# Physical Model of Landslide-Generated Impulse Waves: Experimental Investigation of the Wave-Granular Flow Coupling

Abigaël Darvenne, Sylvain Viroulet, Laurent Lacaze

► To cite this version:

Abigaël Darvenne, Sylvain Viroulet, Laurent Lacaze. Physical Model of Landslide-Generated Impulse Waves: Experimental Investigation of the Wave-Granular Flow Coupling. Journal of Geophysical Research. Oceans, 2024, 129 (9), 10.1029/2024JC021145. http://dx.doi.org/10.1029/2024JC021145

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Submitted on 21 Oct 2025

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Distributed under a Creative Commons Attribution 4.0 International License
**Key Points:**

* A detailed analysis of new impulse wave experiments using a continuous granular supply is presented to highlight the wave-granular coupling

* The wave maximum amplitude is shown to evolve non-monotonically with the deposit runout

* Considering wave-granular interaction, a newly defined Froude number allows to accurately predict the wave characteristics

**Correspondence to:**

A. Darvenne,
abigael.darvenne@toulouse-inp.fr

**Citation:**

Darvenne, A., Viroulet, S., & Lacaze, L. (2024). Physical model of landslide-generated impulse waves: Experimental investigation of the wave-granular flow coupling. *Journal of Geophysical Research: Oceans*, 129, e2024JC021145. https://doi.org/10.1029/2024JC021145

Received 25 MAR 2024
Accepted 7 SEP 2024

**Author Contributions:**

**Conceptualization:** Abigaël Darvenne,
Sylvain Viroulet, Laurent Lacaze

**Data curation:** Abigaël Darvenne

**Formal analysis:** Abigaël Darvenne,
Sylvain Viroulet, Laurent Lacaze

**Funding acquisition:** Abigaël Darvenne,
Sylvain Viroulet, Laurent Lacaze

**Investigation:** Abigaël Darvenne,
Sylvain Viroulet, Laurent Lacaze

**Methodology:** Abigaël Darvenne,
Sylvain Viroulet, Laurent Lacaze

**Project administration:** Sylvain Viroulet,
Laurent Lacaze

**Resources:** Abigaël Darvenne,
Sylvain Viroulet, Laurent Lacaze

**Software:** Abigaël Darvenne,
Sylvain Viroulet, Laurent Lacaze

**Supervision:** Sylvain Viroulet,
Laurent Lacaze

**Validation:** Abigaël Darvenne,
Sylvain Viroulet, Laurent Lacaze

**Visualization:** Abigaël Darvenne,
Sylvain Viroulet, Laurent Lacaze

<figure><img src="image_2.png" /></figure>

# Physical Model of Landslide-Generated Impulse Waves: Experimental Investigation of the Wave-Granular Flow Coupling

Abigaël Darvenne¹ ID, Sylvain Viroulet¹ ID, and Laurent Lacaze¹ ID

¹INP Toulouse, UPS, CNRS, Institut de Mécanique des Fluides de Toulouse, Toulouse, France

**Abstract** Large amplitude and unexpected waves are a regular source of natural disasters. Among them, impulse waves generated by landslides can represent a significant threat. Therefore, predicting and measuring the generation of such waves is essential. In this study, the phenomenon is modeled by a 2D-experimental setup using a steady non-uniform granular flow along a slope as a forcing wave generator. The present device provides a continuous supply of grains to avoid finite volume effects, as the part of the landslide actually involved in the wave generation strongly depends on the configuration and is not necessarily available in geophysical events. This system consists of an energy transfer between the granular flow and the wave generation which is characterized by a Froude number. It is found that the latter cannot be defined only based on the dry flow properties to characterize the wave. In particular, the dynamics underwater influence wave generation during a finite time. Accordingly, the present study shows that the wave maximum amplitude is governed by a newly defined Froude number, based on both dry and underwater granular flow properties. Moreover, it is shown that the granular deposit, specifically its runout, can be thought as a proxy of the immersed granular dynamics as long as the impact properties are still considered.

**Plain Language Summary** Landslides can cause huge waves if they impact the surface of an ocean, a lake or a sea. The properties of the landslide as well as the wave formation are complex features. In this study, an attempt to model this phenomenon experimentally at the laboratory scale is made, using a controlled granular flow impacting water in a 2D wave channel. The rapid evolution of this system requires a camera in order to extract the dynamics of both the wave and the granular material, simultaneously. From these measurements, it is shown that the wave and the granular flow are strongly linked at early stages. Moreover, after the maximum wave amplitude is reached, the wave propagates freely and may influence the granular deposit. This study proposes to link these processes and the wave characteristics to a single dimensionless parameter, which is a combination of Froude numbers.

## 1. Introduction

The term landslide refers to the sliding or movement of a mass of soil or land, which can be associated with a wide range of phenomena such as rock falls, steep slope failures, mudflows and debris flows (De Blasio, 2011; Iverson, 1997). The impact of such landslides on a water surface including a lake, a sea or a fjord, can generate significant waves which can be very destructive for the surrounding environment (Crosta et al., 2016; Fuchs & Hager, 2015). Due to the variety of landslides types, a wide range of impulse waves can be generated and requires different approaches to model (Bullard et al., 2023). Impulse waves differ from earthquake tsunamis in several aspects, as their generation mechanism as well as their length scale of propagation are not the same. Yet, waves generated by landslides can be locally more dangerous than the ones generated by submarine earthquakes. In particular, the wave amplitude can be significant upon generation and may subsequently induce a substantial run-up in a nearby coastal area (Fritz et al., 2009; Synolakis et al., 2002). In this context, predicting the wave behavior after impact is of crucial interest. To have a better global understanding of this phenomenon, many studies have been devoted to its modeling, with a large variety of approaches, either experimental, numerical or field data investigations (see Heller and Ruffini (2023) for a detailed review). Yet, Heller and Ruffini (2023) suggest that the physical understanding of the phenomenon remains partial, even though numerous studies have been conducted over the last two decades.

The first experimental model of impulse wave is known to be the experiment deployed by Russell (1844), who considered the fall of a mass on a water surface. This study is also the first to specifically focus on the maximum
Writing – original draft:

Abigaël Darvenne, Sylvain Viroulet,
Laurent Lacaze

amplitude reached by the wave $a_{max}$. It is found to increase with the mass of the block, and thus marks the starting point for the development of empirical relations to predict the maximum wave amplitude related to forcing characteristics. In a second stage, around 1970, experimental studies evolve toward the use of a piston to generate the impulse wave; $a_{max}$ is then found to increase linearly with the piston velocity (Das & Wiegel, 1972; Noda, 1970).

Subsequently, experimental studies focused on considering the granular nature of landslides, and thus began to generate impulse waves using granular material. In particular, the case of impulse waves induced by a pneumatic landslide generator have been one of the first method to model the dispersed nature of the impact (Fritz et al., 2003; Heller & Hager, 2010; Zweifel et al., 2006). A specificity of this configuration is to allow high impact velocity controlled by the pneumatic generator (Fritz & Moser, 2003). Then, many other devices have been used to generate waves with a granular flow whose dynamics is controlled by gravity. For instance, the collapse of a granular column just above a water surface have been largely studied, whether on an inclined plane (Ataie-Ashtiani & Nik-Khah, 2008; McFall & Fritz, 2016; Mohammed & Fritz, 2012; Viroulet et al., 2014) or on a flat bottom (Huang et al., 2020; Robbe-Saule, Morize, Henaff, et al., 2021; Sarlin et al., 2021, 2022). The collapse of a granular mass over an inclined plane but at a finite distance from the water surface, have also been extensively studied. This setup allows to produce a thin and continuous granular flow impacting water at the velocity induced by the flow. In this configuration, several processes and mechanisms were considered: high mobility induced by fluidization of the granular flow (Bougouin et al., 2020; Lipiejko et al., 2023), the influence of positively buoyant particles (Zitti et al., 2016), the influence of the wave asymmetry on its propagation (Bullard et al., 2019) and the role of the effective granular mass generating the wave (Miller et al., 2017).

