Perturbation and asymptotic methods in mechanics and waves
In this thesis perturbation and asymptotic methods for the solution of three non-linear problems are considered. In Part I approximate methods for the analysis of linear and non-linear waves are developed. Waves in a rod of varying crosssection are examined as are waves propagating through an inhomogeneous region in which the wave speed is continuous but has discontinuous derivative. Iterative procedures are used in both these problems and an estimate is obtained for the region in which these methods converge. In Part II non-linear Rayleigh waves, elastic surface waves of permanent form, are analysed. A straight-forward perturbation about linear sinusoidal waveforms is shown to fail. Retaining the full solution of the linear equations it is found that the surface elevation profiles of non-distorting waveforms must satisfy a certain non-linear functional equation which reduces in the small strain limit to a quadratic functional equation. In Chapter Four, periodic, but non-sinusoidal surface waves on a compressible material with non-linear constitutive law are obtained. Non-periodic waveforms are also considered. Periodic Rayleigh waves on an incompressible material are obtained in Chapter Five. A fibre-reinforced belt stretched round a system of pulleys is analysed in Part III. The general theory, developed in Chapter Six, is applied in Chapter Seven to the case of a belt round two pulleys. A mathematical consequence of using the ideal theory in which the constraints of incompressibility and inextensibility are imposed, is the occurrence of singular sheets of fibres which carry infinite stress, but finite force. The ideal theory also gives an undetermined contribution to the tension carried by the fibres. This is determined by considering the case when the fibres are slightly extensible. The boundary layers are examined and the tension throughout the belt obtained.