Dispersion Management of Propagating Waveguide Modes on the Water Surface

We report on the theoretical and experimental study of the generation of propagating waveguide modes on the water surface. These propagating modes are modulated in the transverse direction in a manner that satisfies boundary conditions on the walls of the water tank. It is shown that the propagating modes possess both anomalous and normal dispersion regimes, in contrast to the extensively studied zero mode that, in the case of deep water, only has normal dispersion with a fixed frequency independent dispersion coefficient. Importantly, by using a carrier frequency at which the group velocity dispersion crosses zero, a linear nonspreading and shape-preserving wave packet is observed. By increasing the wave steepness, nonlinear effects become pronounced, thereby enabling the first observation of linearly chirped parabolic water wave pulses in the anomalous dispersion regime. This parabolic wave maintains its linear frequency chirp and does not experience wave breaking during propagation.

Figure 1

Group velocity dispersion $D_{\omega ,m}$ as a function of the relative angular frequency $\omega /{\omega }_{c,m}$. The parallel dashed line and dotted line mark the zero dispersion point and the maximum positive value of $D_{\omega }$, respectively, while the vertical dashed line marks $\omega /{\omega }_{c,m}=3^{0.25}$.

Figure 2

(a),(b) Illustration of two typical excitation modes of wave maker, denoted as (a) (${\eta }_{-},{\eta }_{+},{\eta }_{+},{\eta }_{-}$) and (b)(${\eta }_{-},{\eta }_{+},{\eta }_{-},{\eta }_{+}$). (c)–(e) The initial waveguide mode patterns: (c) guiding mode with $m=2$, which could be generated using excitation mode (a); (e) guiding mode with $m=4$ generated using excitation mode (b); (d) owing to the limitations of only four wave maker sections, the guiding mode with $m=3$ cannot be excited directly; (f) experimental measurements (data points) for the dispersion relation between $c_{g,m}$ and $\omega$. In experiments, $t_0=3\mathrm{s}$ and $a_0=10\mathrm{mm}$. The solid curves are theoretical predictions based on Eq. (5).

Figure 3

Propagation dynamics of wave packets with a guiding mode of $m=2$. The related parameters were set as: $t_0=3\mathrm{s}$, $a_0=10\mathrm{mm}$, and (a) $\omega =7.3\mathrm{rad}/\mathrm{s}$ (near the 2nd cutoff); (b) $\omega =8.0\mathrm{rad}/\mathrm{s}$; (c) $\omega =9.0\mathrm{rad}/\mathrm{s}$ (zero dispersion frequency). Both the experiments and the simulations show pulse envelope profiles at four different locations of the tank.

Figure 4

Observations of nonlinear parabolic wave packets for two different incident amplitudes: (a)–(c) $a_0=25\mathrm{mm}$ and (d)–(f) $a_0=40\mathrm{mm}$. In experiment, the Gaussian duration was fixed to $t_0=3\mathrm{s}$, and $\omega =8.5\mathrm{rad}/\mathrm{s}$. (a),(d) The measured wave elevations at two different locations; (b),(e) the corresponding pulse intensity profiles; (c),(f) The linear frequency chirp, which could be measured by extracting the local maximum (peak) values of the elevations. Note that these measurements were based on the guiding mode with $m=2$.