3.1. Unbroken and slightly breaking

Tests in the unbroken and slightly breaking regimes exhibit almost no overturning wave crests and air entrapment. The approaching wave is either unbroken or marginally breaking on the plate surface, as shown in images of figure 5(a,c). The pressure time history of test 1 in the unbroken region shows only quasi-hydrostatic pressure on a long time scale, see figure 5(b). Since no impact event occurs, the central time $t_0$ is calculated by integrating the duration of quasi-hydrostatic pressure for this regime. Additionally, all parameters extracted from the impact region are zero. The plate’s response appears quasi-static, with the plate stiffness balancing the external load and resulting in an almost in-phase relationship between pressure and deflection. The smallest measured plate deflection $\Delta x_{{hp}}/l = 0.006$ with a low acceleration of approximately 0.014 m s $^{-2}$ is observed.

Figure 5. Selected time series of pressure (solid line) at PT3 and elastic plate displacement at the free top (dotted line). Each test is centred horizontally about its expected central time $t_0$ . Coloured areas denote the pressure impact time range defined in § 2.4. An enlarged view of pressure oscillations under aerated impacts is shown with the corresponding amplitude spectrum obtained via fast Fourier transform (FFT).

Figure 6. Selected instantaneous wave profile histories of (a) slightly breaking, (b) flip through, (c) low aeration and (d) high aeration. The time interval between successive profiles is 5 ms.

Figure 7. Images captured at key time stamps: incipient impact, $\Delta x_{\textit{imp}}$ and $\Delta x_{{hp}}$ (from left to right) alongside pressure distributions along the vertical elastic plate for representative tests: (a) slightly breaking, (b) flip-through, (c) low-aeration and (d) high-aeration impacts. In the pressure plots, impact pressure maxima $P_{{max}}$ are indicated by red solid squares, while the maximum quasi-hydrostatic pressures $P_{{hp}}$ are shown as black open squares, fitted to a hydrostatic profile. Error bars represent measurement uncertainty, given by the standard deviation of the pressure readings.

The slightly breaking impact for test 3 features appreciable sharp pressure with a low amplitude of 1.9 $\rho gH$ and relatively short duration of 0.16 $\sqrt {g/h}$ , as seen in figure 5(d). The impact pressure maxima $P_{{max}}$ is between 1.0 and 2.5 times the maximum quasi-hydrostatic pressure, which falls within the slightly breaking limit proposed by Bullock et al. (Reference Bullock, Obhrai, Peregrine and Bredmose2007). Figure 7(a) presents the loading processes and associated pressure distributions imposed on the plate for test 3. The approaching wave front forces the run-up jet to rise rapidly at the plate. This can be found in figure 6(a) and the movie of test 3 in the supplementary material. The high impact pressure mainly concentrates below the strike point $z_{\textit{imp}}/h =0.177$ . Afterward, the vertical jet with an acceleration of nearly 15 $g$ forms and induces a rapidly decreasing load along the plate, which is commonly observed in the ELPs introduced by Lafeber et al. (Reference Lafeber, Bogaert and Brosset2012b ). Finally, the quasi-hydrostatic pressure builds up and fades away following the wave drawdown process. To assess the nature of the pressure field after impact, we analyse the spatial distribution of the maximum quasi-hydrostatic pressure measured along the vertical plate. The term quasi-hydrostatic (Bullock et al. Reference Bullock, Obhrai, Peregrine and Bredmose2007; Bredmose et al. Reference Bredmose, Peregrine and Bullock2009) used in this study does not imply a purely hydrostatic pressure distribution, but rather refers to a pressure field that approximately follows a hydrostatic gradient while still allowing for residual dynamic contributions, such as those from wave diffraction, radiation and structural vibration. As illustrated in figure 7, the post-impact pressure profile exhibits a near-linear variation with depth for all tests, which is characteristic of a hydrostatic distribution. Although some spatial fluctuations and dynamic effects remain, their influence becomes secondary, justifying the use of the term ‘quasi-hydrostatic’ to describe the dominant pressure trend in this phase. The relatively low pressure and small spatio-temporal extent of the slightly breaking impact naturally lead to an insignificant plate displacement. The plate deflection caused by the impact $\Delta x_{\textit{imp}}/l$ is 0.009 with an acceleration of approximately 0.14 m s $^{-2}$ , illustrated in figure 5(d). The plate deflects further with the increasing bending moment and reaches its maximum displacement of $\Delta x_{{hp}}/l=0.05$ under the maximum quasi-hydrostatic pressure. The plate eventually vibrates freely with material damping at a frequency of 0.92 $f_n$ . The slight reduction in natural frequency is due to added mass effects in wet conditions.

