Geometrical focusing of surface waves

In this paper we investigate experimentally the waves produced by a parabolically shaped wave maker partially immersed into a water-filled basin and oscillating in the vertical direction. The main motivation of our study is to follow the analogy between light waves that focus on caustics, creating zones of intense light concentration, and surface waves in liquids. Similarly to light focusing, the surface waves undergo spatial focusing leading to the growth of their amplitudes till they reach a cusp singularity. We detect three regimes. In the first one, which corresponds to small forcing amplitudes, the wave field agrees well with geometrical optics and with Pearcey diffraction theory near the caustics. In the second regime, weak nonlinearities are detected. However, waves still focus as predicted. In the third regime, at large forcing amplitude, a large-scale mean flow is generated. This flow is concentrated in two symmetric jets emerging from the wave maker front wall and producing two counter-rotating eddies because of the finite size of the basin. This large-scale flow modifies the shape of the wave fronts and leads to a displacement of the cusp towards the wave maker, modifying the analogy between light and surface waves.

Figure 1

Rays and wave fronts produced by a parabolic wave maker as predicted by geometrical optics. (a) Rays starting at each point $(x_0,y_0)$ of the parabolic wave maker following the normal to the parabola. (b) Wave fronts produced by the parabolic wave maker. They are perpendicular to rays and move towards the top. (c) Picture of the wave maker, the basin, and the shaker. The wave field is visualized by adding white concentrated ink to water. The black dots are small floating beads used to measure the mean surface flow by particle tracking.

Figure 2

Topography of the free surface measured with the synthetic Schlieren method for two frequencies: (a) $f=5\text{Hz}$ and ${\eta }_0=36\mu \text{m}$, (b) $f=10\text{Hz}$ and ${\eta }_0=8\mu \text{m}$. For comparison we draw also some wave fronts (black lines) as predicted by ray theory. These wave fronts are separated from each other by a wavelength $\lambda$. The waves have small amplitudes, so nonlinearities are weak. The shape of the wave fronts and the value of the wavelength $\lambda$ agree very well, respectively, with the prediction of geometrical optics and the linear dispersion relation for surface waves [see Eqs. (3) and (1)]. In addition we also draw the caustics (white line).

Figure 3

Deformation of the free surface $\eta$ versus y along the symmetry axis ($x=0$). (a) $f=5\mathrm{Hz}$, wavelength $\lambda =6.5\text{cm}$, and initial amplitude ${\eta }_0=36\mu \text{m}$. (b) $f=10\text{Hz}$, wavelength $\lambda =2.3\text{cm}$ and initial amplitude ${\eta }_0=8\mu \text{m}$. Both curves are symmetric with respect to $\eta =0$, attesting that nonlinearities are weak. Waves travel from left to right. For comparison we include the prediction of the ray theory for the wave envelope as given by Eq. (5) (blue line), the Pearcey prediction for the wave envelope, calculated from Eq. (8) (black dashed line), and the position of the Huygens cusp $y=\frac{1}{2a}$ (vertical red dashed line). The crests of experimental waves at both frequencies agree with the prediction of the ray theory when $y<20$ cm, but for larger $y$, the Pearcey integral gives a better approximation.

Figure 4

Envelopes of $\eta$ for a monochromatic wave of frequency $f=10$ Hz. (a) Initial amplitude ${\eta }_0=8\mu \text{m}$. (b) Initial amplitude ${\eta }_0=24\mu \text{m}$. The blue curve is the positive branch of the envelope, whereas the red curve is the absolute value of the negative one. For comparison we include the predictions of the Pearcey integral (continuous black line) and the ray theory (dashed black lines). In the first case both branches of the envelopes remain close to each other, so nonlinear effects are undetectable. In the second case, the maximum difference between the two curves is approximately $10\mu \mathrm{m}$, which is approximatively 2.5 times the prediction made by Stokes.

Figure 5

Envelopes of $\frac{\partial \eta }{\partial y}$ for a monochromatic wave of frequency $f=10\mathrm{Hz}$. (a) Initial amplitude ${\eta }_0=8\mu \text{m}$. (b) Initial amplitude ${\eta }_0=24\mu \text{m}$. The blue curve is the positive branch of the envelope, and the red curve is the negative one. For both cases positive and negative branches of envelope collapse, in agreement with the weak nonlinear theory by Stokes.

