nwhm
Effective Rankine-Hugoniot and Lax-entropy conditions on two-dimensional layered medium
This is part of the work done in the Summer of 2014.
- Repository
- Normalizing material parameters
- Influence of sound speed on z-dispersion and impedance on c-dispersion
- Effective Rankine-Hugoniot condition
- Shock speed in general two-dimensional layered media
- Effective Lax-entropy condition
- More on homogenization…
Nonlinear waves in heterogeneous media
This work deals with waves in periodic and random materials, including:
- The 1D \(p\)-system (nonlinear elasticity) and its 2D generalization, with varying density and stress-strain relation
- The shallow water equations with varying bathymetry
- Shallow water scaling
Links to related work by others
Unpublished work
- Instability of homogenized equations
- Symmetry between Z-dispersion and c-dispersion
- Regarding group and phase velocity
- Shocks in c-dispersive media
- Dispersion relations for z-dispersive media
- Propagation of constructed c-solitary waves
- Propagation of short-wavelength linear waves in periodic media
- Riemann invariants in random media
- Some unpublished work on stegotons
- Where stegotons live in terms of velocity
- How to kill a shock wave
- Stegoton model system
- On the nature of second-derivative terms appearing in the homogenized stegoton equations
Open Questions
- Is there some precise symmetry between z-dispersion and c-dispersion? (it seems not)
- Can we generate stegotons with velocity greater than \(c_h\)? Or will they shock? If they shock, then the stegoton velocity must satisfy \(c_{eff} < v < c_h\).
- In Santosa & Symes, an expression is given for the dispersion coefficient due to Z-dispersion in the linear case. Can we show that it matches ours?
- What is the appropriate characteristic condition for shock formation in c-dispersive media? Does it involve \(\rho_h\) instead of \(\rho_m\)? What plays the role of \(\hat{c}\)?
- Can c-dispersion lead to bandgaps?
- Could our high-order homogenization give more accurate predictions for sonic crystals?
Material that appears in (or is superseded by) our submitted papers
- Homogenized equations
- cZ-dispersion
- Scaling of c-solitary waves
- Speed amplitude in c-solitary waves
- Impedance matched media (this is how we discovered c-dispersion)
