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During the past 60 years fluid-structure interaction in a wide range of three dimensional circular cylinder problems have been studied. Initial problems considered a rigid wall structure which were solved using impedance model comparisons. Soon after, further solution techniques were used, such as computer simulation, transfer matrix methods and finite element techniques. However such problems were only valid for low frequencies when compared with experiments, this was because that did not include higher order modes. The importance of higher order modes was then established and studies have since included these modes. More recently, mode matching methods have been used to find the amplitudes of waves in structures comprising two or more ducts. This has been done with using an orthogonality relation to find integrals which occur from the application this method. This methodology is demonstrated in as background information and is applied to prototype problems formed of rigid ducts. The rigid duct theory led to the consideration of elastic shells, of which several shell modelling equations were available from the vibration theory. In this thesis, the Donnell-Mustari equations of motion are used to model thin, elastic, fluid-loaded shells of circular cross-section. It is demonstrated that generalised orthogonality relations exist for such shells. Two such relations are found: one for shells subject to axisymmetric motion and one for shells subject to non-axisymmetric motion. These generalised orthogonality relations are new to the field of acoustics and are specific to shells modelled with the Donnell-Mustari equations of motion. The mode matching method is used to find the amplitudes of waves propagating in prototype problems and the generalised orthogonality relations are used to find integrals which occur through this method. Expressions for energy for all considered structure types are used to find the resulting energy for each prototype problem and results for equivalent problems are compared. In addition, verification of the resulting amplitudes is done by ensuring that the matching conditions are suitably satisfied. It is anticipated that the method will have application to the understanding and control of the vibration of cylindrical casings such as those enclosing turbo-machinery. Another application of the method would be the tuning of cylindrical casings, such as those featured on car exhaust systems or HVAC (heating, ventilation and air conditioning) systems.

There is a good deal of uncertainty and sensitivity in the results for wave impact. In a practical situation, many parameters such as the wave climate will not be known with any accuracy especially the frequency and severity of wave breaking. Even if the wave spectrum is known, this is usually recorded offshore, requiring same sort of (linear) transfer function to estimate the wave climate at the seawall. What is more, the higher spectral moments will generally be unknown. Wave breaking, according to linear wave theory, is known to depend on the wave spectrum, see Srokosz (1986) and Greenhow (1989). Not only is the wave climate unknown, but the aeration of the water will also be subject to uncertainty. This affects rather dramatically the speed of sound in the water/bubble mixture and hence the value of the acoustic pressure that acts as a maximum cutoff for pressure calculated by any incompressible model. The results are also highly sensitive to the angle of alignment of the wave front and seawall. Here we consider the worst case scenario of perfect alignment. Given the above, it seems sensible to exploit the simple pressure impulse model used in this thesis. Thus Cooker (1990) proposed using the pressure impulse P(x, y) that is the time integral of the pressure over the duration of the impact. This results in a simplified, but much more stable, model of wave impact on the coastal structures, and forms the basis of this thesis, as follows: Chapter 1 is an overview about this topic, a brief summary of the work which will follow and a summary of the contribution of this thesis. Chapter 2 gives a literature review of wave impact, theoretically and experimentally. The topics covered include total impulse, moment impulse and overtopping. A summary of the present state of the theory and Cooker’s model is also presented in Chapter 2. In Chapter 3 and Chapter 4, we extend the work of Greenhow (2006). He studied the berm and ditch problems, see Chapter 3, and the missing block problem in Chapter 4, and solved the problems by using a basis function method. I solve these problems in nondimensionlised variables by using a hybrid collocation method in Chapter 3 and by using the same method as Greenhow (2006) in Chapter 4. The works are extended by calculating the total impulse and moment impulse, and the maximum pressure arising from the wave impact for each problem. These quantities will be very helpful from a practical point of view for engineers and designers of seawalls. The mathematical equations governing the fluid motion and its boundary conditions are presented. The deck problem together with the mathematical formulation and boundary conditions for the problem is presented in Chapters 5 and 6 by using a hybrid collocation method. For this case, the basis function method fails due to hyperbolic terms in these formulations growing exponentially. The formulations also include a secular term, not present in Cooker’s formulation. For Chapter 5, the wave hits the wall in a horizontal direction and for Chapter 6, the wave hits beneath the deck in a vertical direction. These problems are important for offshore structures where providing adequate freeboard for decks contributes very significantly to the cost of the structure. Chapter 7 looks at what happens when we have a vertical baffle. The mathematical formulation and the boundary conditions for four cases of baffles which have different positions are presented in this chapter. We use a basis function method to solve the mathematical formulation, and total impulse and moment impulse are investigated for each problem. These problems are not, perhaps, very relevant to coastal structures. However, they are pertinent to wave impacts in sloshing tanks where baffles are used to detune the natural tank frequencies away from environmental driving frequencies (e.g ship roll due to wave action) and to damp the oscillations by shedding vortices. They also provide useful information for the design of oscillating water column wave energy devices. Finally, conclusions from the research and recommendations for future work are presented in Chapter 8.