Whatever the experimental configuration, most studies agreed that among several dimensionless numbers, the Froude number is the main dimensionless number characterizing the process. Yet, this number defined as $Fr = U_c/\sqrt{gH_0}$ still requires attention, and more particularly its characteristic velocity $U_c$. In the case of a piston generator, Noda (1970) referred to the Froude number, whose characteristic velocity was the piston velocity. Following this result, the Froude number based on the slide centroid velocity at impact $U_c \equiv U_{sc}$ was introduced by Fritz et al. (2004) and then used in many studies (Heller & Hager, 2010; Zweifel et al., 2006). Afterward, the slide front velocity at impact $U_c \equiv U_i$ has also been considered to define *Fr* in empirical relations (Bougouin et al., 2020; Lipiejko et al., 2023; Xue et al., 2019). Note that in the case of the pneumatic landslide generator $U_i$ and $U_{sc}$ are strongly linked (Evers et al., 2018). Yet, when using these definitions of *Fr* based on $U_c$ prior impact to predict the wave characteristics, it is always necessary to account for other dimensionless parameters (see Section 3.2). More recently, Robbe-Saule, Morize, Henaff, et al. (2021) showed a high correlation between $a_{max}$ and the Froude number alone when defined from a velocity of the granular flow occurring after impact during the wave generation. Similarly, a local Froude number has been introduced and compared with previous empirical relations (Lipiejko et al., 2023). The latter formulations require knowledge of the dynamics during the wave generation which are associated with a 'wave generation time', all being still largely unknown. For instance, using the impact velocity $U_c = U_i$ suggests an instantaneous wave generation at impact, while other characteristic velocities would suggest a finite time for generation. The crucial question that remains is not whether the generation time is finite, as it is, but more likely whether this finite time depends on the forcing and therefore influences the wave amplitude. To summarize, these recent studies strengthen the idea of focusing on the Froude number alone, refining its definition from physical processes, including time generation among other.

The present study aims to enrich the understanding of impulse wave generation, focusing on Froude number definitions based on physical processes and slide-wave coupled dynamics. For this purpose, the case of a continuous granular inflow is considered. This configuration allows to avoid uncertainties about the role of the granular mass on the wave generation as obtained in case of a finite granular volume system. Moreover, this continuous inflow configuration can be related to geophysical events such as pyroclastic, granular or debris flows, all referred to as landslide for sake of simplicity. The set-up and experimental analysis used to this end are presented in Section 2. In the following, the maximum amplitude of the wave, the time at which this amplitude is reached and the characteristics of the granular deposit are quantified. The purpose of Section 3 is to highlight the strength and the limitation of semi-analytical and empirical relations found in the literature. Moreover, the relevance of *Fr* based on different definitions of $U_c$ is discussed. Afterward, Section 4 is intended to understand the physical mechanisms of the generation process, based on both the wave generation and propagation phases and the dynamics of the immersed granular flow. Finally, results are summarized and discussed in Section 5. In
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Experimental setup and metrology. (a) Wave generation setup and control parameters: *L*, *θ* and *h*₀, whose generation process is recorded using a camera from a side view. (b) Preliminary study of the dry granular flow and fictive position of the shoreline *X*<sub>s</sub>. The dry metrology includes a laser profiler to measure the granular flow thickness *h*(*X*), and a weighing scale to measure the mass flow rate *q*<sub>m</sub>.</figcaption></figure>

particular, a new Fr-based dimensionless number is established to characterize the wave maximum amplitude. Moreover, the relevance to natural hazards is questioned and, accordingly, a valuable correlation between the wave maximum amplitude and the granular flow runout, partly controlled by granular characteristics at impact, is shown.

## 2. Experimental Campaign

### 2.1. Setup and Metrology

A sketch of the experimental setup is shown in Figure 1a. It consists of a 2.20 m-long inclined plane ending in a 4 m-long water tank. Both inclined plane and water tank have a width $W = 20$ cm, resulting in a quasi-2D wave generation system. The tank is filled up with water ($\rho_w = 1000$ kg/m³) to a height $H_0$ and the plane made of smooth PolyVinyl Chloride (PVC) is inclined at angle $\theta$ to the horizontal. A granular reservoir is placed along the inclined plane. The reservoir can be opened along the chute direction using a double gate system. The latter consists of one fixed gate to adjust the granular inflow thickness $h_0$, and a second movable one to open the reservoir and initiate the granular flow. The position of this reservoir is characterized by the length $L$ from the gate to the bottom of the water tank (see Figure 1a). The granular material consists of monodisperse spherical glass beads of density

$\rho_p = 2500$ kg/m³ and mean diameter $d = 1$ mm. In the present study, the total mass added in the reservoir has no influence on the wave as it has been chosen in order to ensure a continuous supply during the generation. The water depth $H_0 = 20$ cm is kept constant, leading to three control parameters: ($h_0$, $\theta$, $L$) which are varied in the range $h_0 = \{2, 4, 6, 8, 10\}$ cm, $\theta = \{20, 22.5, 25, 27.5, 30, 32.5\}^\circ$ and $L = \{1.16, 1.72, 2.20\}$ m. The range of parameters covered here allows to bridge the gap between regimes of capillary driven waves at small Fr and crater-like waves at high Fr. Moreover, changing $L$ allows varying the impact velocity and thickness while keeping a constant granular flow rate for a given ($\theta$, $h_0$). To ensure reproducibility of the experiments, 8 different cases were carried out twice, resulting in a total of $N = 5 \times 6 \times 3 + 8 = 98$ experiments.

Two measurement campaigns were carried out in the study. First, the dry granular flow was characterized independently without the water tank (see Figure 1b). The thickness of the granular flow was measured using a laser profiler (Microepsilon®), while the mass flow rate was measured using a weighing scale (Kern®). Then, the entire process including granular flow and wave generation, was studied in the full set-up. Its description including dry granular impact, underwater granular flow propagation, wave growth and propagation, and the final shape of the granular deposit, as discussed later on, was characterized using a 2000 × 2000-px² Basler® camera, cropped to a 2000 × 500 zone of interest of the camera sensor. The camera is set such as its optical axis is orthogonal to the water tank side wall and records a 1.5 × 0.37 m² spatial field along the horizontal and vertical directions respectively, leading to a resolution of 13 pixels per centimeter (see Figure 1a). The acquisition frequency was 100 fps. The field of view includes part of the dry slope above the shoreline and most of the water tank as sketched in Figure 1a.

### 2.2. Experimental Analysis

#### 2.2.1. Dry Granular Steady Flow

The granular avalanche is first considered disregarding the interaction with water, in order to extract dry-flow characteristics along the X-axis which is parallel to the slope, with $X = 0$ corresponding to the gate. It is more particularly characterized at the fictive position of the shoreline $X_s$ (see Figures 1a and 1b), which depends on $L$ and $\theta$ as $X_s = L - H_0/\sin(\theta)$. Once the granular material is released from the reservoir, it accelerates and eventually reaches a non-uniform steady state along the X-axis. This steady flow can be characterized by its thickness $h(X)$ and granular mass flow rate $q_m$.

