-
1رسالة جامعية
المؤلفون: O'Kane, Aisling
المساهمون: Copley, Alexander
مصطلحات موضوعية: Earthquake Ground Motions, Earthquake Hazards, Foreland Basins, Himalayas, Mountains, Seismic-Wave-Propagation Modelling, Seismology, Tectonics
الوصف: The regions adjacent to tectonically active mountain belts are exposed to significant earthquake hazard, since the range-bounding faults produce large earthquakes, and the underlying geological structure amplifies the resulting ground shaking. The aim of this dissertation is to investigate the regional-scale controls on earthquake ground motions and seismic hazard in these settings. The first part of this dissertation describes models of the seismic wavefield produced by thrust-faulting earthquakes on mountain range fronts. The earthquake source characteristics and foreland basin structure were varied within reasonable geological bounds, and the earthquake-induced ground shaking was calculated. The earthquake source parameters were determined to be the dominant control on the amount of near-source ground shaking. However, the foreland basin structure, in particular the basin depth relative to the dominant wavelength of the seismic waves, determines the importance of dispersion as the waves propagate through the basin. These results highlight the importance of accurately determining earthquake source characteristics (particularly depth), and the underlying geological structure, during hazard assessment. These principles were then applied to study the active tectonics and seismic hazard in the north-west Himalayas. Field, satellite, and seismological observations were used to determine the fault geometry beneath the NW Himalayas and investigate the relationship between thrust faulting and folding. These results were used to construct seismic-wavefield models, to determine earthquake ground motion estimates if the Main Himalayan Thrust in the region were to rupture. These models show that peak ground velocities are extremely sensitive to minor variations in the fault geometry. Finally, the earthquake-induced building damage in foreland basins was investigated. Using seismic-wavefield modelling, alongside fragility curves for generic building types, the relationships between earthquake location, characteristics, and building damage were investigated. The results quantify the previously poorly known trade-off between earthquake location and magnitude in determining damage distributions. Additionally, the results quantify the factors that can cause over- or under-estimates of the magnitudes of historical earthquakes based on reported damage distributions, with important implications for understanding the accumulated slip deficit in continental collision zones.
-
2رسالة جامعية
المؤلفون: Garcia Neefjes, Erik
المساهمون: Parnell, William, Assier, Raphael
مصطلحات موضوعية: Visco-thermal losses, Fluid-structure interaction, Scholte waves, Stress relaxation, Acoustics, Wave propagation, Dissipation, Elasticity, Viscosity
الوصف: This thesis is focused around the theoretical study of attenuation of acoustic and elastic waves due to viscous and thermal effects. The initial focus is on fluid acoustic media, where we employ the well known theory of linear thermo-visco-acoustics (TVA) to study the influence of boundary layer effects on the propagation of sound in narrow channels filled by air and water. In the latter case, the effects of fluid-structure interaction are taken into account by assuming the neighbouring solid is elastic, but only acoustically hard solids are analysed. On an attempt to generalise the type of media in consideration, the possible advantages arising from the development of a theory for thermo-visco-elasticity (TVE) in this context are noticed. We propose a TVE framework which incorporates more general material behaviour such as creep compliance and stress relaxation, and can be reduced to several other physically relevant theories like TVA for Newtonian fluids, in a way that we can accurately study a diverse class of materials ranging from metals and polymers to air and water in a large number of conditions. As for TVA fluids, TVE media accept three families of modes in free-space, namely two coupled thermo-compressional waves and a shear wave whose phase speed and attenuation differ significantly depending on the specific material. Accurate asymptotic approximations to thermo-compressional coupling are provided which highly simplify the initial expressions for the wavenumbers. We consider a canonical scattering problem consisting of a compressional plane wave incident on two TVE half-spaces in perfect contact, where the thermo-viscous effects on reflection/transmissions and conveniences of the developed framework as opposed to standard approaches in the literature are highlighted. We make use of the above framework to extend the initial study by examining fluid-filled slits within soft viscoelastic media, which we find gives rise to very different results to those obtained for hard solids in the initial work. We show that this can partly be attributed to the properties of the Scholte mode which propagates in the interface of a fluid-solid half-space and is analysed thoroughly. Particular emphasis is put on how the stress relaxation effects can influence the results, which we find to be significant under certain conditions that are discussed in detail. Furthermore, given the generality of the framework, we can analyse the problem of fluid-loaded viscoelastic plates under the same set of dispersion equations obtained for the slit. In particular, we find that for sufficiently soft media so that the phase speed of the symmetric coupled plate-Scholte mode becomes dispersive, the mode experiences a global maximum in attenuation which may be of physical interest, particularly if stress relaxation can be exploited.
