duct parameters. Problems may arise when the two turning points lie close together and so we seek to improve on the method used by deriving a solution to trapped modes encompassing both turning points, which will be proposed together with some illustrations in order to justify its use and reliability. Next, the case of mode propagations on a thin elastic shell of varying radius conveying fluid is studied. The acoustic solutions of a straight shell in vacuo are first briefly reviewed and then built up by the addition of radius variation and the presence of a stationary fluid. The work presented first outlines the analysis for wave propagation in a slowlyvarying thin elastic shell in vacuo. It is found that the shell and the fluid terms are coupled through the fluid pressure term, which is added to the equation governing the radial shell displacements since the pressure is assumed to affect radial motion only. Once the newly corrected equation for the radial shell displacements has been obtained, together with the axial and azimuthal displacements equations, this new system of governing equations is then separated into leading order ${\epsilon}^{0}$ and first order ${\epsilon}^{1}$ systems. In order to simplify the calculations, only the zeroth azimuthal order $m = 0$ will be studied here. With this simplification, a notable result is that the solutions of the torsional motion is decoupled from the axial and radial solutions. Once the dispersion equation is extracted from the leading order system, it can be seen that the axial and radial solutions are in fact coupled. The solution to the in vacuo with varying radius problem is first briefly presented and it is then followed by the solution to the fluid inclusion problem with varying radius, which makes up the main part of this section. The solution is studied for various frequencies and at various points along the shell. In addition, the axial and radial components of the first three modes are examined along with their amplitudes and energy distributions. Finally, mean flow is added and the same analysis is carried out, paying particular attention to the differences which arise in comparison to the stationary flow case.
