Applied Maths Seminar: Marchant -- Modulation theory and undular bores

APPLIED MATHEMATICS SEMINAR AT THE UNIVERSITY OF SYDNEY 

Speaker: Assoc/Prof Tim Marchant, University of Wollongong 

Title: Modulation theory and undular bores 

DATE: Wednesday, May 23 
TIME: 2:00pm 
LOCATION: University of Sydney, Eastern Avenue Lecture Theatre (Level 1) 

Abstract: 

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.  

http://www.maths.usyd.edu.au/u/AppliedSeminar
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