-
3رسالة جامعية
المؤلفون: Crawford, Alasdair Ruairidh
المساهمون: Droubi, M. G., Faisal, N. H.
مصطلحات موضوعية: Acoustic emission, Adhesively-bonded joints, Defect detection, Wave propagation
الوصف: The aim of this study is to investigate the application of acoustic emission (AE) techniques to the defect detection and monitoring of adhesively-bonded joints. Pencil Lead Breaks (PLBs) have been used as a simulated AE source to experimentally investigate the characteristics of AE wave-propagation in adhesively-bonded joints, and have been combined with Artificial Neural Networks (ANNs) to provide a novel method of defect detection and sizing. Modal AE analysis has been applied to destructive testing of adhesively-bonded specimens as a novel method to differentiate between fracture modes. Dynamic Finite Element Analysis (FEA) has been utilised to simulate the AE generation and propagation to further investigate the findings of the experimental studies and to assess the applicability of the findings over a broader range of conditions than could be achieved experimentally. PLB tests have been conducted on large (500mm x 500mm x 1mm) aluminium sheet specimens to identify the effects of an adhesive layer on AE wave-propagation. Three specimens were considered: a single sheet; two sheets placed together without adhesive; and an adhesively-bonded specimen. The simulated AE source was applied to the specimens at varying propagation distances and orientations. The acquired signals were processed using wavelet transforms to explore time-frequency features, and compared with modified group velocity curves based on the Rayleigh-Lamb equations to allow identification of wave modes and edge reflections. The effects of propagation distance and source orientation were investigated, while comparison has been made between the three specimens. PLB tests were also used to detect, size and investigate the effect of void-type adhesive defects. Defect-free specimens were used for reference, and specimens with two different void sizes were tested. The PLB source was used to generate simulated AE which would propagate through the defect region and then be recorded with the AE system. Four configurations were tested to assess the effects of source-sensor propagation distance, and source and sensor proximity to the defect. Typical AE parameters of peak amplitude, rise time, decay time, duration, number of counts and AE energy were investigated. Frequency analyses by Fast Fourier Transformation (FFT), partial powers and wavelet transform (WT) were also implemented. ANNs, using the raw Time-Domain signal as an input, were successfully trained and tested to differentiate between the tested specimen types and to estimate the defect sizes. AE-instrumented Double Cantilever-Beam (Mode-I fracture) and Lap-Shear (Mode-II fracture) tests were conducted on similar adhesively-bonded aluminium specimens. Linear source location was used to identify the source-to-sensor propagation distance of each recorded hit. Theoretical dispersion curves were used to identify regions of the signal corresponding to the symmetric and asymmetric wave-modes. Additionally, peak wavelet transform coefficients for the wave modes were compared between the two fracture modes and assessed as an indicator of fracture mode. It was concluded that there is a relationship between the fracture mode and the generated wave modes, with Mode-II fracture typically generating a relatively greater symmetric wave mode than Mode-I fracture. Dynamic FEA was used to replicate both the PLB tests and the destructive tests, and to investigate the effects of a range of parameters that could not all be practically varied in experimental work. Adhesive Young's modulus (representative of different adhesive types), adhesive layer thickness and adhesive void size were varied in the simulated PLB tests. FEA was also used to investigate the effects of fracture mode on the generated acoustic emissions in simulated mixed mode-bending tests, conducted over a range of mode mixities. The FEA results were found to corroborate the results of the experimental work and support a relationship between fracture mode and generated wave modes. It was also identified that a variety of other parameters may also affect the wave modes, and thus need to be considered to achieve effective use of modal analysis to differentiate between fracture modes.
