This thesis aims to resolve some open questions about sonic boom, and particularly secondary sonic boom, which arises from long-range propagation in a non-uniform atmosphere. We begin with an introduction to sonic boom modelling and outline the current state of research. We then proceed to review standard results of gas dynamics and we prove a new theorem, similar to Kelvin's circulation theorem, but valid in the presence of shocks. We then present the definitions used in sonic boom theory, in the framework of linear acoustics for stationary and for moving non-uniform media. We present the wavefront patterns and ray patterns for a series of analytical examples for propagation from steadily moving supersonic point sources in stratified media. These examples elucidate many aspects of the long-range propagation of sound and in particular of secondary sonic boom. The formation of `fold caustics' of boomrays is a key feature. The focusing of linear waves and weak shock waves is compared. Next, in order to address the consistent approximation of sonic boom amplitudes, we consider steady motion of supersonic thin aerofoils and slender axisymmetric bodies in a uniform medium, and we use the method of matched asymptotic expansions (MAE) to give a consistent derivation of Whitham's model for nonlinear effects in primary boom analysis. Since for secondary boom, as for primary, the inclusion of nonlinearities is essential for a correct estimation of the amplitudes, we then study the paradigm problem of a thin aerofoil moving steadily in a weakly stratified medium with a horizontal wind. We again use MAE to calculate approximations of the Euler equations; this results in an inhomogeneous kinematic wave equation. Returning to the linear acoustics framework, for a point source that accelerates and decelerates through the sound speed in a uniform medium we calculate the wavefield in the `time-domain'. Certain other motions of interest are also illustrated. In the accelerating and in the manoeuvring motions fold caustics that are essentially the same as those from steady motions in stratified atmospheres again arise. We also manage to pinpoint a scenario where a `cusp caustic' of boomrays forms instead. For the accelerating motions the asymptotic analysis of the wavefield reveals the formation of singularities which are incompatible with linear theory; this suggests the re-introduction of nonlinear effects. However, it is a formidable task to solve such a nonlinear problem in two or three dimensions, so we solve a related one-dimensional problem instead. Its solution possesses an unexpectedly rich structure that changes as the strength of nonlinearity varies. In all cases however we find that the singularities of the linear problem are regularised by the nonlinearity.
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