Past Industrial and Applied Mathematics Seminar

1 December 2011
16:00
Abstract
Tsunami asymptotics: For most of their propagation, tsunamis are linear dispersive waves whose speed is limited by the depth of the ocean and which can be regarded as diffraction-decorated caustics in spacetime. For constant depth, uniform asymptotics gives a very accurate compact description of the tsunami profile generated by an arbitrary initial disturbance. Variations in depth can focus tsunamis onto cusped caustics, and this 'singularity on a singularity' constitutes an unusual diffraction problem, whose solution indicates that focusing can amplify the tsunami energy by an order of magnitude.
  • Industrial and Applied Mathematics Seminar
24 November 2011
16:00
Alexander Korobkin
Abstract
Initial stage of the flow with a free surface generated by a vertical wall moving from a liquid of finite depth in a gravitational field is studied. The liquid is inviscid and incompressible, and its flow is irrotational. Initially the liquid is at rest. The wall starts to move from the liquid with a constant acceleration. It is shown that, if the acceleration of the plate is small, then the liquid free surface separates from the wall only along an exponentially small interval. The interval on the wall, along which the free surface instantly separates for moderate acceleration of the wall, is determined by using the condition that the displacements of liquid particles are finite. During the initial stage the original problem of hydrodynamics is reduced to a mixed boundary-value problem with respect to the velocity field with unknown in advance position of the separation point. The solution of this problem is derived in terms of complete elliptic integrals. The initial shape of the separated free surface is calculated and compared with that predicted by the small-time solution of the dam break problem. It is shown that the free surface at the separation point is orthogonal to the moving plate. Initial acceleration of a dam, which is suddenly released, is calculated.
  • Industrial and Applied Mathematics Seminar

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