Superoscillations for monochromatic standing waves

On Monochromatic Ascending Waves

A sequence of positive integers w1, w2, . . . , wn is called an ascending wave if wi+1 − wi ≥ wi − wi−1 for 2 ≤ i ≤ n − 1. For integers k, r ≥ 1, let AW (k; r) be the least positive integer such that under any r-coloring of [1, AW (k; r)] there exists a k-term monochromatic ascending wave. The existence of AW (k; r) is guaranteed by van der Waerden’s theorem on arithmetic progressions since an ...

Nash-Moser Theory for Standing Water Waves

Standing Waves: the Equilateral Triangle

It is well known that symmetry considerations can often be a powerful tool for simplifying physical systems. We consider a wave which satisfies the Helmholtz equation in a region with some given symmetry, and note that any solution can be transformed by that symmetry to give another solution. Using this simple observation we describe a number of necessary conditions for wave solutions. For boun...

Bifurcation behavior of standing waves

Two different types of standing waves (SW0 and SWπ) can appear instead of spiral vortices from a supercritical Hopf bifurcation in counter-rotating Taylor-Couette flow for sufficiently small aspect ratios [1,2]. The bifurcation sequence from basic flow to spiral vortices via SW0 can include modulated waves, homoclinic bifurcations, and hysteresis as a consequence of broken translational invaria...

Superoscillations and tunneling times

It is proposed that superoscillations play an important role in the interferences that give rise to superluminal effects. To exemplify that, we consider a toy model that a wave packet to travel in zero time and negligible distortion, a distance arbitrarily larger than the width of the wave packet. The peak is shown to result from a superoscillatory superposition at the tail. Similar reasoning a...