Experimental study of three-wave interactions among capillary-gravity surface waves

In propagating wave systems, three- or four-wave resonant interactions constitute a classical nonlinear mechanism exchanging energy between the different scales. Here we investigate three-wave interactions for gravity-capillary surface waves in a closed laboratory tank. We generate two crossing wave trains and we study their interaction. Using two optical methods, a local one (laser doppler vibrometry) and a spatiotemporal one (diffusive light photography), a third wave of smaller amplitude is detected, verifying the three-wave resonance conditions in frequency and in wave number. Furthermore, by focusing on the stationary regime and by taking into account viscous dissipation, we directly estimate the growth rate of the resonant mode. The latter is then compared to the predictions of the weakly nonlinear triadic resonance interaction theory. The obtained results confirm qualitatively and extend previous experimental results obtained only for collinear wave trains. Finally, we discuss the relevance of three-wave interaction mechanisms in recent experiments studying gravity-capillary turbulence.

Figure 1

(a) Schematic view of the interaction zone between the two mother waves. $O$ is the origin of this zone, $M$ locates a point in this area along the direction $O_{\xi }$ given by ${\mathbf{k}}_{\mathbf{1}}+{\mathbf{k}}_{\mathbf{2}}$. (b) Graphical construction of the triad for $f_1=15$ Hz and $f_2=18$ Hz with ${\alpha }_{12r}=54$ deg (resonant angle). The green circle (dash-dotted line) corresponds to ${\mathbf{k}}_{\mathbf{3}}\mathbf{r}$ satisfying the gravity-capillary dispersion relation and the red circle (continuous line) to all location of the extremity of vector ${\mathbf{k}}_{\mathbf{2}}$. Axes are here defined relatively to ${\mathbf{k}}_{\mathbf{1}}$.

Figure 2

For a fixed mother wave $f_1=15$ Hz, (a) evolution of the angle ${\alpha }_{12r}$ versus $f_2$ from Eq. (5) and (b) evolution of the interaction coefficients versus $f_2$ from Eq. (8).

Figure 3

(a) Top view of the set up for local temporal measurements: the tank is filled with water mixed with ${\mathrm{TiO}}_2$ pigment. The red spot is the beam of the laser vibrometer. (b) Top view of the setup for spatiotemporal measurements with the DLP method. The liquid is a solution of intralipids in water. The blue rectangle depicts the observation area $\mathcal{S}$ of $103\times 94 {\mathrm{mm}}^2$ where the wave field reconstruction is performed. (c) Schematic representation of the side view of the setup for DLP spatiotemporal measurements.

Figure 4

Examples of wave-field reconstruction obtained with the DLP technique, for $f_1=15$ Hz, $f_2=18$ Hz and ${\alpha }_{12}=54$ deg: (a) in the transient and (b) in the stationary regime, with a color scale in mm (total height of fluid in the tank), with $a_1=130$ and $a_2=132$ $\mu \mathrm{m}$ (high forcing). The position of the laser spot for vibrometer measurements is indicated by the black cross symbol located at $x_v=54.6$ mm and $y_v=43.3$ mm.

Figure 5

(a) Local power spectrum $P_{\eta }\left(\omega \right)$ for $f_1=15$, $f_2=18$ Hz and ${\alpha }_{12}=54$ deg. There, $a_1=130$ and $a_2=132 \mu \mathrm{m}$ (high forcing). (b) Spatiotemporal spectrum of wave elevation $S_{\eta }(\omega ,k)$. White dashed line: linear theoretical dispersion relation for ${\sigma }_{\mathrm{fit}}=55$ mN/m. Color scale corresponds to ${log}_{10}\left[S_{\eta }(\omega ,k)\right]$ with $S_{\eta }$ in ${\mathrm{mm}}^3\mathrm{s}$, with $a_1=55$ and $a_2=56 \mu \mathrm{m}$ (low forcing).

