Gravity wave amplification and phase crest re-organization over a shoal

In this experimental work, both wave amplification and phase evolution, due to a submerged mound, are studied. In addition to the classical surface wave measurements, the experimental study takes advantage of photographs that underline crest re-organization above and down-wave the shoal. In particular, together with wave amplification up to more than twice the incident wave, a wave steepening is observed in certain conditions in both the wave direction and in the cross-section. Due to a phase crest separation downstream of the shoal, steepening in the crossshore direction is enhanced (up to 30% above the steepening along the main direction of propagation). Physical aspects are discussed through the analysis of the diffraction effects on the wave properties.


Introduction
An estimate of the wave characteristics near the shore has been one of the major issues in coastal engineering.For the purpose of the maritime navigation, the local increase of wave steepness may become dramatic when huge waves are generated.For wave propagating over varying sea-bed topography, wave propagation numerical models, that are spectral (Booij et al., 1999;Benoit et al., 1996), phase resolving (Berkhoff, 1972;Radder, 1979;Kirby, 1986) or based on Boussinesq equations (Li and Zhan, 2001) have been developed.For their validation, comparisons to experimental studies are often made.A number of experiments on different shoal configurations with various shapes were realised (Arthur, 1946;Berkhoff et al., 1982;Vincent and Briggs, 1989) in order to determine the wave transformation behind the shoal.Numerous numerical Correspondence to: V. Rey (rey@univ-tln.fr)models were tested on the Vincent and Briggs' experiment conducted in CERC's basin (e.g., Suh and Dalrymple, 1993;Holthuijsen et al., 2003).The shoal consisted of an elliptic mound, patterned after Berkhoff et al. (1982), with a major radius of 3.96 m, minor radius of 3.05 m, and a maximum height of 30.48 cm at the centre, on which monochromatic and spectral incident waves were tested.They showed, in particular, wave amplification up to more than twice that of the incident wave at the downstream part of a submerged elliptical mound for regular waves.As a consequence, wave steepness is strongly increased since the wave length is shorter above the mound.If wave amplification and direction of propagation are well-illustrated in these studies, there is no information on the wave phase evolution, especially in the cross-shore direction.This can be of major interest since wave diffraction may lead to steep slopes along the crests.The purpose of this work is to study both wave amplification and phase evolution due to a submerged mound.In addition to the classical surface wave measurements, the experimental study takes advantage of photographs that underline crest reorganization above the shoal.After a presentation of the experimental set-up, results are presented and discussed.

Experimental set-up
The experiment was performed at the wave basin of Toulon University, France.The basin has a useful length of 10 m, an effective width of 2.6 m and a maximum water depth of about 1 m.At one end, a wave generator allowed to generate regular and unidirectional waves.At the other end, a parabolic duckboard played an absorber function.The underwater three-dimensional mound is a 1.50 m long by 1 m wide quasi-elliptic shoal, with a maximum height of 30 cm as shown in  Acquisition was done through the mobile girder every 10cm, from 0.30m upstream the shoal (x=3.20m) to 1.40m downstream (x=6.40m).The sampling frequency was 32 Hz, the duration of the data acquisition was 60 seconds.A digital camera was placed on a footbridge at 8m from the wave generator.Photographs were shot at 1m and 2m above the water level in the total dark.For the wave measurements, a resistive wave gauge was placed in front of the wave generator as reference, and 6 others were fixed linearly crosswise on a mobile girder.In Fig. 2, the location of the shoal and of the gauges are presented from a plan view.Gauges 2-4 were separated from a distance of 15 cm whereas gauges 4-7 were spaced by 20 cm, in order to improve the accuracy of the measure just above the shoal.
Acquisition was done through the mobile girder every 10 cm, from 0.30 m upstream the shoal (x = 3.20 m) to 1.40 m downstream (x = 6.40 m).The sampling frequency was 32 Hz, the duration of the data acquisition was 60 s.A digital camera was placed on a footbridge at 8 m from the wave generator.Photographs were shot at 1 m and 2 m above the water level in the total dark.
The only light came from the flash of the camera that lit the basin in an oblique way.This allowed crests to be illuminated and easily distinguishable on the photographs.In the experiment, incident waves are monochromatic and unidirectional.A summary of all the experimental conditions is provided in Table 1.In the present study focused on wave steepening due to refraction-diffraction over the shoal, runs 6-8 are analysed (water depth h = 0.05 m above the top of the shoal).Runs at this water level for higher wave heights (runs 9-12) did not show any significant changes in the free surface behaviours, except a slight breaking for the highest periods that disturbed observations of the crests location.Detailed results for the other cases are presented  in Jarry (2009).They underline the diffraction phenomenon for the submerged cases and the freely wave crests crossing when the shoal is emerged.

