University of Toronto
                  PDE/Applied Math/Analysis Seminar
                    Thursday, March 30, 12:10-1pm
                          2159 Bahen Center      



SPEAKER:  Allen Tesdall (Fields Institute)

TITLE:    The triple point paradox for the nonlinear 
          wave system.  

ABSTRACT: Experimental observations of the reflection of weak 
          shock waves off a thin wedge show a pattern that 
          closely resembles Mach reflection, in which the 
          incident, reflected, and Mach shocks meet at a triple 
          point.  However, the von Neumann theory of shock 
          reflection shows that a triple point configuration, 
          consisting of three shocks and a contact discontinuity
          meeting at a point, is impossible for sufficiently 
          weak shocks.  This apparent conflict between theory 
          and experiment has been a long-standing puzzle, and 
          is often referred to as the triple point, or 
          von Neumann, paradox.

          We present numerical solutions of a two-dimensional 
          Riemann problem for the nonlinear wave system that is 
          analogous to the reflection of weak shocks off thin 
          wedges.  We develop a new numerical scheme to solve 
          the equations in self-similar variables, and use local 
          grid refinement to resolve the solution in the 
          neighborhood of the apparent triple point.  The 
          solutions contain a remarkably complex structure: 
          there is a sequence of triple points and supersonic 
          patches immediately behind the leading triple point, 
          formed by the reflection of weak shocks and expansion 
          waves between the sonic line and the Mach shock. An 
          expansion fan originates at each triple point, resolving 
          the von Neumann paradox.  These results are consistent 
          with numerical solutions of a shock reflection problem 
          for the unsteady transonic small disturbance equation 
          that describes the Mach reflection of weak shocks, and 
          with recent experimental evidence that shows a sequence 
          of shocks and expansions behind the triple point in a
          weak shock Mach reflection.
                                                                                                           
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University of Toronto   
PDE/Applied Math/Analysis seminar 

Fridays, 3:10-4pm in 6183 Bahen Center, 40 St. George St.
http://www.math.toronto.edu/appmath/

2005-2006 organizers:
	Pieter Blue 		pblue@math.toronto.edu
	Almut Burchard		almut@math.toronto.edu
	Bob Jerrard 		rjerrard@math.toronto.edu
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