The dry granular flow thickness perpendicular to the slope $h(X)$ is obtained using the laser profiler (see Figure 1b). This device allows to extract $h(X)$ over a X length of 12 cm. The experiment is thus repeated four times with the
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Experimental measurements of the dry granular flow for $\theta = 32.5^{\circ}$ and different $h_0$. (a) Bullet points are $h(X)$ measurements while the lines are the extrapolation of $h$ for all $X$, with $X_s$ the fictive position of the shoreline and $h_s$ the extrapolated granular impact thickness. (b) Bullet points are $m(t)$ measurements while the lines are the linear fits whose slopes correspond to the granular mass flow rates $q_m$.</figcaption></figure>

laser device positioned at different distances from the gate (see Figure 1b). Results from such measurement are shown in Figure 2a for $\theta = 32.5^{\circ}$ and different $h_0$. In almost all cases the granular flow thickness decreases with $X$, suggesting an increase of velocity along the slope. This measurement is then used to estimate the flow thickness $h_s$ as an interpolation of $h(X)$ at the position $X_s$, which will be referred to as the granular impact thickness in the following.

The associated velocity at the shoreline $U_s$ is also estimated. For this purpose, the granular mass flow rate $q_m$ is measured using a weighing scale at the outlet of the inclined plane (see Figure 1b). The obtained mass $m$ is plotted as a function of time $t$ in Figure 2b for $\theta = 32.5^{\circ}$ and different $h_0$. The mass $m$ is shown to evolve linearly with time $t$ corresponding to a steady flow of constant mass flow rate $q_m$. The latter is found to increase with $h_0$ and $\theta$. The granular velocity $U_s$ is then defined as $U_s = q_m/(h_s\rho_p W\phi_0)$ where $\rho_p = 2500 \text{ kg/m}^3$ is the grains density, $\phi_0 = 0.64$ is the maximal volume fraction for random close packing of spherical particles and $h_s$ the granular impact thickness defined previously.

### 2.2.2. Wave Generation Process

Wave generation induced by a granular flow is a complex mechanism. Depending on the set of control parameters ($h_0$, $\theta$, $L$), the wave generated by the impact of the granular flow can be very different, from barely observable waves to more energetic ones.

A typical energetic case, as shown in Figure 3, is characterized by a wave amplitude of several centimeters, a significant expansion of the granular flow when immersed and a large entrainment of air. In the following, the reference time $t = 0$ s is defined as the time of impact. For $t > 0$ s, the immersed part of the granular flow continues to propagate along the slope. During this phase (from $t = 0.12$ to $t = 0.36$ s in the example of Figure 3), a bulbous head type front develops, and the wave amplitude increases. The wave crest seems to follow the granular front. When reaching the bottom of the tank ($t = 0.48$ s), the granular material slows down and spreads horizontally ($t = 0.60$ s). Eventually, the wave propagates freely along the flume.

In order to characterize such a complex flow, image analysis have to be simplified into a finite number of measurable quantities to describe the system. These quantities are shown in the sketch of Figure 4, where the x-axis is defined horizontally with its origin $x = 0$ at the shoreline at rest and oriented along the wave propagation direction. The water depth along the x-axis is denoted $H(x)$ and is defined as $H(x) = H_0$ for $x > H_0/\tan\theta$, and $H(x) = x \tan\theta$ for $x < H_0/\tan\theta$. Recall that the wave generation system used here is assumed as quasi-2D.

Images are processed using an in-house Python routine to extract the free surface elevation $\eta(x,t)$ and the vertical thickness of the immersed granular flow $s(x,t)$ (see Figure 4a). From these two measurements, all the other wave and granular flow characteristics depicted in Figure 4a can be derived. In particular, the wave crest is defined as the maximum value of $\eta(x,t)$ at each time $t$. This crest is characterized by its amplitude $a(t)$, its position $x_a(t)$ and its velocity $u_a(t) = dx_a/dt$. The wavelength is defined as $\lambda(t)$ where $\lambda(t)/2$ is the width at mid-height of the wave. On the other hand, the granular flow front is defined as $s(x,t) = 0$ at each $t$ and is characterized by its position $x_f(t)$.
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Time evolution of an experiment for $\theta = 32.5^{\circ}$, $h_0 = 8$ cm and $L = 2.20$ m. Oblique pictures taken using a Nikon® D850 camera allowing a qualitative capture of the 3D process.</figcaption></figure>

and its velocity $u_f(t) = dx_f/dt$. The position of the center of mass of the immersed granular flow is denoted $x_g(t)$. The dry granular flow front velocity prior impact is denoted $U_i$ (Figure 4b). Finally, the granular deposit (measured at $t_{end}$, after the wave left) is characterized by $x_f(t_{end})$ and $x_g(t_{end})$ (Figure 4c).

Typical examples of $\eta(x,t)$ and $s(x,t)$ are shown in Figure 5 for $\theta = 32.5^{\circ}$, $h_0 = 8$ cm and $L = 2.20$ m. This space-time representation of the generation process allows to highlight a clear correlation between the wave crest position $x_a$ and the granular front $x_f$ at early stage. Later on, the wave continues to propagate while the granular flow slows down until it stops. Typical velocities $U_a$ of the wave crest and $U_f$ of the granular front are defined as sketched in Figure 5. The former mostly corresponds to the wave velocity during the second stage of the dynamics while the latter corresponds to the granular front velocity during early stage, both being almost constant during their respective phase.

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Sketch of the experiment with notations of all measured quantities. (a) Is during the propagation of the granular flow under water, (b) is before impact and (c) is when all the grains have reached the bottom. Red and blue notations are for granular and water description, respectively.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Space and time evolution of two measured quantities: the water free surface elevation $\eta(x, t)$ (a) and granular flow thickness $s(x, t)$ (b), for one experiment at $\theta = 32.5^{\circ}$, $h_0 = 8$ cm and $L = 2.20$ m. The dotted horizontal line represents the intersection of the slope with the horizontal bottom of the tank. Typical velocities $U_a$ of the wave crest and $U_f$ of the granular front are represented in red (a) and black (b), respectively.</figcaption></figure>

# 3. Descriptive Model: Wave Amplitude and Granular Deposit

## 3.1. How to Describe the Wave-Deposit System

The maximum amplitude reached by the wave is a universal quantity used to estimate the hazard of a wave generated by the impact of a landslide (Heller & Ruffini, 2023). It cannot be used to predict the propagation in its entirety, but it remains a main indicator of the phenomenon. In the majority of aforementioned studies dealing with empirical prediction of maximum wave amplitude, the water height is measured using wave probes. Thus, the maximum amplitude reported in these studies corresponds to the maximum free surface elevation, which is not always the leading wave and is not necessarily obtained at the first probe. In the present case, the maximum is always associated with the leading wave and located in the field of view of the camera. This maximum can therefore be written as $a_{max} = \max(a(t))$ allowing to define the specific time $t_{max}$ at which this maximum is reached, that is, $a_{max} = a(t_{max})$ (see Appendix A for the detailed calculation of $t_{max}$). Such approach clearly refines the spatial resolution, thus improving the accuracy of the obtained maximum amplitude. This amplitude $a_{max}$ is, however, confronted to a so-called probe amplitude $a_{probe}$, which is the maximum wave amplitude at a given abscissa $x = 80$ cm. As $a_{max}$ and $a_{probe}$ are found to be similar (see Figure 6), relations obtained in the literature can directly be applied to our case using $a_{max}$.

At geophysical scale, the shape of the landslide deposit after impact is often one of the few, if not the only, measurable features of the landslide (Brunet et al., 2017; Poulain et al., 2023; Robbe-Saule, Morize, Bertho,

et al., 2021). Actually, considering moderate slide impact velocity, that is, small Froude regime, this deposit can provide a wide range of information. On the one hand, the immersed volume can approximate the quantity of water displaced, which can be a first estimate of the volume of the wave. On the other hand, the position of the front of the immersed deposit and its center of mass can be used to understand the dynamics of the landslide as it enters the water. In the present experimental set-up, the immersed part of the deposit is extracted. Figure 7 shows an example of this deposit for a set of control parameters ($h_0, \theta, L$). This deposit is characterized by its front and center of mass positions $x_f(t_{end})$ and $x_g(t_{end})$ (dashed line and cross, respectively, in Figure 7).

