Tim Marchant University of Wollongong Modulation theory and undular bores Wednesday 23rd May 14:05-14:55pm, Eastern Avenue Lecture Theatre. Undular bores describe the evolution and smoothing out of an initial step in mean height and are frequently observed in both oceanographic and meteorological applications. Whitham's modulation theory, which describes slowly varying wavetrains, is a popular technique for describing averaged quantities, such as wave amplitude and mean height throughout the bore. The undular bore solution for two equations, a higher-order Korteweg-de Vries (KdV) equation and the modified KdV (mKdV) equation are derived. For the higher-order KdV equation, the undular bore solution is derived using an asymptotic transformation which relates the KdV equation and its higher-order counterpart. Examples of higher-order undular bores, describing both surface and internal waves, are presented. An excellent comparison is obtained between the analytical and numerical solutions. Also, it is illustrated how an asymptotic transformation and numerical solutions can be combined to generate hybrid asymptotic-numerical solutions, thus avoiding the severe instabilities associated with numerical schemes for the higher-order KdV equation. For the modified Korteweg-de Vries (mKdV) equation, two types of undular bore are found. The first, an undular bore composed of cnodial waves, is qualitatively similar to bores found for other integrable equations, with solitons occuring at the leading edge and small amplitude linear waves occuring at the trailing edge. The second, a new type of undular bore, consists of sinusiodal waves, in the form of a rational function, of finite amplitude. At its leading edge is the algebraic solition of the mKdV equation, while small amplitude linear waves occur at the trailing edge. There are three distinct parameter regimes in which a cnodial bore, a combined cnoidal-sinusional bore or a sinusiodal bore combined with a mean height variation can occur. Again an excellent comparison is obtained between numerical and analytical solutions, for a number of different parameter choices.