Amplitude and Phase of Wave Packets in a Linear Potential

We theoretically study and successfully observe the evolution of Gaussian and Airy surface gravity water wave packets propagating in an effective linear potential. This potential results from a homogeneous and time-dependent flow created by a computer-controlled water pump. For both wave packets we measure the amplitudes and the cubic phases appearing due to the linear potential. Furthermore, we demonstrate that the self-acceleration of the Airy surface gravity water wave packets can be completely canceled by a linear potential.

Figure 1

Experimental setup for generating Gaussian and Airy surface gravity water wave packets moving in a time-dependent homogeneous flow created by a water pump.

Figure 2

Kennard phase for Gaussian surface gravity water wave packets. Observed surface elevation $\eta$ (black line) and predicted envelopes (colored lines) given by Eqs. (2) and (4) without (a) and with (b) external flow. In both cases $k_0=20{\mathrm{m}}^{-1}$, $a_0=5.0\mathrm{mm}$ ($ϵ=0.1$), $c_g=0.35\mathrm{m}/\mathrm{s}$, and $t_0=0.72\mathrm{s}$. The thick blue and red lines connect maxima of the envelopes measured for $x=1.4\mathrm{m}$, 1.69 m, 1.89 m, and 2.02 m. (c) The mean value $⟨t⟩(x)$ (blue circles and red squares) of the temporal coordinate, Eq. (6). The corresponding solid lines are quadratic fits, Eq. (7), determining the group velocity $c_g=1/a_1$ and the “force” $F\equiv -({\omega }_0/{ϵ}^3{k}_0^2)a_2=-6.58$ in Eq. (1). (d) The phase offset of the Gaussian wave packet at its maximum without (blue circles) and with (red squares) external flow compared to the prediction by Eq. (8) represented by solid lines. (e) The modified Kennard phase, defined as the difference of the red squares and the blue circles presented in (d), with the black solid line given by $-(2/3)F^2{\xi }^3$, Eq. (8).

Figure 3

Generalized Kennard phase for inverted Airy surface gravity water wave packets. Observed surface elevation $\eta$ (black line) and predicted envelopes (colored lines) given by Eqs. (2) and (13) without (a) and with (b) external flow. In both cases $k_0=23{\mathrm{m}}^{-1}$, $a_0=3.93\mathrm{mm}$ ($ϵ=0.09$), $c_g=0.33\mathrm{m}/\mathrm{s}$, $t_0=0.82\mathrm{s}$, and $\alpha =0.05$. The thick blue and red lines connect maxima of the envelopes measured for $x=0.8\mathrm{m}$, 1.0 m, 1.2 m, and 1.4 m. (c) The mean value (blue circles and red squares) of the temporal coordinate $⟨t⟩(x)$, defined by Eq. (6), with integration performed only over the main lobe, is determined by the surface elevation $\eta$ for the parameter $t_0=0.27\mathrm{s}$. The corresponding solid lines are quadratic fits, Eq. (7), determining $c_g=1/a_1$ together with $F+1/{\tau }_0^3\equiv -({\omega }_0/{ϵ}^3{k}_0^2)a_2$ and thus $F=-1.94$. The black dashed line $x=c_{g}t$ is shown for comparison. (d) The phase offset of the Airy wave packet at the maximum of the main lobe without (blue circles) and with (red squares) external flow, the solid lines are given by Eq. (15). (e) The generalized Kennard phase, obtained by subtracting the linear terms $(F+1/{\tau }_0^3)\tau \xi$ and $({\alpha }^2/{\tau }_0^2)\tau \xi$, Eq. (14), at $\tau ={\tau }_{\max }^{(Ai)}(\xi )$ from the total measured phase presented in (d). The blue and red curves represent the theoretical predictions, $2{\xi }^3/(3{\tau }_0^6)$ and $[F^2/3+F/{\tau }_0^3+2/(3{\tau }_0^6)]{\xi }^3$, accordingly.

Figure 4

Cancellation of the self-acceleration of an inverted Airy wave packet by a linear potential. Experimental results (white dots) and simulations (colored density plot) based on Eq. (1), for an inverted Airy wave packet without (a) and with (b) an external flow, corresponding to $F=0$ and $F=-11.38$, accordingly. In both cases $k_0=23{\mathrm{m}}^{-1}$, $a_0=3\mathrm{mm}$ ($ϵ=0.07$), $t_0=0.32\mathrm{s}$, $\alpha =0.05$, and $c_g=0.33\mathrm{m}/\mathrm{s}$.