## 3.2. Predictive Models of Maximum Wave Amplitude

Several models have been proposed in the literature to link the properties of the granular flow to the maximum wave amplitude. Following the work of Huber and Hager (1997), an empirical relation based on the granular flow characteristics was established (Fritz et al., 2004). Two dimensionless control parameters were then defined: the Froude number $Fr = U_c/\sqrt{gH_0}$ and the relative slide thickness $S = h_c/H_0$ where $U_c$ and $h_c$ are respectively a

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Maximum amplitude $a_{max}$ compared with the amplitude $a_{probe}$ that would have been measured with a wave probe placed at 80 cm from the generation abscissa. The black dashed line corresponds to $a_{max} = a_{probe}$.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 7. Images of the granular deposit (delimited by the red line) in the plane of parameters $h_0$ and $\theta$, for $L = 1.16$ m. The red area is the immersed part of the deposit, the blue line is the slope and flume bottom, the green dashed line corresponds to $x_f(t_{end})$ and the white cross is $x_g(t_{end})$.</figcaption></figure>

characteristic velocity and a characteristic thickness of the granular flow, and $H_0$ is the water depth. In Fritz et al. (2004), these characteristic quantities are defined as $h_c \equiv h_{ms}$ the maximum thickness of the granular flow at impact and $U_c \equiv U_{sc}$ the centroid velocity of the finite volume slide at impact. They established the following empirical relation between maximum wave amplitude and these dimensionless numbers

$$a_{\max}/H_0 = 0.25Fr^{1.4}S^{0.8}. \qquad (1)$$

Afterward, Zweifel (2004) introduced a new dimensionless control parameter: the relative slide mass $M = m_c/(\rho_w WH_0^2)$ also defined as $M = V \times D$, with $V = V_c/(WH_0^2)$ the relative slide volume and $D = \rho_s/\rho_w$ the relative density. Here, $m_c$ is a characteristic mass of the granular slide, $V_c$ its volume and $\rho_s = \phi\rho_p$ its bulk density with $\phi$ the slide volume fraction. In their specific configuration, Zweifel (2004) uses $m_c \equiv m_s$ the total mass of the finite volume slide. Accordingly, a semi-analytical model was developed by Zweifel et al. (2006), by considering a balance between the slide and water fluxes, and assuming a proportionality between the wave amplitude and its celerity. This allows to establish a proportionality between the maximum amplitude $a_{\max}$ and a so-called densimetric Froude number defined as $F_d = Fr(DS)^{1/2}$. Fitting their experimental data led to the following relationship

$$a_{\max}/H_0 = 1/3F_d. \qquad (2)$$

It is important to note that the above mentioned models requires to evaluate expressions for slide and wave momenta. Other approaches to model this term have been proposed. For instance, Mulligan and Take (2017) established two models, one based on a hydrostatic approximation for the fluid phase, leading to

$$a_{\max}/H_0 = \sqrt{1 + DSFr^2 \cos^2 \theta - 1}, \qquad (3)$$

and another controlled by wave inertia when the time scales of generation and of wave crest displacement are similar, leading to

$$a_{\max}/H_0 = DSFr \cos \theta. \qquad (4)$$

Finally, Heller and Hager (2010) introduced a new dimensionless number, called the impulse product parameter $P = FrS^{1/2}M^{1/4} \cos^{1/2}(6\theta/7)$, related to the streamwise slide momentum flux. This parameter is then found empirically to be related to the maximum wave amplitude as

$$a_{\max}/H_0 = 4/9P^{4/5}. \qquad (5)$$

All the previous models, either semi-analytical or empirical, give a relationship between the wave amplitude and the granular characteristics for which the coefficient of proportionality has to be extracted from experimental fit.
These models are applied to the present set of experimental data. Yet, in our configuration, total slide mass $m_s$, maximum slide thickness $h_{ms}$ and centroid slide velocity $U_{sc}$ are not relevant. Instead, the characteristics height $h_c$ and velocity $U_c$ of the dimensionless numbers are chosen as $h_s$ and $U_s$ (see Section 2.2.1) and considered as typical thickness and velocity of a continuous granular flow. Likewise, the characteristic mass $m_c$ was chosen as $m_c \equiv q_m t_c$ where $q_m$ is the dry granular mass flow rate (see Section 2.2.1) and $t_c$ a characteristic time. This characteristic time is not straightforward to estimate and will therefore be discussed in Section 3.3. For sake of simplicity, it is first assumed to be $t_c = 1$ s which is a typical order of magnitude found in experiments (see for instance (Mulligan & Take, 2017)). Results of tested models are shown in Figure 8. In particular, Figure 8a shows the dependence of the maximum amplitude with Fr, while frames (b, c, d, e, f) correspond to models (1), (2), (5), (3), (4), respectively. Gray solid lines correspond to coefficient of proportionalities reported in the literature while black dashed lines show the best fits of the present data.

Using $h_s$ and $U_s$ as characteristic length and velocity, the Froude number $Fr$ alone appears to be a poor parameter to predict $a_{max}$ (Figure 8a), while a better collapse of the experimental data is obtained for the five other models (Figures 8b–8f). In particular, the relative thickness $S$ appears essential for maximum amplitude prediction (see Figure 8b). The four other models provide convincing collapse of the data (see frames (c, d, e, f)). Note that the obtained fit are within a determination coefficient $R^2 > 0.8$ for cases (c, d, f), while case (e) is slightly lower, $R^2 \approx 0.6$ (see Appendix B for the definition of $R^2$). The latter observation is consistent with the regime of moderate impact velocity considered in the present study. Note that other characteristic lengths and velocities to define $Fr$, $S$ and $M$ have been tested, but essentially lead to the same conclusion.

### 3.3. Revisiting Froude Number Models

Behind the semi-analytical models presented in the previous section, lies physical processes yet to be understood. In particular, these models are mostly based on momentum balance between the slide and the water wave. However, such balance is strongly affected by several complex processes during the generation, as for instance, air entrainment, fluid-grains interaction and slide-wave coupling among others. Due to such complexity, slide momentum and its transfer to wave remain difficult to predict and characterize. It is thus essential to tackle the problem by considering measured properties of the granular flow during generation, instead of predicted ones as $(U_s, h_s, q_m)$, to gain in physical understanding of wave generation. However, for sake of simplicity, one still assumes the generation of the wave to be associated with the transfer of kinetic energy from the granular media to the water, which is included in the Froude number. Accordingly, even if previous results show a poor correlation with $Fr$, it only means that the relevant characteristic velocity $U_c$ to be considered shall be refined in the Froude number definition $U_c/\sqrt{gH_0}$. Such idea follows many studies in the past which have focused on its definition, as explained in Section 1. Here, three characteristic velocities have been chosen to represent different aspects of the process: the velocity of the granular flow front at impact $U_c \equiv U_i$, the velocity of the granular flow front underwater $U_c \equiv U_f$, and the flow rate based velocity $U_c \equiv q_{eff}/(WH_0)$. Note that the latter velocity represents the one of the volume $V_{eff}(t)$ (grains + air) entering the water if it were distributed over the entire section of the water channel $H_0W$. This total volume is calculated as $V_{eff}(t) = W \int_0^\infty \eta(x,t) dx$ and the volume flow rate is then deduced as $q_{eff} = dV_{eff}/dt$. This velocity corresponds to the equivalent velocity that a piston-like forcing (Noda, 1970) would have to displace the same volume of water. It shall also be noted that $q_{eff}/(WH_0)$ and $U_f$ are horizontal velocities, while $U_i$ is the velocity along the slope. The latter is defined following previous studies for comparison. Yet, as wave generation probably corresponds to a transfer of horizontal kinetic energy, other velocities are horizontally defined. Figure 9 highlights that these three models characterize $a_{max}$ fairly well, and that the Froude number is a relevant dimensionless number to characterize the process at first order.