-
4رسالة جامعية
المؤلفون: Haindl, Claudia
المساهمون: Nissen-Meyer, Tarje
مصطلحات موضوعية: Earthquake hazard analysis, Seismic prospecting, Equations--Numerical solutions, California--Hayward Fault, Seismic wave propagation
الوصف: Owing to continual improvements in computation power, we are getting closer to modelling seismic waves in ways that reflect the realistic complexity of the Earth. Nevertheless, modelling the true multi-scale structure of the Earth at large scale and/or high frequencies is currently still beyond our reach from a computational cost perspective. Hence, we rely on efficient modelling software to increase the scope of seismic simulations. Azimuthal Complexity Adaptation (ACA) is an innovative way of reducing the computational cost of seismic simulations by exploiting inherent sparsity of the wavefield. In this thesis, AxiSEM3D, an ACA-based modelling solver, is expanded to model wave propagation in local-scale settings. I find that the sparsity of wavefields persists in the presence of complex local-scale structures such as thrust faults and salt bodies and in models with full anisotropy. Further, I develop 3D wavefield scanning to locate complex areas of the wavefield and to investigate how they relate to structures in the subsurface model. The results suggest that high wavefield complexity is typically localised near abrupt wavespeed contrasts, on the low-wavespeed side thereof. I attempt to develop a new modelling solver which overcomes some of the limitations of AxiSEM3D. The resulting method is unstable, but it demonstrates the algorithmic complexity required to perform ACA in settings like coast lines. I conclude that a hybrid method which combines classic 3D modelling and ACA may be a simpler solution for increasing the scope of AxiSEM3D. Finally, I implement fault ruptures to simulate an earthquake scenario in the San Francisco Bay Area. Using ACA, I can include low-wavespeed bay muds which are usually neglected to save computational cost. Neglecting these muds leads to an underestimate of the shaking in the San Francisco Bay, even at low frequencies.
-
5رسالة جامعية
المؤلفون: Caçoilo, Andreia Patrícia Gandarinho
المساهمون: Teixeira-Dias, Filipe, Rush, David
مصطلحات موضوعية: 620.8, protective shelters, safety requirements, shelter configuration, sea containers, blast waves, shock waves, confined spaces, blast wave propagation, pressure history profile, structural analysis, blast loading, pressure-impulse diagrams
الوصف: The safety of both military personnel and equipment in unstable regions has for a long time been a major issue and concern. Protective shelters with multiple configurations have been widely used to meet safety requirements. Since military compounds are subjected to di↵erent types of threats, such as explosive devices, it is of utmost importance a good understanding of the response of such shielding structures to blast waves. Accordingly, propagation of shock waves in partially- or fully-confined environments is a complex phenomenon due to the possibility of multiple reflections, di↵ractions and superposition of waves. Yet, being able to derive valid predictions of such phenomena is highly relevant, e.g. when it comes to the assessment of protection of personnel. This study looks at the propagation of blast waves in confined spaces and its action on structures, such as compound survival shelters. Whilst full-scale tests o↵er useful insights, the time and expenses associated with such experiments renders then unpracticable. Small-scale experimental models in combination with the Hopkinson- Cranz scaling laws, however, represent a viable alternative to the study of blast wave evolution. The experimental set-up was designed as a rigid structure and built to have a geometrical reduction factor across all dimensions. Experimental analyses were performed on a small-scale model of the actual configuration of the compound survival shelter subjected to the detonation of an explosive charge at di↵erent locations close to its entrance. Pressure-time signals were recorded on a number of locations in the model and a numerical model, based on the explicit finite element code LS-DYNA, was also developed to complement the experimental programme. The recorded experimental data were compared with the numerical predictions to validate the FE model. The proposed numerical model predicts and captures the relevant stages of the propagation of the shock wave. The study of blast wave propagation, which is di↵erent from the propagation of a shock wave in free-field scenarios, is not completely described in literature, especially when it comes to structural response. A numerical analysis of a single corrugated member was performed to evaluate the influence of several wave related parameters on its structural response, e.g. impulse, multiple positive and negative pressure profiles and signal simplifications. Results indicate that the negative impulse train in the pressure-time history plays a significant role in obtaining an accurate performance of the structure. It was also found that a complex pressure history profile can be reduced to a simplified pulse for structural analysis purposes. The consequence of blast events, namely terrorist attacks, warfare scenarios or accidental explosions, usually means severe damage of structures and loss of life. Pressure-impulse diagrams are widely used as a rapid and intuitive tool to investigate the blast response of structural elements under a number of di↵erent blast scenarios. In this study, a numerical model of a 20 ft steel ISO container was developed using LS-DYNA and the accuracy of its response to blast loading is verified against experimental full-scale test data available in the literature. The results show a good agreement between the experimental data and numerical results. Pressure-impulse diagrams were also derived to correlate the damage criteria under di↵erent blast loading scenarios.