Figure 6

Spatiotemporal spectra of wave elevation $S_{\eta }(\omega ,k_x,k_y)$, is displayed for the triad frequencies: (a) $f_1$, (b) $f_2$, and (c) $f_3$. Circles correspond to the linear theoretical dispersion relation at the given frequency. The arrows indicate the vector ${\mathbf{k}}_{\mathbf{i}}$ extracted from the maxima of the experimental spectra. We observe ${\mathbf{k}}_{\mathbf{3}}\approx {\mathbf{k}}_{\mathbf{1}}+{\mathbf{k}}_{\mathbf{2}}$. Color scale ${log}_{10}\left(S_{\eta }\right)$ arbitrary unit, with $a_1=55$ and $a_2=56 \mu \mathrm{m}$.

Figure 7

Comparison between the experimental values of ${\mathbf{k}}_{\mathbf{1}}+{\mathbf{k}}_{\mathbf{2}}$ and ${\mathbf{k}}_{\mathbf{3}}\exp .$ as a function of the product $a_1{a}_2$ of the mother wave amplitudes. (a) in norm and (b) in direction, when ${\alpha }_{12}=54$ deg. In (a) the value of $k_3$ given by the linear dispersion relation is represented by a black line. The black dashed line provides the acceptable bounds due to the finite resolution ${\delta }_k=30.5 {\mathrm{m}}^{-1}$ of the spectra. In (b) $\theta$ is the angle between the horizontal axis of Fig. 6 and ${\mathbf{k}}_{\mathbf{3}}$. The black dashed lines indicate the angular accuracy ${\delta }_{\theta }$ around the average orientation $〈\theta 〉$ of ${\mathbf{k}}_{\mathbf{1}}+{\mathbf{k}}_{\mathbf{2}}$. The product $a_1{a}_2$ has been measured with the laser vibrometer, in equivalent experimental conditions.

Figure 8

(a) ${\tilde{a}}_1(x,y)$ providing the spatial distribution of wave mode at the frequency $f_1$, when ${\alpha }_{12}=54$ deg. The black cross indicates the corresponding position of the laser beam for the measurements performed with the vibrometer. Color scale ${\tilde{a}}_i$ in mm. Here $a_1=104$ and $a_2=110 \mu \mathrm{m}$. (b) ${\tilde{a}}_2(x,y)$ spatial mode at $f_2$. (c) ${\tilde{a}}_3(x,y)$ spatial mode at $f_3$.

Figure 9

(a) Evolution of wave amplitudes ${\tilde{a}}_i$ (averaged between $49<x<55$ mm) along the propagation direction of the wave 3 as a function of the distance $d$ from the bottom of the image. The behavior of ${\tilde{a}}_3$ is compared to the model (black solid line) given by Eq. (10) with $\xi \simeq d$. The vertical black dashed line gives the position $y_v$ of the laser beam for vibrometer measurements. (b) Rescaled amplitude of ${\tilde{a}}_3/\left({\tilde{a}}_1{\tilde{a}}_2\right)$ as a function of the distance $d$ for different measurements ($6.5\times {10}^{-9}<a_1{a}_2<1.7\times {10}^{-8}{\mathrm{m}}^2$). We notice that the spatial amplitude modulations are not in phase in the different measurements. The average evolution of ${\tilde{a}}_3$ on the measurements, green (light gray) line, is compared with the model given by Eq. (10), black solid line. The linear limit for negligible dissipation [black dashed line from Eq. (12)] and the saturated value [light-gray dotted line from Eq. (13)] are also depicted.

Figure 10

(a) Amplitudes $a_i$ and (b) $sin\varphi$ versus time for $f_1=15$ Hz, $f_2=18$ Hz, and ${\alpha }_{12}=54$ deg. For these parameters, $a_1=130$ and $a_2=132 \mu \mathrm{m}$. (c) Mean value of $sin\varphi$ versus $a_1{a}_2$ in stationary regime, with in the inset: the standard deviation of $sin\varphi$, quantifying fluctuations around the mean value. We observe that the phase locking occurs for the considered forcing amplitudes, but at low forcing $sin\varphi$ departs slightly from 1 and fluctuations are stronger. (d) In stationary regime $\langle a_3\rangle \times {\delta }_3/\left(K\left(\xi \right)\langle sin\varphi \rangle \right)$ is found proportional to $a_1{a}_2$ for small and moderate forcing. Solid line is the linear fit valid for not too high values of $a_1{a}_2$.

Figure 11

Decrease of the amplitude of the wave with frequency $f=33$ Hz versus the distance from the wave maker, for a small excitation forcing amplitude.