While $U_f$ and $q_{eff}/(WH_0)$ provide information on the propagation of the granular flow in water, $U_i$ only characterizes the dry granular flow prior impact. Figure 9 shows that $U_f$ and $q_{eff}/(WH_0)$ allow a better collapse of the maximum amplitude $a_{max}$ data than using $U_i$. This suggests that the maximum amplitude reached by the wave is strongly controlled by underwater flow propagation, in this range of parameters. As this propagation is intrinsically dependent on the dry flow characteristics at impact, this explains why empirical equations using Froude number based on dry flow conditions also require complex dependency with other parameters as $S$ and $M$ for instance. Finally, note that even if both definitions $U_c \equiv U_f$ and $U_c \equiv q_{eff}/(WH_0)$ give a convincing relationship with $a_{max}/H_0$, the later one allows to provide a simple linear model as
<figure><img src="image_10.png" /><figcaption>Figure 8. Relative maximum wave amplitude $a_{max}/H_0$ as a function of relations of the granular flow dimensionless numbers from the literature defined here as $Fr \equiv U_s/\sqrt{gH_0}$, $S \equiv h_s/H_0$, $D = \rho_s/\rho_w$, $M \equiv q_m t_c/(\rho_w H_0)$, $\theta$. Panel colors correspond to classification of the relation type, as Fr-model comparison in white panel, empirical in red panel and semi-analytical in blue panel. For each, the black dashed line is the best linear fit and $R^2$ is the associated determination coefficient, while the solid gray line is the relation obtained in the literature. (a) $a_{max}/H_0$ as a function of $Fr$; (b) $a_{max}/H_0$ as a function of $Fr^{1.4}S^{0.8}$, empirical model from Fritz et al. (2004); (c) $a_{max}/H_0$ as a function of $F_d$, semi-analytical model from Zweifel et al. (2006), (d) $a_{max}/H_0$ as a function of $P^{4/5}$, semi-analytical model from Heller and Hager (2010), and (e, f) $a_{max}/H_0$ as a function of $\sqrt{1+DSFr^2 \cos^2 \theta} - 1$ and $DSFr \cos \theta$, respectively, both semi-analytical models are from Mulligan and Take (2017). Legend for colors and markers is in Figure 6.</figcaption></figure>

$$ \frac{a_{max}}{H_0} \approx 2 \frac{q_{eff}}{WH_0\sqrt{gH_0}}. \qquad (6) $$

(see dashed line in Figure 9c), with a determination coefficient $R^2 = 0.844$.
<figure><img src="image_11.png" /><figcaption>Figure 9. Relative maximum wave amplitude $a_{max}/H_0$ as a function of different Froude numbers, where the black dashed lines correspond to the best linear fits, with their corresponding prefactor and determination coefficient $R^2$. (a) Is for $U_c \equiv U_i$, (b) is for $U_c \equiv U_f$ and (c) is for $U_c \equiv q_{eff}/(WH_0)$. Legend for colors and markers is in Figure 6.</figcaption></figure>

Given that underwater dynamics of the granular flow, and not only the instantaneous dry impact, influences the maximum wave amplitude, the interaction process during the finite time of wave generation has thus to be considered. Then, to better understand the maximum amplitude reached by the wave $a_{max}$, it is necessary to consider $t_{max}$, the time at which the wave reaches this amplitude. This time characterizes the wave generation process during which the amplitude of the wave grows. In most of the experiments carried out, the amplitude reaches a plateau on a relatively short time scale prior decreasing, all while the wave is propagating. Here, $t_1$ and $t_2$ are defined as the beginning and end of the plateau reached by the amplitude (see Appendix A for the detailed calculation of $t_1$ and $t_2$). In Figure 10, $t_{max} \sqrt{g/H_0}$ is plotted as a function of the previously defined Froude numbers, with errorbars accounting for the plateau interval $[t_1, t_2]$.

Figure 10 shows that the time scale of the plateau mentioned previously (errorbars) is not negligible compared to the range of variation of $t_{max}$. However, $t_{max}$ is observed to mostly increase with the Froude number, whatever its specific characteristic velocity $U_c$. This is clearly obtained when considering $U_c \equiv U_i$ and $U_c \equiv U_f$ (see Figures 10a and 10b). Unlike the maximum amplitude correlation, $U_c \equiv q_{eff}/(WH_0)$ does not allow a convincing collapse of $t_{max}$ data. Note that surprisingly, the best relationship is obtained with $U_c \equiv U_i$, suggesting that the generation time $t_{max}$ is mainly dependent on the velocity of the granular flow front prior impact $U_i$ as (see solid line in (a))

$$t_{max} \approx 3 \frac{U_i}{g} \qquad (7)$$

### 3.4. Deposit Description

As mentioned previously, the wave characteristics ($a_{max}$ and $t_{max}$) are strongly related to the granular flow dynamics (immersed or at impact). As the deposit is a signature of the flow evolution, it is considered in the following. The deposit is characterized by two typical lengths: the horizontal positions of its front and center of mass ($x_f(t_{end})$, $x_g(t_{end})$) respectively. Both are shown in Figure 11, normalized by the water depth $H_0$, as a function of a Froude number based on the different $U_c$ discussed previously. As $x_f(t_{end}) > x_g(t_{end})$, their position

<figure><img src="image_12.png" /><figcaption>Figure 10. Relative time of maximum amplitude $t_{max}/\sqrt{H_0/g}$ as a function of different Froude numbers, where the black dashed lines correspond to the best linear fits, with their corresponding prefactor and determination coefficient $R^2$. (a) Is for $U_c \equiv U_i$, and the solid gray line corresponds to $t_{max} = 3U_i/g$, (b) is for $U_c \equiv U_f$ and (c) is for $U_c \equiv q_{eff}/(WH_0)$. Legend for colors and markers is in Figure 6, with errorbars accounting for the plateau interval $[t_1, t_2]$.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Abscissa of the front $x_f(t_{end})$ and center of mass $x_g(t_{end})$ of the final immersed granular deposit, as a function of the three different Froude numbers. Legend for colors and markers is in Figure 6.</figcaption></figure>

on these figures are clearly distinguished. It should be noted that $x_g(t_{end})$ is roughly invariant with the Froude number, while $x_f(t_{end})$ is not, as was already visible in Figure 7 in the control parameter space. In all cases, $x_f(t_{end})$ mostly increases with the Froude number, at least above a critical value. Yet, a better collapse of the data is obtained for $U_c \equiv U_i$, as observed for $t_{max}$.

Moreover, Figure 11a highlights a clear transition, represented by the yellow area, which occurs for $U_i/\sqrt{gH_0} \sim Fr_c = 1.25$, suggesting two distinct regimes. The first regime corresponds to $U_i/\sqrt{gH_0} < Fr_c$, that is, impact velocity is small compared to the characteristic wave velocity in the channel. In this case, as the wave travels mostly faster than the granular medium, it is observed that it does not influence its final deposit. In the range of parameters considered here, this leads to a nearly constant $x_f(t_{end})$ in this regime. On the other hand, in the second regime, $U_i/\sqrt{gH_0} > Fr_c$, the granular front propagates faster than a typical water wave. In this case, the granular spreading is enhanced with increasing $U_i/\sqrt{gH_0}$. This suggests that the granular flow is in return influenced by the generated wave. It is therefore suggested that the range of variation of $x_f(t_{end})$ is governed by an interaction between the granular front and the surrounding fluid flow induced by the wave. Such local process, not observed on $x_g(t_{end})$, influences the shape of the deposit close to the front. This can be seen in Figure 7 where the granular front deposit becomes thinner from case to case.

