Coastal Walls May Face Unexpectedly Powerful Surges Even When Waves Do Not Break
A wave does not need to collapse into foam to become a serious threat to a coastal structure. New numerical research suggests that smooth, non-breaking surges can generate substantial forces when they encounter sloping sea defenses, with the geometry of the wall playing a larger role than engineers might expect. Using high-resolution Smoothed Particle Hydrodynamics (SPH) simulations, researchers examined how “undular surges”—long, steep-fronted waves followed by a train of smaller waves—interact with inclined coastal walls. The study tested five wall inclinations ranging from −30 to 30 degrees and six different incoming surge amplitudes under still-water conditions. The results reveal a counterintuitive combination: the highest water excursion up a slope can resemble the run-up produced by a solitary wave striking a vertical wall, while the total force imposed on an inclined structure can be greater. The findings could influence the design of seawalls, flood barriers, harbor entrances and other infrastructure exposed to sudden water-level changes.
Undular surges are a special type of long wave that arise when a rapid change in water depth or flow produces a moving step in the free surface. Unlike a breaking wave, whose crest overturns and dissipates energy through turbulence and air entrainment, an undular surge retains a relatively coherent wavefront. Behind that front, dispersive effects can generate a sequence of secondary oscillations. Such waves may occur in rivers as tidal bores, in channels following abrupt releases of water, or in coastal settings affected by rapidly displaced water. Their appearance can be deceptively mild: the surface may remain organized rather than exploding against a wall. Yet the water particles beneath the crest do not move in exactly the same way as they do in a solitary wave, even when the two wave types have similar profiles at the front. That distinction matters because pressure and force depend not only on the shape of the surface, but also on the velocity field—the magnitude and direction of water motion throughout the fluid.
To capture this complex behavior, the researchers used the Smoothed Particle Hydrodynamics method, a mesh-free computational technique designed to follow moving fluids with a free surface. In SPH, the water is represented by a large collection of interacting numerical particles. Each particle carries properties such as density, pressure and velocity, while its motion is calculated from the influence of neighboring particles. Because the method does not rely on a fixed grid, it can naturally represent large deformations of the water surface, run-up along a wall and rapidly changing contact zones that are difficult to track with conventional grid-based methods. The simulations modeled non-breaking undular surges approaching inclined walls in an initially quiescent body of water. By varying both the surge amplitude and the wall angle, the team could isolate how wave intensity and structural geometry altered the resulting run-up, pressure distribution and integrated force.
The simulations showed that undular surges and solitary waves cannot be treated as interchangeable simply because their leading profiles look alike. A solitary wave is a localized, permanent-form disturbance in idealized shallow-water theory, with its nonlinear steepening balanced by dispersion. An undular surge, by contrast, is a bore-like transition whose front is linked to a wavetrain and whose structure evolves as it propagates. In physical terms, the water behind the front contains a different pattern of horizontal and vertical velocities. Those differences influence how momentum is redirected when the surge meets a barrier. The study’s comparison nevertheless produced a striking result for run-up: the maximum height reached by an undular surge on an inclined wall was close to previously reported solitary-wave results for a vertical wall. The largest discrepancy was 7.24 percent, a similarity that could make existing engineering estimates useful—but only for certain aspects of the impact.
Run-up is the maximum vertical or slope-directed rise of water above the undisturbed level. It is one of the most visible consequences of a wave striking a coastal defense, but it is not the same as the mechanical load on the structure. A steep surge can climb high while transferring momentum in a way that differs from a lower-looking but more forceful impact. In the new simulations, the maximum pressure consistently appeared around the still-water level, regardless of the wall’s inclination. This result is important because it identifies a persistent zone of peak loading: the most dangerous pressure was not necessarily concentrated at the highest point reached by the water. The still-water line is where the advancing surge’s dynamic action and the wall’s submerged resistance combine most effectively. Engineers evaluating reinforcement, joints or foundations may therefore need to focus on this elevation even when the principal visual concern is water rushing far up the slope.
The inclination of the wall altered the overall force in a way that could surprise designers who assume that a sloped face always reduces wave loading. According to the simulations, inclined walls experienced greater surge forces than the vertical-wall comparison under the studied conditions. A wall’s angle changes the direction in which the incoming water is deflected, the area over which pressure acts and the duration of contact between the moving fluid and the structure. The load on a wall is obtained by integrating pressure over its wetted surface, so a reduction in pressure at one location does not automatically mean a reduction in total force. A slope can encourage the surge to travel upward while maintaining substantial contact with the structure, allowing momentum to be redistributed over a larger or differently oriented surface. The study used these computed pressure and force responses to develop predictive models for estimating maximum surge pressure and total surge force on inclined walls.
The work also identified two distinct dispersion regimes during the evolution of the undular surges. Dispersion describes the tendency of wave components with different wavelengths to travel at different speeds. In shallow water, this effect competes with nonlinearity, which tends to sharpen and steepen a wave as its faster, higher portions catch up with slower parts. The balance between these processes determines whether a surge maintains one type of wavetrain or evolves into another pattern. The researchers found a transition at a wave steepness of 0.028. Wave steepness is commonly expressed as a characteristic wave height divided by wavelength, so the threshold marks a change in how strongly the wave’s geometry promotes nonlinear steepening relative to dispersive spreading. This boundary does not mean that every real-world surge abruptly changes behavior at one universal value, but it provides a quantitative indicator for classifying the simulated evolution and improving models of long-wave transformation.
The findings arrive as coastal engineers confront hazards that do not fit the familiar picture of a breaking ocean wave. Sudden releases from reservoirs or impounded channels, rapidly advancing flood fronts, landslide-generated disturbances and tidal bores can all create surge-like motions. In many locations, protective structures are deliberately built with inclined faces to manage erosion, reduce reflection or allow a more gradual transition between water and land. Their performance is often assessed using simplified wave types and idealized loading assumptions. The new study suggests that such simplifications may underestimate the force generated by a non-breaking surge, particularly when the structure is sloped. At the same time, the close agreement in run-up between undular surges on inclined walls and solitary waves on vertical walls hints that some established results may be transferable, provided that engineers do not extend that similarity to pressure or force without additional analysis.
The authors emphasize a computational investigation rather than a set of newly generated laboratory measurements; the article reports that no datasets were generated or analyzed during the study. As a result, the predictive models should be understood within the tested range of surge amplitudes, wall inclinations and still-water conditions. Real coastlines add complications, including bottom friction, irregular bathymetry, turbulence, sediment movement, porous or stepped defenses, currents and waves arriving from multiple directions. Even so, the simulations provide a detailed view of quantities that are difficult to observe everywhere at once in an experiment, especially the full velocity field beneath a rapidly advancing front. By linking that hidden flow structure to pressure peaks, run-up and integrated force, the research offers a framework for more targeted physical validation and for refining numerical tools used in coastal-risk assessments. The central warning is simple: a wave that remains unbroken can still deliver a powerful, highly organized push, and a sloping wall may not be the safer shape engineers instinctively expect.
