Pauline Letortu
email: pauline.letortu@univ-brest.fr

Stéphane Costa
email: stephane.costa@unicaen.fr

Olivier Maquaire
email: olivier.maquaire@unicaen.fr

Robert Davidson
email: robert.davidson@unicaen.fr

Stéphane Costa

Marine and subaerial controls of coastal chalk cliff erosion in Normandy (France) based on a 7-year laser scanner monitoring

Keywords: Marine processes, Subaerial processes, Cliff erosion, Terrestrial laser scanning (TLS)

published or not. The documents may come

Introduction

The retreat rates, the modalities of erosion (frequency, intensity, and spatial distribution), and the cliff profile depend on the morpho-structural conditions and the efficacy of marine and subaerial agents and processes [START_REF] Emery | Sea cliffs: their processes, profiles, and classification[END_REF]. Improving understanding of erosion is necessary because coastal cliff dynamics are inherently non-linear. Nothing happens for several years and suddenly a fall can occur and may cause fatalities and damage to infrastructure close to cliff line. One of the challenges in the research is to foresee coastal evolution to ensure the resiliency of coastal communities (e.g. [START_REF] Gilham | Marine control over negative power law scaling of mass wasting events in chalk sea cliffs with implications for future recession under the UKCP09 medium emission scenario[END_REF].