Experimental results
The experimental results, hereafter presented, correspond to runs 6 to 8.They correspond to the moderate incident wave steepness ξ = H /L and to a water depth h = 0.05 m above the top of the shoal (see Table 1).

Wave evolution in the direction of propagation
The amplification factor along x-axis in the centre of the wave basin (Y = 2 m) is presented in Fig. 3, respectively, for T = 0.4 s, 0.5 s and 0.6 s.The same trend is observed, with an increase of the amplitude up to X = 4.5-4.7 m, near the end of the top part of the shoal (see Table 2) and a decrease downstream.
For T = 0.6 s, an increase of a factor 2.5 of the wave amplitude is observed.The relative wave slope RWS can be calculated as follows: Where H 0 and L 0 are, respectively, the incident wave height and wave length, H is the maximum wave height and L the local distance between two successive maxima.RWS for runs 6-8 are reported in Table 2.We can observe (see Sect. 3.2) that the local distance between two successive maxima downstream the shoal is slightly higher than for the incident wave.It can be explained by the wave diffraction phenomenon at the origin at the wave acceleration in the central part of the shoal (see Sect. 4.1).Table 2: Maximum of the amplification factor and wave slope along the x-axis For T=0.6s, an increase of a factor 2.5 of the wave amplitude is observed.The relative wave slope RWS can be calculated as follows: Where H0 and L0 are respectively the incident wave height and wave length, H is the maximum wave height and L the local distance between two successive maxima.RWS for runs 6 to 8 are reported in Tab. 2. We can observe (see section 3.2) that the local

Wave amplitude evolution in a cross-section
The amplification factor along y-axis down-wave the shoal (X = 5 m) is presented in Fig. 4 for T = 0.4, 0.5 and 0.6s.We can observe an oscillating behaviour of the amplification with decreasing values of the maxima from the centre to the sides of the wave basin.Assuming an almost constant phase in the cross-section, the maximum slope is given by H /L c where H is the difference between the maximum (at Y = Y max = 2) and the first minimum (at Y = Y min ) and L c = Y min -Y max .The RSW can be expressed as follow: Results are given in Table 3.We can see that the wave slope remains quite mild compared to the wave slope in the direction of propagation.

Wave crests evolution
Photographs of the free surface for runs 6 to 8 are presented in Figs.5-7.
As a general trend, we can observe a refraction-diffraction process due to the shoal.Downstream of the shoal, a tridimensional pattern is observed due to segmentation onto 3 parts of the wave crest: a central crest with a quasilinear increase of its width and two symmetrical crests of an opposite phase on both sides.The quantitative results are calculated thanks to the synchronized wave gauges 2-7 and       the use of the gauge 1 as a reference for successive locations of the girder along the basin.Wave phases along the basin (diamonds, crest extrapolations are in grey) and amplitudes above and after the shoal are presented in Figs.8-10 for runs 6-8.
In Figs.5-7, the boundary between the crest at the centre and on both sides is depicted by two dotted lines moving apart with a certain angle.This opening angle is logically due to a diffraction effect since refraction does not occur after the shoal passage, because of a flat bottom.The phase lag between the crest at the centre of the basin and the crest on both sides is about one half-period, as observed in Figs.8-10.Within this transitional zone, the signal from the wave gauges was completely flat and then did not allow the determination of the location of the crests.Let us note that for the run 8 (T = 0.6 s), the basin was quickly subject to transverse oscillations that disturbed wave propagation.
However, for the three cases, both photographs and wave fields show that despite the phase lag at the crests in the cross direction, a behaviour of the crests nearly rectilinear is observed down-wave of the shoal.For T = 0.4 s (Figs. 5  and 8), the central crest is rectilinear with an almost constant amplitude in the cross direction after down-wave from the shoal after a transition zone diffusion of the energy along the crests (see the three first wavelengths in Fig. 10).We can also observe a diminishing of the phase lag (from X = 5.8 m), which may also be due to sidewall effects.Similar trends are observed for T = 0.5 s and T = 0.6 s with a more pronounced curvature of the crests.
Right downstream (X = 5 m), central and side lines are in opposite phases.The maximum slope is then given by H /L c where H is the sum of the maximum (at Y = Y max = 2) and the first minimum (at Y = Y min ) and L c = Y min −Y max .The RSW are given by expression (2), results are presented in Table 4.
We can observe that the wave slope is of the same order as observed in the direction of propagation.For T = 0.6 s, the slope is 30% steeper than observed in the direction of propagation.In figures 5 to 7, the boundary between the crest at the centre and on both sides is depicted by two dotted lines moving apart with a certain angle.This opening angle is logically due to diffraction effect since refraction does not occur after the shoal passage, because of a flat bottom.The phase lag between the crest at the centre of the basin and the crest on both sides is of about one half-period, as observed in Figs. 8 to 10. Within this transitional zone, the signal from the wave gauges was completely flat and then did not allow the determination of the location of the crests.Let us note that for the run 8 (T=0.6s), the basin was quickly subject to transverse oscillations that disturbed wave propagation.
However, for the three cases, both photographs and wave fields show that despite the phase lag at the crests in the cross direction, a behaviour of the crests nearly rectilinear is observed down-wave of the shoal.For T=0.4s (Figs. 5 and 8), the central crest is rectilinear with an almost constant amplitude in the cross direction after down-wave the shoal after a transition zone diffusion of the energy along the crests (see the three first wavelengths in Fig. 10).We can also observe a diminishing of the phase lag (from X=5.8m), which may also be due to sidewall effects.Similar trends are observed for T=0.5s and T=0.6s with a more pronounced curvature of the crests.
Right downstream (X=5m), central and side lines are in opposite phases.The maximum slope is then given by H/ L c where H is the sum of the maximum (at Y= Y max =2) and