-
6رسالة جامعية
المؤلفون: Abdullahi, Mustapha
المساهمون: Oyadiji, Sunday
مصطلحات موضوعية: 621.8, FEA, Acoustic Reflectometry, Time of Flight, CFD, Computational Fluid Dynamics, Finite Element Analysis, Pipeline integrity testing, Non-Destructive Testing, Acoustic, Pipeline damage, Leakage detection, Blockage detection, Acoustic Wave Propagation, NDT
الوصف: Pipelines are considered to be the safest and most economical way of transporting liquid and gas over a long distance. They are used for transporting crude oil and natural gas in process industries, for water transportation and distribution systems in urban and rural areas, for transporting aviation fuel and hydraulic piping as well as air pressure chambers in aircraft. Leaking pipelines can lead to spillage of hydrocarbon fluid into the surrounding environment, and consequently result in fire or outright explosion of the pipeline system as well as production loss, damage to the environment and potential loss of lives. Therefore, accurately identifying and predicting the location and growth of defects are essential and enable the operator to assess and actively manage the integrity of the pipelines. Current leakage detection methods use several non-destructive testing methods such as magnetic particle inspection, radiographic, ultrasonic, pressure transient and acoustic signal methods. This thesis presents acoustic wave propagation-based methods to identify leakage and blockage in pipe systems. In the study, the acoustic finite-element analysis (AFEA) method is employed to simulate acoustic wave propagation in fluid-filled pipes with leakage and blockage but without flow. Computational fluid dynamics (CFD) analysis was also employed to simulate acoustic wave propagation in fluid-filled pipes with leakage and blockage under flow conditions. In addition, experimental testing was conducted to validate some of the numerical results. The experiments consisted of the measurement of acoustic wave propagation in an air-filled pipe with leakage and blockage. The main investigations and contributions of the thesis are summarised as follows, 1) An efficient finite element analysis (FEA) procedure to simulate acoustic wave propagation in fluid-filled pipes is developed. The developed technique overcomes the large errors that occur in the numerical predictions of acoustic wave propagation. The use of different acoustic element types of 2D-like and 3D formulations, and of linear and quadratic interpolations, are investigated in order to deduce the most accurate acoustic FEA approach, and thereby, to minimise the errors. 2) The CFD technique is applied to simulate acoustic wave propagation and reflectometry in a fluid-filled pipe with flow. This is a novel approach of using CFD to predict the acoustic wave propagation in a pipeline in the presence of flow. 3) AFEA and CFD were employed to simulate acoustic wave propagation and reflectometry in a fluid-filled pipe with and without flow. The Time of Flight approach was used on the predicted acoustic waveforms for the pipes with leakage in order to identify the presence of leakage and to predict the size and location of the leakage in the fluid-filled pipe in the presence of flow. 4) Detection of blockage from the predicted data obtained from the CFD and FEA simulation of acoustic wave propagation/reflection in air-filled pipelines without mean flow. Also, the CFD technique was also used to simulate the acoustic wave propagation/reflection in air-filled pipelines with mean flow. Furthermore, the Time of Flight approach was used to identify size and locate the blockage from the simulated data. 5) The experimental measurements of acoustic wave propagation in a straight air-filled intact pipe and air-filled pipes with leakage. The Time of Flight approach was employed to identify size and locate the blockage from the simulated data. Furthermore, the measured and simulated results are compared to validate the simulated results.