Research contributions into the dynamics of rocky coasts in recent years allow for a better understanding of retreat rates in a variety of locations worldwide [START_REF] Brooks | Temporal and spatial variations in recession rates and sediment release from soft rock cliffs, Suffolk coast, UK[END_REF][START_REF] Bezore | The drowned Apostles: the longevity of Sea stacks over Eustatic cycles[END_REF][START_REF] Sajinkumar | A composite fall-slippage model for cliff recession in the sedimentary coastal cliffs[END_REF][START_REF] Young | Decadal-scale coastal cliff retreat in southern and central California[END_REF], the identification of factors responsible for erosion (for a review, Kennedy et al., 2014a) and of some precursor signs before failure [START_REF] Rosser | Patterns of precursory rockfall prior to slope failure[END_REF], but knowledge of the respective contributions of triggering factors responsible for erosion are still difficult to determine [START_REF] Costa | Falaises à recul rapide et plages de galets: de la quantification des dynamiques d'un système complexe à la caractérisation des risques induits[END_REF][START_REF] Naylor | Rock coast geomorphology: Recent advances and future research directions[END_REF][START_REF] Lim | Quantifying the controls and influence of tide and wave impacts on coastal rock cliff erosion[END_REF][START_REF] Kuhn | Coastal cliff monitoring and analysis of mass wasting processes with the application of terrestrial laser scanning: A case study of Rügen, Germany[END_REF]Letortu et al., 2015a) even though some research has been successful in prioritizing factors [START_REF] Bernatchez | Seasonal quantification of coastal processes and cliff erosion on fine sediment shorelines in a cold temperate climate, north shore of the St[END_REF][START_REF] Bernatchez | Development of an automated method for continuous detection and quantification of cliff erosion events[END_REF]. In the northeastern part of Normandy (NW France), along the Channel, the plateaus of the Parisian Basin abruptly end in subvertical cliffs made up of Upper Cretaceous chalk with flint bands. The cliff line in Seine-Maritime had an average retreat rate of 0.15 m/year between 1966 and 2008 [START_REF] Letortu | Vitesses et modalités de recul des falaises crayeuses de Haute-Normandie (France): méthodologie et variabilité du recul[END_REF] and has been increasingly monitored for a better understanding of erosion (e.g. [START_REF] Prêcheur | Le littoral de la Manche, de Sainte-Adresse à Ault: étude morphologique[END_REF][START_REF] Evrard | La stabilité des falaises du Pays de Caux (Normandie)[END_REF][START_REF] Costa | Dynamique littorale et risques naturels : L'impact des aménagements, des variations du niveau marin et des modifications climatiques entre la baie de Seine et la baie de Somme[END_REF][START_REF] Hénaff | Le recul des falaises crayeuses du Pays de Caux: détermination des processus d'érosion et quantification des rythmes d'évolution/Retreat of chalk cliffs in the Pays de Caux: processes and rates[END_REF][START_REF] Costa | Quantification of the Normandy and Picardy chalk cliff retreat by photogrammetric analysis[END_REF]Mortimore et al., 2004b;[START_REF] Regard | Late Holocene seacliff retreat recorded by 10Be profiles across a coastal platform: Theory and example from the English Channel[END_REF][START_REF] Rohmer | On the deviation of extreme sea-cliff instabilities from the power-law frequency-volume distribution: Practical implications for coastal management[END_REF]Letortu et al., 2015b). On this chalky coast, there is a broad consensus on the role of lithological control and intense rainfall events as the main drivers of erosion. The trigger for falls often coincides with heavy winter rainfall [START_REF] Duperret | Coastal rock cliff erosion by collapse at Puys, France: The role of impervious marl seams within chalk of NW Europe[END_REF]Letortu et al., 2015a) particularly during wet years that cause the water table to rise in the chalky massif [START_REF] Lageat | The retreat of the chalk cliffs of the Pays de Caux (France): erosion processes and patterns[END_REF] or during heavy rainfall during dry periods [START_REF] Duperret | Coastal chalk cliff instability in NW France: role of lithology, fracture pattern and rainfall[END_REF]. However, a significant erosion factor on both the cliff and shore platform is the role of freeze/thaw cycles [START_REF] Moses | Predicting rates of shore platform downwearing from rock geotechnical properties and laboratory simulation of weathering and erosion processes[END_REF][START_REF] Dewez | Shore platform lowering due to frost shattering during the 2009 winter at Mesnil Val, English channel coast, NW France[END_REF]Letortu et al., 2015a). [START_REF] Lageat | The retreat of the chalk cliffs of the Pays de Caux (France): erosion processes and patterns[END_REF] concluded that subaerial processes were the main processes responsible for the triggering of chalk cliff falls on the French Channel coast and that marine influence was limited to the removal of falls. This has been questioned in recent years because numerous researchers have highlighted the role of marine factors, both as an eroding agent and as a removal agent to reactivate erosion [START_REF] Hénaff | Le recul des falaises crayeuses du Pays de Caux: détermination des processus d'érosion et quantification des rythmes d'évolution/Retreat of chalk cliffs in the Pays de Caux: processes and rates[END_REF][START_REF] Earlie | Coastal cliff ground motions and response to extreme storm waves[END_REF]Letortu et al., 2015a;2015b). To feed the debate, our previous research used statistical analyses (adjustment to power law scaling [START_REF] Letortu | Vitesses et modalités de recul des falaises crayeuses de Haute-Normandie (France): méthodologie et variabilité du recul[END_REF] and multivariate analysis as PCA) and empirical analyses to identify the triggers of coastal chalk cliff failure based on an inventory of 331 rock falls collected weekly between 2002 and 2009 from Veules-les-Roses to Le Tréport (37.5 km, Seine-Maritime). The results highlighted that 'cold and dry weather', 'high rainfall and high wind' are the conditions most likely to trigger rock falls (Letortu et al., 2015a). To go one step further and identify the dominance of marine or subaerial forcing, the current research analyzes retreat rates, modalities of retreat (frequency, intensity, and spatial distribution) and cliff profile evolution based on a 7-year monitoring project using Terrestrial Laser Scanning (TLS) of three cliff faces with close structural characteristics (lithology, stratigraphy, and tectonics) but different exposures to marine processes (abandoned and active cliffs).

Study area

Regional settings

The study area is of three Norman cliff sections along the Channel (Seine-Maritime): one cliff face in Varengeville-sur-Mer and two others in Dieppe (Fig. 1). The climate is a marine temperate west coast climate with mild temperatures throughout the year (4.9°C in January and 16.9°C in August according to Météo-France (1971-2000)). Temperatures in winter are positive (the coldest month is January with 4.9°C) but freeze/thaw cycles can be numerous (an average of 26 daily freeze/thaw cycles per year) and frost can be intense (up to -15°C).