3.2. Low and high aerations

The aerated impact on the plate is characterised by a distinct overturning wave crest accompanied by the run-up of the wave front, which can be found in figure 6(c,d) and movies of tests 5–7 in the supplementary material. The degree of the crest overturning determines the amount of air entrapment. The low-aeration impact experiences relatively little air enclosed by the wave front adjacent to the plate, as seen in figure 5(e) for test 5, while high aeration has an appreciable air-pocket entrainment between the plunging jet and plate prior to impact, as shown in figure 5(g) for test 7. Moreover, the flip-through impact (Peregrine Reference Peregrine2003; Lugni et al. Reference Lugni, Brocchini and Faltinsen2006) with a wave front almost parallel to the plate is observed in the surface profile history in figure 6(b) and the image in figure 7(b) for test 4, with little, if any, air entrapment. An arc with upward curvature forms between the wave crest and the run-up jet, gradually contracting to a point, as demonstrated in the movie of test 4 in the supplementary material. This process is similar to the ‘focusing surface’ phenomenon described by Wang et al. (Reference Wang, Ikeda-Gilbert, Duncan, Lathrop, Cooker and Fullerton2018). The rapidly focusing surface at the contact region leads to an upward splash with an acceleration of approximately 466 g close to 481 g observed in Wang et al. (Reference Wang, Ikeda-Gilbert, Duncan, Lathrop, Cooker and Fullerton2018). The pressure distribution along the plate exhibits the most spatially localised behaviour in all tests.

The pressure history of low aeration presents a shorter rise time and longer fall time compared with the slightly breaking impact, see figure 5(f) for test 5. The impact duration and pressure maxima significantly increase to 0.2 $\sqrt {g/h}$ and 8.9 $\rho gH$ , respectively. The pressure oscillations during the impact process are associated with a dense cloud of bubbles formed by the fragmentation of the entrapped air pocket. When a wave impacts a vertical wall, the natural frequency $f_b$ of an entrapped 2-D gas pocket with a semicircular cross-section of radius $r$ in water can be estimated using the theoretical expression (Faltinsen & Timokha Reference Faltinsen and Timokha2009, p. 499)

(3.1) \begin{equation} f_b = \frac {1}{2 \pi } \sqrt {- \frac {2 \gamma P_0 \left ( 1 + \frac {1}{2} \lambda _d^2 r^2 \right )} {\rho r^2 \left [ \log { \left (\frac {1}{2} \lambda _d r \tan {\lambda _d d_b} \right )}+ \frac {1}{4} \lambda _d^2 r^2\right ]} }, \end{equation}

where $\gamma =1.4$ is the ratio of specific heats for air, $P_0$ is the atmospheric pressure and

(3.2) \begin{equation} \lambda _d=\frac {\pi }{2\left (d_a + d_b\right )}, \end{equation}

where $d_b$ and $d_a$ represent the distances from the centre of the gas pocket to the free surface and the bottom boundary, respectively. These distances can be determined based on the bubble locations near the PT. This formulation accounts for boundary effects and better reflects the hydrodynamic conditions of low-aeration scenarios near solid walls. The bubble radii, identified from images in figure 5(f), range from approximately 2.5 to 7.5 mm, yielding a natural frequency between 195 and 517 Hz as estimated by (3.1). This range aligns well with the pressure oscillation frequencies obtained from the Fourier analysis of the pressure history after the impact peak. While FFT is limited in resolving transient signals, it is employed here in a complementary manner to identify dominant frequency content associated with the bubble dynamics. For this case with more pronounced impact characteristics, the plate response exhibits a distinct two-stage process, as illustrated in figure 4(b). The plate deflects to $\Delta x_{\textit{imp}}/l=0.048$ with an acceleration of 0.55 m s $^{-2}$ due to the high impact pressure and long duration of test 5. High-frequency vibrations are observed, accompanied by pressure oscillations. The falling droplets during wave run-down also contribute to the plate deflection, as shown in figure 7(c). The impact pressure along the plate becomes less confined as the presence of air extends the impact zone. Following the maximum deflection $\Delta x_{{hp}}/l=0.101$ , the elastic plate vibrates at a frequency of 0.93 $f_n$ .