Finally, two regimes of impulse wave generation are observed, depending on the ratio of the granular flow front impact velocity and the celerity of shallow water waves. These regimes have a significant influence on the deposit spreading. This suggests a strong two-way coupling dynamics of the system, at least for $U_i/\sqrt{gH_0} > Fr_c$. In this context, the dynamics of generation becomes crucial in predicting the wave structure. In the following, the dynamics of wave generation will thus be examined.

# 4. Dynamics: A Coupled Generation Process

In order to describe the wave generation, the temporal evolution of the dimensionless wave amplitude $a(t/t_{max})/a_{max}$ is shown in Figure 12a. For $t/t_{max} < 1$, the wave grows during a so-called generation phase while for $t/t_{max} > 1$ its amplitude decreases during the propagation phase. Figure 12b focusses on the generation phase, which will be discussed in Section 4.2. The fairly good collapse of all data using this dimensionless representation ($a_{max}$, $t_{max}$) suggests that the evolution of the wave amplitude during both phases is similar whatever the control parameters. Note that the data scattering observed in this figure is partly due to small values of $a_{max}$ and $t_{max}$ at small $U_i/\sqrt{gH_0}$, increasing measurement errors. In order to link this evolution of the wave amplitude to the predictive model described in the previous section, the propagation phase and the generation phase, respectively, are now investigated.

## 4.1. Propagation Phase

In the propagation phase, $t/t_{max} > 1$, Figure 12a shows an amplitude decay rate close to $(t/t_{max})^{-1/3}$ (see dashed line). This corresponds to the theoretical 2D free propagation of a wave induced by a localized pulse perturbation, in the weakly dispersive and linear shallow water approximation (Pelinovsky et al., 2000; Viroulet et al., 2013; Whitham, 2011). Yet, several studies have shown that the amplitude decay rate in the propagation phase strongly depends on the type of wave such as weakly non-linear oscillatory wave, non-linear transition wave, solitary like wave or dissipative transient bore (Fritz et al., 2004; Heller & Hager, 2011; Zweifel et al., 2006). Here, it mostly
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Amplitude *a* normalized by the maximum amplitude *a*<sub>max</sub> as a function of time *t*, (a) is for the dimensionless time *t*/*t*<sub>max</sub> and the dashed line represents (*t*/*t*<sub>max</sub>)<sup>-1/3</sup>. (b) Is *a*/*a*<sub>max</sub> = *f*(*t*/*t*<sub>1</sub>) in logspace and only during the generation phase.</figcaption></figure>

follows the linear and weakly dispersive solution $t^{-1/3}$. This can be explained by the limited wave-type range obtained in the present configuration. Nevertheless, this amplitude trend suggests that the wave evolves freely in this regime, that is, is independent of the granular dynamics.

In order to verify this suggestion, the temporal evolution of the dimension-
less wave velocity $u_a/u_\phi$ is plotted in Figure 13b, with
$u_\phi = \sqrt{g\lambda/(2\pi)}\tanh((2\pi H(x)/\lambda)}$ the theoretical wave velocity in the linear and
finite depth approximation. As observed here, for $t/t_{max} > 1$, all data tend
toward $u_a = u_\phi$. As this is clearly not the case during the generation phase,
$t/t_{max} < 1$, this strengthens the idea of a free wave in the propagation phase.

Even if free propagation of the wave is obtained at long times, the link with the granular forcing remains unclear. The evolution of the wave velocity with respect to the granular front velocity can be considered as a signature of this link. Therefore, the time evolution of the dimensionless wave velocity $u_a/\sqrt{gH_0}$ as a function of the immersed granular front velocity $u_f/\sqrt{gH_0}$ is shown in Figure 13b. At late stage, that is, in the propagation phase, the wave velocity tends toward the long wave approximation velocity $\sqrt{gH_0}$ (solid line) as expected. However, at early times (blue symbols), the wave velocity is found to be correlated to the granular front velocity, $u_a \approx u_f$ (dashed line).

More interestingly, the wave velocity is still found to be directly correlated to the granular flow velocity at t_max,
with u_a ≈ 1.75u_f (dashdotted line). Altogether, this figure shows a relationship u_a ~ αu_f with α ∈ [1, 1.75]
for t ∈ [0, t_max].

Finally, this suggests the relevance of *t*<sub>max</sub> as the transition time from the two-way coupling generation phase toward the free-wave propagation phase. This generation phase is thus discussed in the next section.

4.2. Generation Phase

The wave amplitude growth phase, for $t/t_{\text{max}} < 1$, is the most important stage to understand the mechanical process at the origin of the generated wave, and thus to allow the proposition of expected predictive models. This phase is shown in Figure 12b, where $t_1$ has been used here instead of $t_{\text{max}}$ to disregard the finite time transition from generation to propagation phases. Dashed lines correspond to different power law of the dimensionless amplitude with the dimensionless time. The power law exponents are found in the range [0.25, 1], mostly close to 0.5 as obtained from numerical simulation in Viroulet et al. (2013).

In order to link the amplitude growth rate with the immersed granular flow, the evolution of their volumes,
$V_{\text{wave}} = Wa\lambda/2$ and $V_{\text{eff}}$ respectively, is now considered. Figure 14 shows $V_{\text{wave}}/V_{\text{wave}}(t_{\text{max}})$ (a) and
$V_{\text{eff}}/V_{\text{eff}}(t_{\text{max}})$ (b) as a function of the dimensionless time $t/t_{\text{max}}$. In Figure 14a power law exponents are found in

<figure><img src="image_15.png" /><figcaption>**Figure 13.** Temporal evolution of the wave crest velocity $u_a$. (a) Devided by the theoretical phase velocity $u_\phi$ over normalized time $t/t_{\text{max}}$ and (b) Time evolution (represented by the colorbar) of the wave crest velocity $u_a$ in function of the granular flow front velocity underwater $u_f$, both normalized by $\sqrt{gH_0}$. Solid line is $u_a = \sqrt{gH_0}$, dashdotted line is $u_a = 1.75u_f$ and dashed line is $u_a = u_f$.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Figure 14.** Temporal evolution of the volumes, (a) time evolution of $V_{wave}$ normalized by $V_{wave}(t_{max})$ and (b) time evolution of the effective volume of the immersed flow $V_{eff}$ normalized by $V_{eff}(t_{max})$. The dashed lines correspond to power law evolution with time.</figcaption></figure>

the range [1, 2], centered around 1.5. Note that this result can appear surprising: if the granular flow rate was constant, the volume entering water should evolve as $t$ and so would the wave volume. Hence, the volume effectively entering water is plotted in Figure 14b. Interestingly, the range of power law exponents is the same as the ones for the wave volume (see Figures 14a and 14b). The time evolution of the wave volume is thus directly linked to the effective volume of the immersed granular flow in the range of parameter covered here. Then, the power law exponent between 1 and 2 must be explained by more complex dynamics of the granular flow when crossing the free surface. As already mentioned, a steady granular flow would lead to 1. On the other hand, if the granular flow were stopped at the interface prior entering water, then the flow would have an accelerated underwater dynamics. The time evolution of the position for a constant acceleration of the front from rest is $t^2$, which would lead, for a constant flow thickness, to the power law exponent 2 for the volume. The complex flow considered in the present study lies in between these idealized dynamics (see Figures 14a and 14b).