Rainfall occurs throughout the year (about 800 mm) but the wettest month is November, with 93.8 mm, and the driest is August, with 50.6 mm. However, in August, thunderstorms may occur and lead to a maximum daily rainfall of 100.8 mm. Wind is frequent with a total of 102.8 days per year with gusts greater than or equal to 16 m/s, especially in December and January. The average tidal range is 8 m and swell is limited. However, wave height due to wind sea can reach 4 m in Dieppe (annual return period) [START_REF] Augris | Evolution morpho-sédimentaire du domaine littoral et marin de la Seine-Maritime[END_REF].

The topography of the Seine-Maritime is characterized by the Pays de Caux plateau (average altitude of 150 m) extending to the Channel, an Upper Cretaceous cliff line made up with chalk with flint bands, oriented SW/NE. This plateau is dissected by valleys and dry valleys. The valleys are quite wide (kilometer extent) and are former alluvial plains. This plateau presents different deformations with a succession of anticlines and synclines, in a NW/SE direction (Armorican direction), sometimes prolonged by flexures and faults [START_REF] Cavelier | Le Bassin de Paris[END_REF][START_REF] Mégnien | Synthèse géologique du Bassin de Paris[END_REF][START_REF] Cazes | Structure de la croûte hercynienne du Nord de la France : premiers résultats du profil ECORS[END_REF][START_REF] Pomerol | Histoire géologique du Bassin de Paris[END_REF][START_REF] Cavelier | Aspect et évolution géologiques du Bassin de Paris[END_REF] (Fig. 1). A topographic saddle, which corresponds to the northwestern termination of the "Norman gutter" (the exhumed anticlinal inlier of the Pays de Bray, in the hinterland of Dieppe, constitutes a geographical entity that extends south to Picardy) follows the syncline of the Vexin and extends to that of the Scie.

Cliffs from cap d'Antifer to Le Tréport (100 km) have an average height of 60-70 m but the structural variations explain the variations of height (from 20 m to more than 100 m; Fig. 1) of chalk stage outcrop (from the oldest to the newest stages: Cenomanian, Turonian, Coniacian, Santonian and Campanian). The subtle resistance contrasts of chalk outcrop explain different profiles of the cliffs [START_REF] Pomerol | Lithostratigraphy and biostratigraphy of the Lewes and Seaford chalks: A link across the Anglo-Paris Basin at the Turonian-Senonian boundary[END_REF][START_REF] Costa | Dynamique littorale et risques naturels : L'impact des aménagements, des variations du niveau marin et des modifications climatiques entre la baie de Seine et la baie de Somme[END_REF]Mortimore et al., 2004b) and mostly explain the high spatial variability of the retreat rates (from 0.09 to 0.23 m/year (± 0.03 m/an; 1966-1995-2008)) obtained by photo-interpretation [START_REF] Costa | Quantification of the Normandy and Picardy chalk cliff retreat by photogrammetric analysis[END_REF][START_REF] Letortu | Vitesses et modalités de recul des falaises crayeuses de Haute-Normandie (France): méthodologie et variabilité du recul[END_REF]. The retreat rate is the highest where, above Santonian and Campanian chalk strata, the residual flint formation [START_REF] Pomerol | Lithostratigraphy and biostratigraphy of the Lewes and Seaford chalks: A link across the Anglo-Paris Basin at the Turonian-Senonian boundary[END_REF][START_REF] Laignel | Les altérites à silex de l'ouest du Bassin de Paris: caractérisation lithologique, genèse et utilisation potentielle comme granulats[END_REF][START_REF] Costa | Quantification of the Normandy and Picardy chalk cliff retreat by photogrammetric analysis[END_REF] has been replaced by a bed of clay and sand sediments about 10-30 m thick from the Paleogene period [START_REF] Bignot | Etude sédimentologique et micropaléontologique de l'Eocène du Cap d'Ailly (près de Dieppe-Seine-Maritime)[END_REF]. This bed is mainly along the cap d'Ailly (Sainte-Margueritesur-Mer, Varengeville-sur-Mer), and Sotteville-sur-Mer (Fig. 1).