The pressure history of high aeration shows the measured highest pressure of 13.8 $\rho gH$ and longest duration of 0.24 $\sqrt {g/h}$ in all tests, see figure 5(h) for test 7. The negative (sub-atmospheric) pressure due to the trapped air expansion (Bullock et al. Reference Bullock, Obhrai, Peregrine and Bredmose2007; Steer et al. Reference Steer, Kimmoun and Dias2021) is not obvious herein because of the negligible air compressibility (Mach number $\mathrm{Ma}\ll 0.3$ ) and air leakage effects as will be discussed in § 3.5. The image from test 7 shows a large air pocket breaking into a cluster of bubbles that move along with the vertical jet upon wave impact, as also observed in figure 7(d) for test 6. The bubbles generated from the fragmentation of the initial air pocket contribute to high-frequency pressure oscillations (210–370 Hz) even during high-aeration impacts. However, the oscillation frequency is lower than in low-aeration impacts due to the increased bubble radii. Note that the excited plate modes cause an elongated air cavity to form beneath the jet tongue following the impact shown in figure 7(d), which may influence the integrity of the structures involved. During the drawdown phase, the increase in kinetic energy (reflected in dynamic pressure) causes the measured pressure to slightly deviate from the hydrostatic assumption. In addition, the spatial extent of the impact pressure along the elastic plate further increases, similar to wave impacts on rigid structures (Bredmose et al. Reference Bredmose, Peregrine and Bullock2009). The plate deflection of test 7 reaches a maximum of $\Delta x_{\textit{imp}}/l=0.095$ with an acceleration of 1.73 m s $^{-2}$ during the impact and $\Delta x_{{hp}}/l=0.145$ under the maximum quasi-hydrostatic force, as shown in figure 5(h). Finally, the elastic plate vibrates at a frequency of 0.94 $f_n$ .

3.3. Effects of flexural rigidity

We have presented pressure characteristics and structural responses for an elastic plate under multimodal breaking wave impacts in §§ 3.1 and 3.2. On the fluid side, how hydroelasticity in turn influences impact pressure is debated. For isolated breaking wave impacts, previous laboratory experiments have shown that structural elasticity can reduce peak pressure (Mai et al. Reference Mai, Mai, Raby and Greaves2020). However, Shen et al. (Reference Shen, Wei, Ji and Zhang2024) reported that elasticity has little influence on the impact load but modifies the impact duration and impulse. Attili, Heller & Triantafyllou (Reference Attili, Heller and Triantafyllou2023) indicated that the flexible plate does not necessarily lead to a smaller wave force compared with the rigid one. For impacts induced by successive wave breaking, Hu & Li (Reference Hu and Li2023) showed that the impact variability caused by hydroelasticity is mainly due to the changes in wave crest shape caused by different residual motions between rigid and flexible structures, e.g. wave reflections and turbulence left from the preceding wave.

Figure 8. Illustration of the (a) 2-D Wagner model (ELP2) and (b) 1-D Bagnold solution (ELP3).

Figure 9 compares the maximum impact and quasi-hydrostatic pressure between rigid and elastic plates. The flexural rigidity $EI$ of the rigid plate is approximately 300 times that of the elastic plate. The pressure coefficient $C_p$ derived from the impact pressure maxima $P_{{max}} = ({1}/{2})\rho C_p U_0^2$ is depicted in figure 9(a). The wave front velocity $U_0$ is estimated from the high-speed video recordings using a cross-correlation algorithm for image analysis (Raffel et al. Reference Raffel, Willert, Scarano, Kähler, Wereley and Kompenhans2018), which tracks the displacement of the wave front between successive frames before impact. The accuracy of this method depends on the frame rate, resolution and visual contrast in the video. Under our experimental conditions, the uncertainty in $U_0$ is estimated to be within $\pm$ 5 %. It is observed that $C_p$ increases with $H/h$ for the slightly breaking regime. Given that the flexural rigidity of the plate had negligible effects on the measured impact pressure, the model proposed by Wagner (Reference Wagner1932) for a 2-D rigid wedge provides an estimate of the pressure coefficient

