Note: my latest code for this is in ~/Research/Projects/rotated_stegotons/.

Here I discuss the use of impedance-matched layered media, i.e. those for which \(Z=\sqrt{K(x)\rho(x)}\) is constant. In the linear case, impedance matched materials lead to no reflection at all; this is discussed in LeVeque & Yong 2003. In the nonlinear case, the impedance is a function of strain: \[Z(\epsilon,x) = \sqrt{\sigma_{\epsilon}(\epsilon,x)\rho(x)} = \sqrt{K(x)\rho(x)(\sigma(\epsilon,x)+1)}\] If \(\sigma\) is smooth function of \(x\), it seems that little reflection will occur. This case is discussed in LeVeque & Yong (see their Figure 3). They claim that an “effective viscous medium” arises in this case, which isn’t correct (it’s still dispersive).

I’ve conducted experiments in 1D to investigate, using the following material parameters: \[ K_A = 5/8, \ \ K_B = 5/2\] \[ \rho_A = 8/5, \ \ \rho_B = 2/5\] The code is here. The result seems non-dispersive:

Indeed, as in the homogeneous medium case we see N-wave formation and decay, although there is a very slight tail. I tried varying the amplitude, the width, and the grid spacing – I got the same kind of results each time. This also agrees with LeVeque & Yong, Figure 3.

Manuel has run simulations with a 2D checkerboard medium with the same material parameters and observes solitary wave formation: His code is here.

Update (Sept. 4, 2012)

Manuel has discovered solitary-like waves form even when the medium is “striped” rather than checkerboard; i.e., homogeneous in one direction and periodic in the other. Here is a figure showing his results for a medium with vertical stripes:

vertical stripes

As expected (based on 1D results) a shock forms in the x-direction. Surprisingly, dispersive behavior occurs in the y-direction.

I was able to reproduce the same behavior, also when the medium is horizontally-striped (I did this to check for bugs):

Next I tried using a long, thin domain with periodic boundaries in the y-direction and a horizontally striped medium. I made the initial condition uniform in the y-direction and increased the initial amplitude to 20. I got this:

This result seems quite surprising, since the initial condition is uniform in \(y\) and the medium is uniform in \(x\). My initial thought as to what may be going on is as follows.

Consider two adjacent stripes, layer \(A\) with small sound speed \(c_A\) and layer \(B\) with large sound speed \(c_B>c_A\). The solution propagates much faster in layer \(B\), so after some time it gets ahead of the corresponding bit of wave in layer \(A\). Now there is a big difference in the stress at the \(A\)-\(B\) interface, so the nonlinear impedance changes abruptly there. Consequently, there will be transmission and strong reflection.

Also, clearly the solution is periodic in \(y\) with period the same as that of the medium, so we only need a domain that includes one period in \(y\) (which dramatically reduces the computational cost).

Update Sept. 5, 2012

It seems that there is dispersion even in the linear case when the medium is uniform in x (horizontally striped) and the initial condition is uniform in y. Here is a slice plot of such a simulation, where the dispersive tail is clearly visible:

The last few results above were computed with amplitude \(A=20\) in order to see effects quickly. But it seems that such large amplitudes are not necessary in order to produce solitary waves. Here is a result using \(A=1\) and a horizontally-striped medium. The domain is made to be periodic so that it doesn’t have to be so large. The first plot is a slice along the middle of layer A:

The second is along the middle of layer B:

The structure is almost the same, but the amplitude is much smaller in layer B. The waves are also a bit wider in layer B.

It seems that entropy is conserved to numerical accuracy in the last case. These waves should be very amenable to analysis since they don’t change shape as they propagate (unlike stegotons).

Here’s a slice from a more highly-resolved simulation(\(\Delta x = 1/160\)) that show slices through both the \(A\) and \(B\) layers:

Homogenization

The homogenization for this system has already been carried out by Manuel, resulting in equations (43)-(45) of his report. The framework there assumes that the medium is homogeneous in y (vertically-striped), which corresponds to a 90-degree rotation of my experiments above. Furthermore, the initial condition considered will be uniform in x, and in the homogenized solution it seems that it will remain so. In that case, the homogenized equations reduce to (dropping all terms with \(x\)-derivatives):

\[ K^{-1} \sigma_t - (\sigma+1) v_y = c_1 (\sigma+1) v_{yyy} + c_3 \sigma_y v_{yy} \] \[ \rho v_t - \sigma_y = 0. \] Notably, the system becomes 1-dimensional. Furthermore, in the impedance-matched case, the coefficients \(c_1\) and \(c_3\) vanish, reducing the system even further; it’s just the lowest-order homogenized system. It seems that all the dispersive terms of order \(\delta^2\) vanish and the dispersion we see is due to even higher order terms.