A method for solving boundary-value problems where a boundary parameter varies continuously in space is applied to some canonical waveguide problems. The method, previously employed by Roseau and Evans, generates a functional difference equation, the solution to which enables fluid velocity potential to be written as an explicit integral transform. The two main problems to which the method is applied are a two-dimensional waveguide with a varying impedance condition and an elastic-walled waveguide with varying bending stiffness. Limiting versions of the two problems are solved using the Wiener-Hopf technique. This provides a check on the varying-parameter solutions as well as being of interest in itself.

Eigen-mode matching techniques offer a versatile approach for solving acoustic scattering problems in ducts. However, until recently, these techniques have been restricted to problems in which the boundary conditions contain at most one derivative, that is, Neumann, Dirichlet or Robin's conditions. Here a method is developed to solve scattering problems in ducts that are discontinuous in height and have at least one surface described by a high order boundary condition. Attention is focussed on the membrane condition, but the method can be extended to elastic plates and other higher order conditions. An original orthogonality condition is derived and used to solve two problems. Limiting cases of the results are compared with some special cases solveable by standard Fourier techniques and (for the case of no height discontinuity) the Wiener-Hopf technique.

The two-dimensional problem of hydrodynamic interaction of the horizontal circular cylinders with a free-surface is investigated both analytically and numerically. The fully nonlinear initial boundary-value problem is described and a numerical solution for it is presented. The free motion of a cylinder rising towards, or moving away from the free-surface or moving horizontally is studied. The numerical calculations are compared with a simple analytic theory in which we take the low- and high-frequency limit of the added mass, and the constant added mass of the submerged cylinder in the coefficients of the equation of motion. Further numerical calculations of an initially displaced, spring-loaded cylinder undergoing slow motions are compared with a simple analytic theory in which we also take the low-frequency limit of the added mass of the submerged cylinder. The aim is to provide a useful approximate method for simulation of various offshore operations. Fully nonlinear calculations of the free-surface deformations of the initially calm water caused by forced constant velocity motion of a totally submerged circular cylinder are compared with small-time asymptotics due to Tyvand & Miloh (1995). Their analytic results, which are taken to third order5 when gravity terms first appear in the expansion, are in excellent agreement with the numerical calculations for small times, beyond which only the numerical method will give accurate results, valid until the free-surface breaks. The breaking of the surface as a result of vertical downward motion is further investigated with the aim of establishing when and how this happens, since the phenomena causes the breakdown of the numerical calculations. The free motion of a cylinder entering a free-surface, initially half-submerged in calm water and having specific gravity of 1.2 is also investigated. This motion is pursued beyond the complete submergence stage, giving rise to interesting free-surface deformations and body dynamics. This study is complemented by a further investigation involving impulsively started and forced constant motion of a cylinder entering a free-surface at various angles and Froude numbers, and is also taken beyond the complete engulfment stage. Hydrodynamic forces on the cylinder obtained for various angles at the same Froude number are compared. Also, the hydrodynamic forces for the motion in the same direction at various Froude numbers are compared.