(3.3) \begin{equation} C_p = \frac {\pi ^2}{4\tan ^2\beta }, \end{equation}

where $\beta$ represents the effective deadrise angle between the wetted and free surfaces, i.e. the angle between the wave front and vertical wall, as illustrated in figure 8(a) and listed in table 1. Here, $C_p$ remains almost unchanged for aerated impacts, where the approaching fluid velocity $U_0$ mainly determines the impact pressure maxima. Using a one-dimensional (1-D) piston compressed air model to simplify the air-pocket phenomenon, the air cushion theory proposed by Bagnold (Reference Bagnold1939) provides an estimate of $C_p$ within $\pm 10\, \%$ error for the aerated impacts under adiabatic conditions

(3.4) \begin{equation} P_{max }=2.7 \frac {L_w}{l_a} \rho U_0^2, \end{equation}

where $L_w$ is the length of the water column compressing the air, and $l_a$ is the mean initial thickness of the air cushion, as displayed in figure 8(b) and listed in table 1. In accordance with Bagnold (Reference Bagnold1939) formulation, $L_w$ is approximated as half the vertical width of the air cushion, based on the rationale that the mass of water contributing to the impact is equivalent to that of a horizontal column with a cross-section equal to the frontal area of the cushion and a length equal to half the vertical cushion width. This estimation provides a practical way to quantify the water mass involved in the pressure generation mechanism. Wagner’s model and Bagnold’s theory give acceptable predictions for $C_p$ in the present study, as shown by the diamond in figure 9(a). Note that $C_p$ is indistinguishable between rigid and elastic plates, suggesting that the elasticity has little effect on the impact pressure maxima, which is also seen in Shen et al. (Reference Shen, Wei, Ji and Zhang2024), who studied the performance of four materials with different Young’s moduli under the flip-through impact.

Figure 9. Breaking-wave-induced (a) pressure coefficient at the impact pressure maxima and (b) height of the water column corresponding to the maximum quasi-hydrostatic pressure on the rigid and elastic plates, where the error bars represent the pressure measurement uncertainty calculated as the standard deviation. The vertical dotted lines denote the boundaries separating the four distinctive impact types.

Figure 10. Breaking-wave-induced (a) temporal variance of impact pressure at PT3 and (b) impulse on the rigid and elastic plates, where the error bars represent the pressure measurement uncertainty calculated as the standard deviation. The vertical dotted lines denote the boundaries separating the four distinctive impact types.

The temporal deviation of impact pressure, $\sigma _P$ , termed the characteristic impact duration, is shown in figure 10(a). The increase in aeration extends the impact duration $\sigma _P$ . Consequently, the integrated impulse $I_{\textit{imp}}$ on the plate increases as the impact varies from slightly breaking to high aeration, as shown in figure 10(b). Likewise, the elasticity has little effect on the impact duration and impulse for distinctive breaking wave impact types. This observation is closely tied to the specific characteristics of the tested cantilever plate, whose relatively high stiffness and short structural response time result in minimal deformation during the brief impact phase, as shown in figure 5. The structural time scale is significantly longer than the fluid loading time scale, thereby reducing the dynamic amplification effects typically associated with hydroelastic impacts. However, it is also observed that the maximum quasi-hydrostatic pressure acting on the elastic plate is higher than that on the rigid plate, as shown by $z_{{hp}}$ in figure 9(b). This is more pronounced for higher $H/h$ because the increased deformation could create a convex shape to enhance the local run-up, which is also seen in a large-scale computational fluid dynamics simulation (Hu & Li Reference Hu and Li2023). This indicates that elasticity modifies the post-impact wave–structure interaction, potentially through radiation and diffraction effects during the longer-duration quasi-hydrostatic phase.

3.4. Predictive law for plate deflections

For hydroelastic impacts, the strains in the clamped-end plate are numerically determined using a beam model (Faltinsen & Timokha Reference Faltinsen and Timokha2009, pp. 533–544). This study aims to develop a simplified predictive law for plate deflections following a breaking wave impact. The analysis relies on non-dimensional quantities derived from Euler–Bernoulli beam theory, assuming small deflections. The unsteady Euler–Bernoulli beam theory with constant flexural rigidity $EI$ reads

(3.5) \begin{equation} \rho _s b \frac {\partial ^2 x_p}{\partial t^2} + c_s \frac {\partial x_p}{\partial t} + \textit{EI} \frac {\partial ^4 x_p}{\partial z^4} = P. \end{equation}

The left-hand side describes the acceleration, damping and internal elastic forces per unit width of the plate, based on a beam model formulation. Here, $x_p$ is the plate deflection relative to $x=0$ m, $c_s$ is the damping coefficient, $E$ is Young’s modulus, $I=b^3/12$ is the moment of inertia and $z$ is the vertical coordinate in the coordinate system (figure 2). The right-hand side describes the external distributed pressure.

