Experimental investigation of the Peregrine Breather of gravity waves on finite water depth

A series of laboratory experiments were performed to study the Peregrine Breather (PB) evolution in a wave flume of finite depth and deep water. Experimental cases were selected with water depths $k_{0}h$ ($k_0$ is the wave number and $h$ is the water depth) varying from 3.11 to 8.17 and initial steepness $k_0{a}_0$ ($a_0$ is the background wave amplitude) in the range 0.06 to 0.12, and the corresponding initial Ursell number in the range 0.03 to 0.061. Experimental results indicate that the water depth plays an important role in the formation of the extreme waves in finite depth; the maximum wave amplification of the PB packets is also strongly dependent on the initial Ursell number. For experimental cases with the initial Ursell number larger than 0.05, the maximum crest amplification can exceed three. If the initial Ursell number is nearly 0.05, a shorter propagation distance is needed for maximum amplification of the height in deeper water. A time-frequency analysis using the wavelet transform reveals that the energy of the higher harmonics is almost in-phase with the carrier wave. The contribution of the higher harmonics to the extreme wave is significant for the cases with initial Ursell number larger than 0.05 in water depth $k_{0}h<$ 5.0. Additionally, the experimental results are compared with computations based on both the nonlinear Schrödinger (NLS) equation and the Dysthe equation, both with a dissipation term. It is found that both models with a dissipation term can predict the maximum amplitude amplification of the primary waves. However, the Dysthe equation also can predict the group horizontal asymmetry.

Figure 1

Temporal and spatial evolution of the PB [Eq. (7)]. $a_0$ = 0.04 m, $f_0$ = 0.8 Hz, $k_0$ = 2.59 ${\mathrm{m}}^{-1}$, $h$ = 1.2 m.

Figure 2

The expected location as a function of $k_{0}h$.

Figure 3

Experimental setup (not to scale). Note that not all of the 46 wave probes are depicted in the sketch.

Figure 4

Temporal evolution of the surface elevations at 12 different locations along the flume (Case 3: $k_{0}h$ = 3.11, $U_r$ = 0.052). Dashed red and dashed dot black lines correspond to propagation with the linear group velocity $c_g$ = 0.995 m ${\mathrm{s}}^{-1}$ and the third-order nonlinear group velocity $c_g$ = 1.012 m ${\mathrm{s}}^{-1}$, respectively.

Figure 5

Temporal evolution of the surface elevations at 12 different locations along the flume (Case 8: $k_{0}h$ = 4.83, $U_r$ = 0.061). Dashed red and dashed dot black lines correspond to propagation with the linear group velocity $c_g$ = 0.781 m ${\mathrm{s}}^{-1}$ and the third-order nonlinear group velocity $c_g$ = 0.798 m ${\mathrm{s}}^{-1}$, respectively.

Figure 6

Comparison of the measured surface height at the first gauge ($x$ = 3.7 m) and the position of maximum wave amplitude with the theoretical PB solution for (a) Case 3; (b) Case 8. (These wave trains have been shifted in time so that the maximum peaks are shown to occur simultaneously.)

Figure 7

The wave amplitude spectra at various locations along the flume. The vertical dashed lines represent the boundaries of the frequency range ($f_0$, $1.8f_0$), the red dash dotted lines denote the spectra of the maximum wave in the evolutionary process. (a) Case 1; (b) Case 4; (c) Case 6; (d) Case 3.

Figure 8

The wave amplitude spectra at various locations along the flume. The vertical dashed lines represent the boundaries of the frequency range ($f_0$, $1.8f_0$), the red dash dotted lines denote the spectra of the maximum wave in the evolutionary process. (a) Case 5; (b) Case 11; (c) Case 8; (d) Case 10.

Figure 9

The corresponding wavelet spectrum of Case 3 at three locations: (a) $x$ = 3.7 m; (b) $x$ = 39.2 m; (c) $x$ = 50.1 m. (Each spectrum is nondimensionalized by the maximum value at each location.)

Figure 10

The corresponding wavelet spectrum of Case 11 at three locations: (a) $x$ = 3.7 m; (b) $x$ = 21.2 m; (c) $x$ = 30.2 m. (Each spectrum is nondimensionalized by the maximum value at each location.)

Figure 11

Spatial variation of maximum wave crest (left) and maximum wave height (right) in the cases (Cases 1 through 11 shown via $U_r$) with different wave steepness and relative water depth; hollow symbols denote the filtered recording, and the solid symbols denote the original data.

Figure 12

The measured $E/E\left(0\right)$ and the fitted curves [Eq. (12)] versus the distances from the wave maker: (a) Case 3; (b) Case 8.

Figure 13

Calculated [NLS and Dysthe equations with the dissipation term, the surface elevations $\zeta (x,t)$ are related to the variable $A(x,t)$ to the leading order] and measured maxima of the absolute values of the wave train envelope of the series (band-pass filtered $[0.6f_0$, $1.8f_0]$): (a) Case 8; (b) Case 11.

Figure 14

The variations of the maximum crests along the propagation direction for the selected cases (a) Case 8; (b) Case 11. The results obtained by the numerical models are corrected to the third Stokes order.

Figure 15

The variations of the maximum crests along the propagation direction for the selected cases (a) Case 3; (b) Case 5; (c) Case 8; (d) Case 11. The results obtained by the numerical models are corrected to the leading Stokes order.

Figure 16

Comparison of the computed and measured experimental shapes (band-pass filtered $[0.6f_0$, $1.8f_0]$) of the modulus of the complex wave train envelope for Case 8: $U_r$ = 0.061, $k_{0}h$ = 4.83.

Figure 17

Comparison of the computed and measured experimental shapes (band-pass filtered $[0.6f_0$, $1.8f_0]$) of the modulus of the complex wave train envelope for Case 11: $U_r$ = 0.051, $k_{0}h$ = 8.17.