Fig 1.
Spatial variation of thickness over the surface of a cylindrical waveguide.
The radius of the cylindrical shell is R, axial length Lx, thickness H that varies spatially as per H(x, s) = H0(1 − h(x, s)), where 〈h〉 = 0, 〈h2〉 ≪ 1. Heatmap shows the spatial variation of h(x, s). Inset on the top left is an “unwrapped” view of h(x, s). In the middle of the inset, a circle of radius Lc, equal to the correlation length, is shown.
Fig 2.
Comparison of thin shell ray equations and equations obtained from making the paraxial approximation.
Left: Ray propagation using full thin shell theory and paraxial approximation of thin shell theory. Circular and cross markers indicate the location of the first caustic. The axis has been “unwrapped” for representational purposes. Middle: Same ray propagation shown on the cylindrical shell. Right: Plot of the temporal evolution of quantities describing the one of the rays. Here, markers indicate temporal location of the first caustic. It can be seen that, at higher values of 〈h2〉, the location of the first caustic detected from the paraxial approximation differs from thin shell.
Fig 3.
Emergence of branched flow in for an initially plane wavefront in a thin cylinder of non-uniform thickness.
Temporal evolution (a → b → c → d) of an initially plane flexural wave front in a thin elastic cylinder of non-uniform thickness. The full cylinder has been shown on the top of each panel. The regions of high amplitude at each time instant is indicated with colored lines and they been zoomed into and shown at the bottom of each panel. The emergence of branching leading to locations of extreme amplitudes and widening of the wavefront consistent with the dispersive character of this class of elastic waves is also observed.
Fig 4.
Scaling of location of the expected location of the first focusing event with “severity” of randomness.
Locations of first caustic as obtained from numerical ray integration (left) and FE elastodynamics simulations (right). Circular markers indicate locations of first caustic from individual simulations. Square markers show the expected location of the first caustic. Vertical lines indicate 2 standard deviation (centered around the mean). The expected locations of the first caustic show the expected power law scaling with 〈h2〉.
Fig 5.
Scaling of the expected location of the first focusing event with correlation length of the randomness.
Expected locations of the first caustic from numerical ray integration (left) and FE elastodynamics simulations (right) for different correlation lengths. Taking the curve corresponding to to be the reference, dashed lines show the predicted behavior at other correlation lengths assuming
. The actual simulations agree well this prediction hence confirming the linear scaling.
Fig 6.
Scaling of the expected location of the first focusing event wavelength of the initial wavefront.
Expected locations of the first caustic from numerical ray integration (left) and FE elastodynamics simulations (right) for different wavelengths. As long as λ ≪ Lc, is independent of wavelength. The power law scaling seems to break down at higher 〈h2〉, especially in FE simulations. This is consistent with the expectation that the scaling holds only for weak scattering and higher 〈h2〉 corresponds to higher scattering.
Fig 7.
Scaling of the expected location of the first focusing event with the radius of the cylinder.
Scaling of the expected locations of the first caustic with radii from numerical ray integration. The expected location of the first caustic is independent of the radius in the parameter ranges of interest.
Fig 8.
Plot of the temporal evolution of the curve obtained from numerical ray simulations.
Since values of are centered around zero, a constant positive offset is added for the purpose of visualization. The first caustic is detected by finding the location where the
curve folds over itself (shown in red).
Fig 9.
Plot of integrated intensity (top) and scintillation index (bottom) of an exemplar FE simulation.
Note that the integrated intensity has been normalized along by the mean along the circumferential direction for visually emphasizing the locations of extreme amplitudes. The first prominent peak (circular marker) of the scintillation index (SI(x)) curve indicates the location of the first caustic (lf).
Fig 10.
Histograms of angular locations of the first focusing events obtained from numerical ray integration.
Polar histograms of the angular locations, shown in blue, show no consistent appreciable angular bias. The green circle indicates the outline of a polar histogram with perfectly zero angular bias. The individual data-points used to construct the histograms are shown with orange dots. Their angular location is the one obtained from simulations; their radial position in the above plots is randomized in the interest of visualization.