nwhm/stegoton_velocity

In my thesis, it is shown that stegotons must move faster than \[c_{eff} = \sqrt{K_h/\rho_m}.\] In KL, it is postulated that shocks form when a front moves faster than \(c_h\), the harmonic average of the sound speeds. It now seems that stegotons can exist only with velocity between these two speeds:

\[ c_{eff} \le v \le c_h.\]

Notice that when the impedance contrast vanishes, we have \(c_{eff}=c_h\), and precisely in that limit stegotons cannot exist.

I’ve done some crude experiments to test this theory, just by taking the LY experiment and gradually increasing the wall velocity in order to generate larger-amplitude stegotons. Some code is here. It seems that whenever there are waves moving faster than \(c_h=1\), there is also high-frequency incoherence – a signature of shock formation. I haven’t yet checked this more carefully.

Update: June 25, 2013

I’ve now performed some careful experiments as follows code here:

  1. Run the LY stegoton simulation and extract the leading stegoton (from gauge data).
  2. Rescale the stegoton amplitude by \(a\) and rescale time by \(1/\sqrt{a}\).
  3. Use the result as a moving wall boundary condition and propagate the resulting stegoton through the LY medium. Of course, the result is not a perfect stegoton and generates some small trailing oscillations.
  4. Measure the relative energy (entropy) loss and the stegoton velocity, for different choices of \(a\). Results are in the figure below.

The three curves correspond to:

  • Blue: Classic with 48 cells per layer
  • Red: Classic with 96
  • Black: SharpClaw with 96

The results seem to support the hypothesis above, though there is some (probably numerical) dissipation for stegotons with velocity slightly less than 1.

It would be worthwhile to run this for a different medium (with larger impedance contrast). Also to run with higher resolution/order on Shaheen.

Here is the amplitude versus velocity:

The different curves are obtained depending on how/where the velocity and amplitude are measured.

It would probably be better to transpose the axes in the last figure.