P h y s i c s   o f   S t r i n g s

VIGRE Seminar, CAAM 499.005

Asymptotic Methods in Analysis

Spring 2010, Wednesday 4pm, HZ 120

Participants: Noel Cavazos, Sharmaine Jennings, Aaron Levine, Shuyi Li, Devin Taylor, Brian Wigianto


Inverse Problems for Boundary Damping

Fall 2009, Wednesday 4pm, HZ 120

Participants: Noel Cavazos, Sharmaine Jennings, Aaron Levine, Shuyi Li, Devin Taylor, Brian Wigianto


Uncertainty Quantification for Spring Experiments

Summer 2009, MWF 1-2pm, DH 2014

Participants: Jordon Cavazos, Aneesh Mehta, Matthew Broussard, Jeff Bridge, Heather Williamson


  • Calvetti and Somersalo Text
  • Pfeiffer Applied Probablity Text
  • paper of Bal and Ren

    The Mathematics of Drag

    Spring 2009, Tuesday 2:30pm, HZ 212

    Participants: Nikolay Kostov, Aneesh Mehta, Avery Cate, Jordon Cavazos, Carter Wang, William Li, David Garland, Anthony Austin, Aaron Tallman, Jared Ye, Kyung Oh, Russell Carden, Jeffrey Hokanson, James Doty, Brian Leake, Bonnie Brinkerhoff


    I. Introduction to Low R (Reynold's number) Flow
       A. Overview: The Navier--Stokes Equations
       B. Mathematical formulation: Flow past a cylinder and a sphere, 
          Calculating the drag coefficient.
    II. History of Low R Flow Studies
       A. Experiments and numerical calculations: Measuring the drag 
          on a sphere and a cylinder
       B. Theory: Stokes and paradoxes, Oseen's equation, Matched asymptotics, 
          Uniformly valid approximations.
    III. Renormalization Group Applied to Low R Flow
       A. Introduction to the renormalization group
       B. Flow past a cylinder and a sphere: Rescaling, Naive perturbation analysis, 
          Secular behavior, Renormalization, Truncation, Meeting boundary conditions, 
          Calculating the drag coefficient.


    J Veysey and N Goldenfeld, Simple viscous flows: From boundary layers to the renormalization group, Rev. of Modern Physics 79, 2007, pp. 883--927.

    L D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed., 1987.


    Mar 24: Applying the method of matched asymptotic expansions to the (incompressible) Navier-Stokes problem of steady flow past a solid body.

    Mar 17: An example of the method of matched asymptotic expansions, continued.
    Maple worksheet: matching_asymp_example.mw
    As html: matching_asymp_example.html

    Mar 10: The method of matched asymptotic expansions; a motivating example (2nd order, 1-D boundary value problem).

    February 24: More on Stokes's paradox; axisymmetric flow and the stream function; Whitehead's paradox.

    February 17: Flow at small Reynolds number past a cylinder. Stokes' paradox.
              stokesSolnAssymp.m (asymptotic claims from last week)
              A more convincing presentation of Stokes' paradox: stokes_paradox.pdf
              Accompanying Maple code: stokes_paradox.html
              The actual Maple worksheet: stokes_paradox.mws

    February 10: Flow at small Reynolds number past a sphere.
              lecture 5 notes, part 1 (drag force computation)
              lecture 5 notes, part 2 (Stokes solutions far from the sphere)

    February 3: Flow at small Reynolds number past a sphere.
              lecture 4 notes
              vfield.m: produces a quiver plot of the velocity field

    January 27: Energy dissipation and the Reynolds number.
              lecture 3 notes

    January 20: Derivation of the Navier-Stokes equations.
              lecture 2 notes

    January 13: Conservation of mass, Euler's equations and Momentum flux.

    Fall 2008

    How do strings vibrate and what forces slow them? How can one still a vibrating body as quickly as possible? Can one deduce material properties of a body from measurements of its vibrations?

    Participants: Anthony Austin, Jared Ye, Hunter Gilbert, Aaron Tallman, Toni Tullius, Russell Carden, Jeffrey Hokanson, Derek Hansen, Mark Embree and Steve Cox.

    Seminar 14:
    Simple Viscous Flows (Cox)
    Seminar 13:
    Strings in Fluid, Part V (Hansen)
    Seminar 12:
    Strings in Fluid, Part IV (Hansen)
    Seminar 11:
    Strings in Fluid, Part III (Hansen)
    Seminar 10:
    Strings in Fluid, Part II (Hansen)
    Seminar 9:
    Strings in Fluid, Part I (Hansen)
    Seminar 8:
    Eigenvalues for the wave equation on a network, Part III ( Carden)
    Seminar 7:
    Fall Break
    Seminar 6:
    Eigenvalues for the wave equation on a network, Part II (Carden)
    von Below (Linear Algebra and Its Applications, 1985)
    Seminar 5:
    Eigenvalues for the wave equation on a network, Part I (Carden)
    Tritare information and sound samples: 1, 2, 3, 4, 5
    Seminar 4:
    Recovering a damped two beaded string: A concrete example of BP Theorem 5.1 (Cox)
    Notes: bp2damp.pdf
    Seminar 3:
    Eigenvalues and energy for damped systems (Embree)
    For extended content, see paper 1 and paper 2 by Krein, and Langer (IEOT, 1978)
    Seminar 2:
    Energy decay for damped beaded strings (Cox)
    See Hunter's MATLAB movie: bp4eig.m, bp4eigmovie.m
    Seminar 1:
    Introduction to damped beaded strings (Cox)
    Boyko and Pivovarchik (Inverse Problems, 2008)
    Report on (undamped) beaded strings, theory and experiment
    Preamble: Several areas are ripe for investigation this semester:
    • Explication and experimentation of the discrete three-spectra inverse problem of Boyko and Pivovarchik (Inverse Problems, 2008).
      Excellent progress on this problem was made by the Summer 2008 students; see their VIGRE poster.
    • Eigenvalues for wave equations on networks.
      Following the work of last year's seminar, we are ready to consider thermoelastic strings, viscous damping, etc.
    • Physical experiments with simple string networks: tritars and spider webs;
    • Mathematical modeling of strings moving in viscous fluids.
    • Inverse eigenvalue problems for damped strings.