### 4.3. Consequences on the Wave Structure

The wave evolution has been shown to be directly linked to the effective volume of the granular flow entering the water. Following this result, the maximum wave amplitude shall be related to the effective volume at $t_{max}$. Note that similar approaches have been used in the literature based on the final deposit. In particular, the maximum amplitude has been shown to be a power law of the immersed deposit volume by Robbe-Saule, Morize, Bertho, et al. (2021). Yet, it shall be important to note that the entire volume of the deposit has not been necessarily involved in the wave generation (Mulligan & Take, 2017; Rauter et al., 2021). Therefore, the immersed volume at $t = t_{max}$ is used here instead.

First, the relationship between the effective volume $V_{eff}$ at the end of the generation phase $t_{max}$ and the wave characteristics is discussed. In particular, Figures 15a and 16b show the volume $V_{wave}(t_{max})$ of the wave and the maximum amplitude $a_{max}$ as a function of $V_{eff}(t_{max})$ in their dimensionless forms, respectively. Figure 15a shows a linear relationship as $V_{wave}(t_{max}) \approx V_{eff}(t_{max})$. The obtained power law is different from the one obtained with final deposit in the literature. This suggests that the dynamics during the generation phase are crucial and, consequently, the final deposit is not necessarily a signature of wave characteristics. Moreover, it is interesting to note the linear relationship (gray solid line) obtained between the volumes, highlighting that all the volume entering water is transmitted to the first wave. As a consequence the volume of the wave train $V_{train}$ is zero as the troughs compensates for the following crests (see insert of Figure 15a).

Finally, if one assumes that the wave volume is equally divided between the wave amplitude and its wavelength, that is, $a_{max} \propto \lambda(t_{max})$, then previous results would suggest a scaling of the form $a_{max} \propto \sqrt{V_{eff}(t_{max})}$. Figure 15b shows the dimensionless wave amplitude as a function of the dimensionless effective volume at $t_{max}$. The obtained results lead to

$$ \frac{a_{max}}{H_0} \approx \sqrt{0.1 \frac{V_{eff}(t_{max})}{WH_0^2}}, \qquad (8) $$
<figure><img src="image_17.png" /><figcaption>**Figure 15.** (a) $V_{wave}(t_{max})/(WH_0^2)$ as a function of $V_{eff}(t_{max})/(WH_0^2)$, the black dashed line corresponds to the best linear fit with the corresponding determination coefficient $R^2$, and the solid gray line corresponds to $V_{wave}(t_{max}) = V_{eff}(t_{max})$, (insert: sketch of the characteristic volumes). (b) $a_{max}/H_0$ as a function of $V_{eff}(t_{max})/(WH_0^2)$ in logspace, the dashed/solid line corresponds to the best linear fit which is $a_{max}/H_0 = (0.1 V_{eff}(t_{max})/(WH_0^2))^{0.5}$ with the corresponding determination coefficient $R^2$.</figcaption></figure>

with a determination coefficient of $R^2 = 0.933$. Surprisingly this correlation is even better than the one obtained considering the wave volume ($R^2 = 0.88$). This suggests that the wavelength has a slightly more complex behavior than the amplitude.

**5. Discussion**

Waves generated by the impact of a granular flow on a water surface are investigated at the laboratory scale in order to mimic landslide-generated waves. The specificity of this experimental study lies in the relatively low impact velocity of the flow front and in the unlimited source of incoming granular material provided by a steady flow along a constant slope. A detailed description of the generation process is presented by analyzing the evolution of both the generated wave and the granular flow from the impact to the beginning of the propagation.

<figure><img src="image_18.png" /><figcaption>**Figure 16.** Relative maximum wave amplitude $a_{max}/H_0$ as a function of the final position of the granular flow front $x_f(t_{end})$. Colors represent $U_i/\sqrt{gH_0}$, where the yellow area spots the regime transition $Fr_c \approx 1.25$.</figcaption></figure>

**5.1. Summary of the Experimental Investigation**

The wave is characterized by its maximum amplitude $a_{max}$ for which the data shows a good agreement with relations from the literature. However, it turns out that the Froude number $Fr = U_c/\sqrt{gH_0}$ alone, with $g$ the gravity and $H_0$ the still water depth, can be a relevant characteristic dimensionless number of this study, as long as the characteristics velocity of the granular flow $U_c$ is appropriately defined. Three characteristic velocities are investigated: $U_c \equiv (U_i, U_f, q_{eff}/(WH_0))$ the granular flow front velocity at impact, the granular flow front velocity underwater and an equivalent velocity associated with the effective flow rate $q_{eff}$ and the channel width $W$, respectively. It is found that $U_c \equiv U_f$ and $U_c \equiv q_{eff}/(WH_0)$ allow a remarkable collapse of $a_{max}$ with $Fr$, highlighting the role of underwater granular flow dynamics in wave generation. It therefore suggests that the generation process occurs on a finite time scale during which the coupled granular-fluid flow evolves. To investigate the link between the granular flow properties and this finite time scale, the time at which $a_{max}$ is reached ($t_{max}$) is considered. Quite
surprisingly, a clear relationship with the impact velocity $U_c \equiv U_i$ is found. This means that the timescale over which the generation occurs is only related to the gravity-driven dynamics of the dry impacting granular flows.

To further explore the link between the generated wave and the immersed granular flow, the wave dynamics is then described in more details. In the range of parameters studied, the wave shows a self-similar time evolution of its amplitude, normalized by $a_{max}$ and $t_{max}$. Two different phases are observed: the generation phase for $t < t_{max}$ and the propagation phase for $t > t_{max}$. The latter is characterized by a freely propagating wave, whose velocity tends toward the theoretical phase velocity in the linear approximation and whose amplitude decays as $t^{-1/3}$. On the contrary, during the generation phase for $t < t_{max}$, a strong influence of the granular flow on the wave is expected as mentioned previously. In particular, a linear relation is observed between the effective volume of flow underwater and the volume of the leading wave at $t_{max}$, while their temporal evolution follows the same trend prior $t = t_{max}$. The initial velocity of the leading wave is thus equal to the one of the immersed part of the flow at impact. Interestingly a linear relation between the underwater granular flow front velocity and the leading wave velocity is still observed until $t_{max}$ showing again that the interactions between the wave and the granular flow are strong during the entire generation process. Beyond this point, the wave then propagates freely, reaching the celerity of linear surface waves.

## 5.2. Relations for the Wave Amplitude

In some geophysical applications, it can be useful to link the observable final deposit to the generated wave characteristics. In this context, the deposit is often characterized by its volume. Yet, in the case of a continuous inflow, and according to the obtained results, this volume is not a relevant parameter. Then, in a first attempt to provide such information from the present study, the runout of the granular flow is now considered, $x_f(t_{end})$ defined in Section 2.2.2. In Section 3, it was shown that $x_f(t_{end})$ can be related to the landslide impact velocity $U_i$ only, depending on the regime of impact, $U_i/\sqrt{gH_0} < Fr_c$ or $U_i/\sqrt{gH_0} > Fr_c$, with $Fr_c \approx 1.25$ (see Figure 11a). As the wave maximum amplitude $a_{max}$ increases with $U_i$ (see Figure 9a), a link between $a_{max}$ and $x_f(t_{end})$ can be expected. Hence, $a_{max}$ is plotted as a function of $x_f(t_{end})$ in Figure 16.