The boundary conditions for the fixed bottom end on $z=0$ are

(3.6) \begin{equation} \left . \begin{array}{ll} \displaystyle \frac {\partial x_p}{\partial z}=0 \quad \mbox{on }\quad z=0,\\[10pt] \,\,\,\,\displaystyle x_p = 0 \quad \mbox{on }\quad z=0 ,\end{array}\!\!\right \} \end{equation}

and for the free top on $z=l$ are

(3.7) \begin{equation} \left . \begin{array}{ll} \displaystyle \frac {\partial ^2 x_p}{\partial z^2} = 0 \quad \mbox{on }\quad z=l,\\[10pt] \displaystyle \frac {\partial ^3 x_p}{\partial z^3} = 0 \quad \mbox{on }\quad z=l. \end{array}\!\!\right \} \end{equation}

We begin with a few simplifying assumptions to derive a practical prediction law for the maximum deflection of the elastic plate under breaking wave impact. First, the time derivative of the plate deflection, $\partial x_p / \partial t$ , is considered negligible at the moment the maximum deflection is reached. Given the relative stiffness of the plate, its response time is short, and the inertial effects are minor compared with the elastic restoring force. Accordingly, the maximum deflection can be reasonably approximated using a quasi-static approach, corresponding to the first peak in the transient response rather than a fully static equilibrium deflection. Second, the maximum impact pressure is assumed to be uniformly distributed over the impact zone. Although figure 7 reveals some spatial variation in the impact pressure, particularly under flip-through impact scenarios. The assumption of pressure uniformity within the impact zone, analogous to the uniform impact velocity assumption (Faltinsen & Timokha Reference Faltinsen and Timokha2009, p. 497), although idealised, facilitates the development of a simplified and generalisable relationship for estimating plate deflection. Third, the spatial distribution of the maximum quasi-hydrostatic pressure is assumed to follow a hydrostatic distribution, which is supported by the observed spatio-temporal pressure distributions (see figure 7).

The high impact pressure remains localised and short in duration ( $\sim$ 25 ms), see figures 5 and 7. The spatial extent of the impact zone is defined by the vertical coordinate of the impact point, denoted as $z_{\textit{imp}}$ , which is measured from the still water level. It is identified through inspection of the high-speed video recordings, as shown in figure 7. When water impacts a flat surface at a velocity $U_0$ , the induced plate deflection, $\Delta x_{\textit{imp}}$ , can be estimated by assuming the external load corresponds to the pressure maxima, $P_{{max}} = ({1}/{2})\rho C_p U_0^2$ , within the impact zone

(3.8) \begin{equation} P = \left \{ \!\!\begin{array}{ll} \tfrac {1}{2}\rho C_p U_0^2, & z \leqslant z_{\textit{imp}}, \\[8pt] 0, & z\gt z_{\textit{imp}}, \end{array} \right . \end{equation}

where $C_p$ is the pressure coefficient of the maximum impact pressure. After the impact, a quasi-hydrostatic pressure distribution is observed on the elastic plate, as shown in figure 7. While additional sensors at higher locations could provide further insights, the current set-up effectively captures the quasi-hydrostatic pressure distribution. Moreover, placing sensors higher up could influence the plate’s response, as demonstrated in the impact hammer test. For the estimation of plate deflection under the maximum quasi-hydrostatic pressure, $\Delta x_{{hp}}$ , the external load can be expressed as the hydrostatic pressure $ P = \rho g (z_{{hp}}-z)$ over the fluid–solid interface

(3.9) \begin{equation} P = \left \{ \!\!\begin{array}{ll} \rho g (z_{{hp}}-z), & z\leqslant z_{{hp}}, \\[2pt] 0, & z\gt z_{{hp}}, \end{array} \right . \end{equation}

where $z_{{hp}}$ is the height of the water column obtained by linearly fitting the measured maximum quasi-hydrostatic pressure from all pressure sensors, assuming a hydrostatic pressure distribution.