-
7مورد إلكتروني
مستخلص: The Maxwell equations may be viewed as evolution equations which develop an initial state of the electromagnetic field forward in time. Such evolution can be simulated numerically, that is modeled on a computer, in which case the domain of simulation is typically finite in extent. Nonetheless, one is often interested in the electromagnetic waves which reach infinity (of course which is outside of the simulation domain). Thus we are interested in near-to-far field signal propagation, that is a mathematical process where a signal or solution recorded at a finite radius r = r1 can be converted to a signal at r = r2 > r1. We achieve such a conversion via application of convolution kernels in the time-domain, although the derivation of the appropriate kernels relies on Laplace transform arguments. Decomposing the wave and Maxwell equations using scalar and vector spherical harmonics respectively, we have solved the equations on the assumption that the source and initial data are compactly supported. We further assume that we work at a large distance outside of the supports. We develop from a theoretical standpoint signal-conversion formulas for the 3d wave and Maxwell equations and these generalize the simple time delay associated with the propagation between two radii of a solution to the 1d wave equation.
الموضوعات: wave propagation
مصطلحات الفهرس: wave propagation, Maxwell equations, far field, Applied Mathematics
-
8مورد إلكتروني
مستخلص: The Maxwell equations may be viewed as evolution equations which develop an initial state of the electromagnetic field forward in time. Such evolution can be simulated numerically, that is modeled on a computer, in which case the domain of simulation is typically finite in extent. Nonetheless, one is often interested in the electromagnetic waves which reach infinity (of course which is outside of the simulation domain). Thus we are interested in near-to-far field signal propagation, that is a mathematical process where a signal or solution recorded at a finite radius r = r1 can be converted to a signal at r = r2 > r1. We achieve such a conversion via application of convolution kernels in the time-domain, although the derivation of the appropriate kernels relies on Laplace transform arguments. Decomposing the wave and Maxwell equations using scalar and vector spherical harmonics respectively, we have solved the equations on the assumption that the source and initial data are compactly supported. We further assume that we work at a large distance outside of the supports. We develop from a theoretical standpoint signal-conversion formulas for the 3d wave and Maxwell equations and these generalize the simple time delay associated with the propagation between two radii of a solution to the 1d wave equation.