As expected, a clear correlation is obtained between $a_{max}$ and $x_f(t_{end})$, which seems strongly linked to the value of $U_i/\sqrt{gH_0}$. In particular, it appears that $a_{max}$ mostly increases with $x_f(t_{end})$. Yet, two different trends are observed which can be associated with two regimes. For energetic enough granular impact, that is, for $U_i/\sqrt{gH_0} > Fr_c$, $a_{max}$ linearly increases with $x_f(t_{end})$, while for $U_i/\sqrt{gH_0} < Fr_c$, $x_f(t_{end})$ remains almost constant for different values of $a_{max}$.

Even though the previous result provides insights on the link between the landslide deposit and the wave amplitude, the quantitative correlation remains low. This is due to the strong coupling between the wave maximum amplitude and the immersed granular dynamics, as reported in this study. Unfortunately, this coupling has been shown to be very complex, including air entrainment and outgassing, grains/fluid flow interaction, among others. Yet, it is useful to extract simpler effective quantities which contains these complexities, as already reported in similar studies (Bullard et al., 2023; Miller et al., 2017; Zitti et al., 2015). In the present study, the role of the generation time $t_{max}$ through the effective volume entering water $V_{eff}(t_{max})$ has been evidenced in Equation 8. Based on this result, assuming $V_{eff}(t_{max}) \approx q_{eff}t_{max}$ with $t_{max}$ predicted using Equation 7, then substituting into Equation 8 leads to:

$$ \frac{a_{max}}{H_0} \approx \sqrt{0.3 \left( \frac{q_{eff}}{WH_0\sqrt{gH_0}} \right) \left( \frac{U_i}{\sqrt{gH_0}} \right)}. \quad (9) $$

In Figure 17, a comparison between Equation 9 and experimental results is shown.

Note that this dependency of $a_{max}$ with $(q_{eff}U_i)^{1/2}$ is not in contradiction with Equation 6, as $q_{eff}$ and $U_i$ are necessarily linked. Then Equation 9 refines the previous result. In particular, a very good agreement between data and model Equation 9 is obtained ($R^2 = 0.94$). This new relation accounts for both the dry flow characteristics prior impact $U_i$ and the granular flow propagation underwater $q_{eff}$. Unfortunately, this relation does not provide a predictable model from the dry inflow only. Yet, it is interesting to note that replacing $q_{eff}$ with the dry flow rate
<figure><img src="image_19.png" /><figcaption>**Figure 17.** Relative maximum wave amplitude $a_{max}/H_0$ as a function of $q_{eff}/(WH_0\sqrt{gH_0})$ and $U_c/\sqrt{gH_0}$. Solid gray line corresponds to Equation 9, and black dashed line corresponds to the best fit with a prefactor of 0.5 and a determination coefficient of $R^2 = 0.94$.</figcaption></figure>

$q_m/\rho_p$, would lead to a relation similar to Equation 2 with appropriate characteristic velocity $U_c$ and thickness $h_c$. Nevertheless, Equation 9 remains the most convincing relation, with $R^2 = 0.94$.

## 6. Conclusion

As shown in the present study, providing predictive models for impulse wave characteristics remain challenging. The coupling between the granular flow and the wave has been shown to be highly complex. In particular, the granular flow impact and its dynamics underwater influence wave generation and dynamics. Moreover, depending on the dry granular velocity at impact, the wave can influence backward the deposit front. From those observations, models based only on the granular deposit or the properties of the granular flow at impact could then miss part of the processes of generation. However, considering both simultaneously is shown to lead to an interesting signature of the deposit runout as a marker of the wave amplitude. Moreover, following a similar approach, a link between the dry granular flow properties with the immersed ones is shown to predict the wave characteristics. In particular, the study of the wave-granular coupling during the generation phase led to an empirical fit of the wave maximum amplitude as a function of the product of two Froude numbers, one characterizing the impact and the other the granular flow propagation underwater. These new results provide a good basis for the development of more accurate models in future works.

However, remaining questions need to be addressed. The experimental analysis presented here is necessarily limited by the 2D-configuration of the experimental setup, which likely induced an overestimation of the wave amplitude (Heller & Spinneken, 2015). An extrapolation to 3D configurations is required. Moreover, predictive models require an estimation of the maximum amplitude based on the inflow characteristics. It therefore requires to link generic configuration of dry granular flows to the flow properties at impact, and their transition at the interface with water. Finally, the range of parameter covered in the present study is aimed at modeling moderate impact velocity applications. The link between moderate Fr configuration and high Fr configuration still require attention. In particular, the relation between wave volume and slide volume found here can not be extrapolated to high Fr configuration for which the wave volume can be found larger than any granular volume impacting the water.

## Appendix A: Measures Deduced From $\eta$

The extraction of $a(t)$, $a_{max}$, $t_{max}$, $x_{max}$, $t_1$ and $t_2$ is summarized in Figure A1. First, the amplitude along time $a(t)$ is measured as the maximum free surface elevation at each time (see red bullets in Figure A1a), with its corresponding abscissa $x_a(t)$. Then, to estimate the maximum amplitude of the wave $a_{max}$ and $t_{max}$, a third degree polynomial fit is applied on $a(t)$ (see Figure A1c), only on the region around the maximum amplitude. From $x_a(t)$, $x_{max}$ is then deduced by interpolation at $t_{max}$ (see Figure A1d). The plateau interval ($t_1$; $t_2$) is estimated by applying
<figure><img src="image_20.png" /><figcaption>**Figure A1.** Methodology of the extraction of important features from the free surface elevation $\eta(x,t)$, for one experiment corresponding to $\theta = 20^{\circ}$, $h_0 = 10$ cm and $L = 1.16$ m. (a) Shows the free surface elevation $\eta(x)$ at each time $t$ represented by the gray scale. The maximum elevation $a(t)$ at each time $t$ is represented in red on this graph, with its associated position $x_a(t)$. (b, c) Both shows $a(t)$ in red bullets, black curve in (b) is the fit from which $t_1$ and $t_2$ are extracted. Black curve in (c) is a third degree polynomial fit with which $a_{max}$ and $t_{max}$ are extracted. (d) Is the time evolution of $x_a(t)$ where $t_{max}$ is reported and allows to deduce $x_{max}$ by interpolation.</figcaption></figure>

a fit on the curve $a(t)$. This fit is applied with a non-linear least square algorithm to fit $a(t)$ data to a function. The latter is defined as a positive power law before $t_1$, a negative one after $t_2$ and a constant value between $t_1$ and $t_2$. The power law exponents, $t_1$, $t_2$ and the constant plateau value are ajustable coefficients for the least square fit.

As $a_{max}$ is deduced from a fit, it is clear that $a(t)$ can sometimes reach values higher than $a_{max}$, it is visible in Figure A1c, and it explains some standing out points in Figure 12a where $a/a_{max}$ is plotted along time and can in some cases be quite higher than 1.

## Appendix B: Computation of the Determination Coefficients $R^2$

All the determination coefficient $R^2$ from this article are calculated for the best linear fit (black dashed lines), following the equation:

$$R^2 = 1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2}, \qquad (B1)$$

where $y_i$ are the measured values and $\hat{y}_i$ are the predicted values from the linear fit.

## Data Availability Statement

Three typical videos of the experiments are available. Moreover, most data from the 98 experiments conducted to realise this study are available. In particular, the control parameters, the dry granular flow impact characteristics, the characteristics of the wave and of the granular flow deposit, the characteristics of the granular flow dynamics and the effective immersed granular flow volume at the time of maximum amplitude are available at: https://doi.org/10.5281/zenodo.12070742 (Darvenne, 2024).

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## Acknowledgments

This study has been partially supported through the grant EUR TESS N°ANR-18-EURE-0018 in the framework of the Programme des Investissements d'Avenir.
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