Substituting the above boundary conditions (3.6)–(3.9) to (3.5) can yield the plate displacements caused by the impact and maximum quasi-hydrostatic pressure, respectively. The maximum plate deflections at the free top $z=l$ induced by the impact and the maximum quasi-hydrostatic pressure are given as

(3.10a) \begin{align}& \Delta x_{\textit{imp}}= \frac {\rho C_p U_0^2z_{\textit{imp}}^3(4l-z_{\textit{imp}})}{48EI},\\[-10pt]\nonumber \end{align}

(3.10b) \begin{align}&\quad\,\,\, \Delta x_{{hp}} = \frac {\rho g z_{{hp}}^4 (5l-z_{{hp}})}{120EI}. \end{align}

Assuming $4l \gg z_{\textit{imp}}$ and $5l \gg z_{{hp}}$ , the plate deflection (3.10a , b ) can be reduced to

(3.11a) \begin{align} \frac {\Delta x_{\textit{imp}}}{l} &\sim \frac {Ca_{\textit{imp}}}{6},\\[-10pt]\nonumber \end{align}

(3.11b) \begin{align} \frac {\Delta x_{{hp}}}{l} &\sim \frac {Ca_{{hp}}}{12}. \end{align}

The deflections of the plate are essentially controlled by the Cauchy number defined in this study

(3.12a) \begin{align}& Ca_{\textit{imp}} = \frac {1}{2}\frac {\rho C_p U_0^2z_{\textit{imp}}^3}{EI},\\[-10pt]\nonumber \end{align}

(3.12b) \begin{align}&\quad\,\, Ca_{{hp}} = \frac {1}{2}\frac {\rho g z_{{hp}}^4}{EI}. \end{align}

The Cauchy number represents the relative magnitude of the fluid force and the restoring force due to structural stiffness. For constant $EI$ , the impact Cauchy number $Ca_{\textit{imp}}$ is linearly related to the product of the impact pressure maxima and the cube of $z_{\textit{imp}}$ , while the quasi-hydrostatic Cauchy number $Ca_{{hp}}$ is linearly correlated with the fourth power of $z_{{hp}}$ . It is seen that both $\Delta x_{\textit{imp}}/l$ and $\Delta x_{{hp}}/l$ are proportional to the Cauchy number. This predictive law will be verified in the following.

Figure 11. The plate deflections induced by (a) impact pressure with a parabolic best-fit curve and (b) maximum quasi-hydrostatic pressure with a linear best-fit line. The vertical dotted lines denote the boundaries separating the four distinctive impact types.

Figure 11 shows the plate deflections versus the initial wave height to water depth ratio $H/h$ under different types of breaking wave impact. As the wave height increases from the unbroken to high-aeration range, the plate deflections induced by impact and maximum quasi-hydrostatic pressures tend to increase parabolically and linearly, respectively. The parabolic increase in $\Delta x_{\textit{imp}}/l$ is because of the combination of increased impact spatial extent $z_{\textit{imp}}$ and pressure maxima $P_{{max}}$ , as seen in figure 12(a,b). $z_{\textit{imp}}$ gradually increases with $H/h$ in the impact regimes. In contrast, $P_{{max}}$ shows strong sensitivity between slightly breaking and low aeration, which is aligned with a recent experimental study of breaking wave impacts on a monopile (Moalemi et al. Reference Moalemi, Bredmose, Kristiansen and Pierella2024). The highest impact pressure, typically expected during low-aeration impact (Hattori et al. Reference Hattori, Arami and Yui1994; Jensen Reference Jensen2019), is instead observed in the high-aeration impact. This discrepancy may be attributed to the limited spatial resolution of the pressure measurements in this experiment, which may have failed to capture the localised peak pressure accurately. The impact Cauchy number $Ca_{\textit{imp}}$ then increases with $H/h$ parabolically, as shown in figure 12(c). The predictive law (3.11a ) for $\Delta x_{\textit{imp}}/l$ presents satisfactory agreement against the measured values shown in figure 13(a). The linear fit with a slope of 0.87 in the dashed line suggests that $\Delta x_{\textit{imp}}/l$ is slightly overpredicted due to the overestimation of the bending moment as we simplified the external load (3.8). Furthermore, the simplification from the plate deflection (3.10a ) to the predictive law (3.11a ) leads to an overestimation by 3 % to 5 % for the conditions studied.