Figure 6

(a) Wave profile $\eta$ versus $y$ along the symmetry axis (blue line) for a monochromatic wave of frequency $f=6.5\text{Hz}$ and an initial amplitude ${\eta }_0=200\mu \text{m}$. For comparison we include the prediction of both the ray theory (continuous black line) and the Pearcey integral (black dashed line) for the envelope of the waves. The wave exhibits an asymmetry with respect to $\eta =0$, which is a signature of a nonlinear behavior. In addition, wave crests are larger than the prediction of the wave envelope for $y<20\text{cm}$. (b) Topography of the free surface. For comparison we include some wave fronts as predicted from Eq. (3) (black lines) and the caustics (white lines). Despite the existence of a nonlinear behavior, there is still an agreement on the shape of the wave fronts between experiments and the ray theory prediction.

Figure 7

(a) Wave profile versus $y$ along the symmetry axis of a monochromatic wave of frequency $f=6.5\text{Hz}$ and an initial amplitude ${\eta }_0=500\mu \text{m}$ (blue line). For comparison we include the prediction of both the ray theory (continuous black line) and the Pearcey integral (black dashed line) for the wave's envelope. The wave profile is modified with respect to the case presented in Fig. 6. First, the position of maximal amplitude moves towards the wave maker. Second, the discrepancy between the amplitude of the crests and the envelopes is even greater. (b) Wave field in the domain $-12\mathrm{cm}<x<12$ cm and 5 cm $<y<$ 35 cm. For comparison, some wave fronts as predicted by the ray theory [deduced from Eq. (3)] are included (black lines) together with their corresponding caustics (white lines). The agreement with geometrical optics holds only near the plunger. Two differences emerge: first, the wavelength is no more equal to 4.2 cm, and, second, the wave fronts shape differs from the parabolic shape when moving away from the wave maker.

Figure 8

(a) Wave profile versus $y$ along the symmetry axis of a monochromatic wave of frequency $f=6.5\text{Hz}$ and an initial amplitude ${\eta }_0=900\mu \text{m}$ (blue line). For comparison we include the prediction for the waves envelope of both the ray theory (continuous black line) and the Pearcey integral (black dashed line). The position of maximum amplitude stands approximately at $y=15$. This point is closer to the wave maker than those obtained in the experiments with initial amplitudes ${\eta }_0=200\mu \text{m}$ and ${\eta }_0=500\mu \text{m}$. (b) Wave field in the domain $-12$ cm $<x<$ 12 cm and 5 cm $<y<$ 35 cm. For comparison, some wave fronts as predicted by the ray theory [deduced from Eq. (3)] (black lines) and the caustics (white lines) are included. The wave field is very different to the prediction from geometrical optics.

Figure 9

Large-scale velocity field produced by a monochromatic wave of frequency $f=6.5\text{Hz}$ and two different initial amplitudes: (a) ${\eta }_0=500\mu \text{m}$ and (b) ${\eta }_0=900\mu \text{m}$. These amplitudes are the same as those used for experiment shown in Figs. 7 and 8. The wave maker is drawn in red on the left. Two vortices and two jets appear, symmetrically located with respect to the center line. When measuring the vorticity field, it appears to be different from zero, which is in contrary to the assumptions made to deduce the Stokes drift.

Figure 10

Wave fronts in the presence of the large-scale flow. Black lines are the calculated wave fronts resulting from the transport induced by the large-scale velocity field. These wave fronts are superimposed to the experimental results and are in good agreement. (a) Wave field for $\eta =500\mu \text{m}$ and (b) wave field for $\eta =900\mu \text{m}$. The presence of the large-scale flow modifies the shape of wave fronts. The Huygens cusp is located at $y=21.5$ cm in the first case and at $y=16$ cm in the second case. The resulting caustics are represented by the white lines.

Figure 11

Maximum value of the transversal component of the large-scale flow velocity field $u_{\max }$ as a function of the wave amplitude ${\eta }_0$. At weak wave amplitude, we observe a linear relation. However, $u_{\max }$ saturates for large values of ${\eta }_0$ and then slightly decreases. The overall behavior does not match with the Stokes drift but agrees with the results by Punzmann et al. [19].

Figure 12

Longitudinal velocity $v$ profile along the $x$ direction, for two different waves amplitudes, ${\eta }_0=250\mu \text{m}$ (continuous line) and ${\eta }_0=900\mu \text{m}$ (dashed line), for $y=26.4$ cm and 20.2 cm, respectively. The maximum value of the velocity happens at the symmetry axis. We observe also negative values inside two intervals located symmetrically along the side of the container and corresponding to the back flow.

Figure 13

Vorticity distribution of the large-scale flow on the free surface. (a) Initial amplitude ${\eta }_0=500\mu \text{m}$ and (b) initial amplitude ${\eta }_0=900\mu \text{m}$. The vorticity is concentrated in the two recirculating cells and in the two jets. The flow is rotational, meaning that it does not fulfill the assumptions of the Stokes drift theory.