الموضوعات: wave propagation
مصطلحات الفهرس: wave propagation, Maxwell equations, far field, Applied Mathematics
-
9رسالة جامعية
المؤلفون: Baksh, Ahmad
المساهمون: Savvaris, Al
مصطلحات موضوعية: ZigBee, wireless sensor network, radio wave propagation, aircraft wing, wing structure, wing materials, efficient energy routing, optimization
الوصف: The applicability of wireless sensor networks (WSNs) has dramatically increased from the era of smart farming and environmental monitoring to the recent commercially successful internet of things (IoT) applications. Simultaneously, diversity in WSN applications has led to the application of specific performance requirements, such as fault tolerance, reliability, robustness and survivability. One important application is structural health monitoring (SHM) in airplanes. Airborne Wireless Sensor Network (AWSN) have received considerable attention in recent times, owing to the many issues that are intrinsic to traditional wire-based airplane monitoring systems, such as complicated cable routing, long wiring, wiring degradation over time, installation overhead, etc. This project examines the SHM of aircraft wing and WSN design (ZigBee), and aspects such as node deployment and power efficient routing, vis-à-vis energy harvesting. Node deployment and power efficient routing protocol are related problems, and so this thesis proposes solutions using optimization techniques for Ant Colony Optimization (ACO), and power transmission profiling using Computer Simulation Technology software (CST). There are three wing models; namely NACA64A410 model, Empty NACA64A410 model for the Wing, and Empty Prismatic model of the wing was specified and simulated in CST software. A simulation was carried out between the frequencies of 100 MHz to 5 GHz, and identified significant variations in the Sij parameter between the frequency range 2.4GHz and 2.5GHz. Critical analysis of the obtained results revealed the presence of a significant impact from wing shape and the wing's inner structure on possible radio wave propagation in the aircraft wing. The different material composition of aircraft wings was also examined to establish the influence of aircraft wing material on radio wave propagation in an aircraft wing. The three materials tested were Perfect Electrical Conductor (PEC), Aluminium, and Carbon Fibre Composites (CFCs). For power transmission profiling (Sij parameter), 130 nodes were deployed in regular and periodic compartments, created by ribs and spars, usually at vantage points and rib openings, so that a direct line of sight could be established. However, four sink nodes were also placed at the wing root, as presented in NC37 and NC38 simulations for aluminium and CFC wing models respectively. The evaluation of signal propagation in aluminium and CFC aircraft wing models revealed CFC wing models allow less transmission than aluminium wing models. A multiple Travelling Salesman (mTSP) problem was formulated and solved, using Ant Colony Optimization in MATLAB to identify optimal topology and optimal routes to support radio propagation in ZigBee networks. Then solving the mTSP problem for different regular deployments of nodes in the wing geometry, it was found that an edgewise communication route was the shortest route for a large number of nodes, wherein 4 fixed sink nodes were placed at the wing root. For a realistic wing model, the different possible configuration of ZigBee units were deduced using rational reasoning, based on results from empty wing models. Besides the determined S-parameter, aircraft wing materials and optimal nodes, the residual energy of each sensor node is also considered an essential criterion to improve the efficiency of ZigBee communications on the aircraft wing. Therefore, a novel hybrid protocol called the Energy-Opportunistic Weighted Minimum Energy (EOWEME) protocol can be formulated and implemented in MATLAB. The comparative results revealed the energy saving of EOWEME protocol is 20% higher compared to the Ad Hoc On-Demand Distance Vector (AODV) routing protocol. However, the need for further energy savings resulted in development of an improved EOWEME protocol when incorporating the clustering concept and the previously determined S-parameter, a number of nodes, and their radiation patterns. Critical evaluation of this improved EOWEME protocol showed a maximum of 10% higher energy savings than the previous EOWEME protocol. To summarize key insights and the results of this thesis, it is apparent that the thesis addresses SHM in aircraft wings, using WSNs from a holistic perspective with the following major contributions, • CST simulations identify power transfer (S-parameter) profiling in various wing models, with no internal structural elements to identify realistic wing with spars, and bars. With an average S-parameter of -107 dB at around 3 m, the communication or transmission range of 1 m was identified to minimize loss of transmitted power. A range less than 1m would cause issues such as interference, reflection etc. • Using a transmission range of 1 m, WSN nodes were assessed for shortest route commensurate with energy efficient packet transmission to sink node from the farthest node; i.e. near the wing tip. The shortest routes converged to travel along the length of the wing in the case of an empty wing model, however it was also observed in a realistic wing model, where internal structural elements constrained node deployment. An average distance of nearly 13 m required data transmitted from the farthest nodes to reach the sink nodes. Increasing the nodes however increased the distance required to up to 20 m in the case of 240 nodes. • A new routing protocol, EOWEME was formulated, showing 20% greater energy savings than AODV in the realistic wing model.
-
10رسالة جامعية
المؤلفون: Rasolonjanahary, Irina
المساهمون: Peake, Nigel
مصطلحات موضوعية: 532, resonance, wave propagation, trapped mode, slowly-varying radius, compressible flow, thin elastic shell, acoustic
الوصف: 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 hard-walled 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 slowly-varying 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.