Figure 12. Impact pressure, maximum quasi-hydrostatic pressure and corresponding Cauchy number. (a) Spatial extent of the impact zone captured from the high-speed movie recordings, (b) impact pressure maxima and (c) impact Cauchy number with a parabolic best-fit curve. (d) Height of the water column with a quarter-power function fit, (e) maximum quasi-hydrostatic pressure and ( f) quasi-hydrostatic Cauchy number with a linear best-fit line. The error bars represent the pressure measurement uncertainty calculated as the standard deviation. The vertical dotted lines denote the boundaries separating the four distinctive impact types.

Figure 12(d,e) displays the maximum quasi-hydrostatic pressure $P_{{hp}}$ and corresponding height of the water column $z_{{hp}}$ . Here, $z_{{hp}}$ conforms to a quarter-power law with respect to $H/h$ . Accordingly, the normalised $P_{{hp}}$ gradually decreases with $H/h$ from slightly breaking to high aeration. The quarter-power law suggests the linear relationship between the quasi-hydrostatic Cauchy number $Ca_{{hp}}$ and $H/h$ as well, according to the definition (3.12b ), as presented in figure 12(f). The predictive law (3.11b ) for $\Delta x_{{hp}}/l$ agrees well with the measured values, as shown in figure 13(b). The linear fit has a slope of 1.05, indicating $\Delta x_{{hp}}/l$ is slightly underpredicted due to the abovementioned pressure deviation from the hydrostatic assumption. The dynamic pressure, which is neglected in the simplification of the quasi-hydrostatic pressure, can also contribute to the deflection of the elastic plate.

Figure 13. Plate deflection predictions: (a) $\Delta x_{\textit{imp}}/l$ versus $Ca_{\textit{imp}}/6$ linearly fitted with a zero intercept and (b) $\Delta x_{{hp}}/l$ versus $Ca_{{hp}}/12$ linearly fitted with a zero intercept. The error bars represent the pressure measurement uncertainty calculated as the standard deviation. The dashed line is the linear regression, and the solid line is the 1 : 1 line.

3.5. Error and uncertainty

In all tests, the elastic plate considered upright slightly tilts toward the shore by 0.35 $^\circ$ at the initial position, as seen in figure 2(c). The tiny plate tilt may cause a minor increase in the shear internal force $E I (\partial ^4 x_p/\partial z^4)$ in the plate-normal direction, leading to a slightly underestimated plate displacement, especially for the plate response under a relatively lower impact impulse. Throughout this experimental campaign, the pressure measurements along the plate were conducted at least twice via the single transducer. The standard deviation across different runs for each impact is calculated to represent the pressure measurement uncertainty in §§ 3.1–3.4. The low deviation indicates that isolated impacts are reproducible, unlike successive wave breaking, which is shown to have significantly varying impact behaviour under nominally identical conditions (Bullock et al. Reference Bullock, Obhrai, Peregrine and Bredmose2007; Bredmose et al. Reference Bredmose, Peregrine and Bullock2009). To allow the elastic plate to move freely under the impact of breaking waves, we have preset a narrow gap of approximately 2.5 mm between both sides of the plate and the side walls of the flume. The gap in the 3-D nature of laboratory experiments allows the entrapped air to escape along the sides of the plate (Steer et al. Reference Steer, Kimmoun and Dias2021), which potentially results in less air compression/expansion and faster decay of pressure oscillations compared with vertical wall impacts (Hattori et al. Reference Hattori, Arami and Yui1994; Bullock et al. Reference Bullock, Obhrai, Peregrine and Bredmose2007; Hu & Li Reference Hu and Li2023). Additionally, water flowing through gaps depicted in the images of figure 7 inevitably reduces plate deflection, especially under the maximum quasi-hydrostatic pressure. Finally, the current set-up focuses on a 2-D analysis, with PTs aligned along a single vertical plane, limiting the ability to capture lateral pressure variations and 3-D effects such as irregular wave breaking and oblique impacts. To overcome these limitations, future research should incorporate advanced measurement techniques, such as multi-plane pressure arrays and volumetric particle image velocimetry, to enhance the understanding of 3-D hydroelasticity in breaking wave impacts.