Discussion and conclusion
In the experiments presented above, we observed that not only the wave slope may be steep downstream of the shoal in the main wave direction, but also in the cross-section.In the following, the role of the diffraction in the wave crest evolution is presented and then discussed for the present experiment conditions.

Role of the diffraction on the wave properties
The potential associated to a progressive surface wave is given by: where H (x,y,ω) the complex representation of the crestto-trough height of the surface elevation, ω the pulsation, h(x,y) the water depth, g the gravity acceleration and k the wave number given by the dispersion relation: H can be written where Ĥ (x,y,ω) is the height envelope and S (x,y,ω) its phase.In the linear theory of surface gravity waves propagating over a mild sloping bottom, the equation of propagation is given by the mild-slope equation (Berkhoff, 1972): where C = ω/k is the phase celerity and C g = ∂ω/∂k is the group velocity.
Taking the mild-slope equation, following the height envelope and phase, one obtains after the separation of real and imaginary parts: and div C C g Ĥ 2 − − → gradS = 0 (8) In the pure refraction case, where the amplitude variation is considered as negligible, Eq. ( 7) leads to the eikonal equation: k = − − → gradS.When diffraction effect becomes preponderant, the second term of the right-hand side of Eq. ( 7) can not be neglected since k cannot be directly assimilated to the wave number of a progressive wave.Writing and introducing this relation in Eqs. ( 7) and ( 8), one obtains the following diffraction parameter (Holtuijsen and al., 2003): This parameter, which can be either positive or negative, indicates that in the presence of the diffraction effect, the Fig. 1.It consisted of a half-sphere, widened with a 50 cm rectilinear part on its centre.The extension of the shoal in the wave propagation direction (x-axis) had been numerically tested by using the model REF/DIF1 (Kirby Published by Copernicus Publications on behalf of the European Geosciences Union.N. Jarry et al.: Gravity wave amplification and phase crest re-organization over a shoal view.Gauges 2 to 4 were separated from a distance of 15cm whereas gauges 4 to 7 were spaced by 20cm, in order to improve the accuracy of the measure just above the shoal.

Fig. 2 :
Fig. 2: Schematic plan view of the shoal and location of the wave gaug

Fig. 2 .
Fig. 2. Schematic plan view of the shoal and location of the wave gauges.

Fig. 3 :
Fig. 3: Amplification factor along x-axis in the centre of the basin

Fig. 3 .
Fig. 3. Amplification factor along x-axis in the centre of the basin.

Fig. 4 .
Fig. 4. Amplification factor in the cross-section x = 5 m downstream of the shoal.

3. 3
Photographs of the free surface for runs 6 to 8 are presented in Figs. 5 to 7.