The pre-existing fracture network influences the rock mass strength and the mode of failure [START_REF] Middlemiss | Instability of Chalk cliffs between the South Foreland and Kingsdown, Kent, in relation to geological structure[END_REF][START_REF] Evrard | La stabilité des falaises du Pays de Caux (Normandie)[END_REF][START_REF] Duperret | Coastal chalk cliff instability in NW France: role of lithology, fracture pattern and rainfall[END_REF]Mortimore et al., 2004a).

The dominant fracture set collected on the cliff face and on the shore platform is N110-N130E followed by N025E [START_REF] Evrard | La stabilité des falaises du Pays de Caux (Normandie)[END_REF][START_REF] Genter | Multiscale fracture analysis along the French chalk coastline for investigating erosion by cliff collapse[END_REF]Costa et al., 2006b). [START_REF] Genter | Multiscale fracture analysis along the French chalk coastline for investigating erosion by cliff collapse[END_REF] highlight the N110-130E fracture set is almost ubiquitous throughout the Norman chalk coastline, made mainly of master-joints (large nearly vertical fractures that cross the whole cliff with an apparent extension of tens to hundreds of metres) and normal faults (typical apparent vertical offsets of cm to metre scale). Movements of various magnitudes (from a few m 3 to hundreds of thousands of m 3 ) are a ubiquitous phenomenon along the Seine-Maritime coastline. At the cliff foot, the shore platform has a fairly homogeneous appearance with a rocky surface and is slightly inclined towards the sea (from 0.2 % gradient at Le Tréport to 2 % at cap d'Antifer). Its width is between 100 and 750 m. In contact with the cliff foot and the shore platform, the average width of the gravel beach is about 15-20 m, with a thickness of 2-3 m.

At the valley outlets, where they are supported by longitudinal sea defense structures (dikes and piers), their width may exceed 100 m and their thickness 11 m at the Dieppe jetty (Costa et al., 2006a).

Site selection

The active cliff at Varengeville-sur-Mer is along the cap d'Ailly (6 km from Dieppe) (Fig. 2a), on either side of the Petit Ailly dry valley. This cliff is 250 m long, 40 m high, facing 010°N, with a subvertical profile (from 70° to overhang). The historical annual retreat rate is of 0.38 m/year (1966[START_REF] Hoyez | Les Falaises du Pays de Caux: lithostratigraphie des craies turonocampaniennes[END_REF]( from Letortu et al., 2014)).

The abandoned cliffs at Dieppe are located on the right bank of the Arques river mouth, behind an extension of the northeastern part of the harbor, land reclaimed from the sea. Dieppe 1 W is 45 m long, 35 m high, facing 310°N, with a subvertical profile. This cliff, in front of the Newhaven-Dieppe ferry port, has a historical annual retreat rate of 0.04 m/year (1966[START_REF] Hoyez | Les Falaises du Pays de Caux: lithostratigraphie des craies turonocampaniennes[END_REF]( from Letortu et al., 2014)). Dieppe 2 N is 80 m long, 35 m high, facing 010°N, with a subvertical profile. The historical annual retreat rate of Dieppe 2 N is of 0.06 m/year (1966[START_REF] Hoyez | Les Falaises du Pays de Caux: lithostratigraphie des craies turonocampaniennes[END_REF]( from Letortu et al., 2014)). Dieppe 1 W became abandoned during the nineteenth century whereas Dieppe 2 N became abandoned in the 1980s (Fig. 2a). with FACETS in Cloudcompare® [START_REF] Dewez | Facets : a Cloudcompare Plugin to Extract Geological Planes from Unstructured 3d Point Clouds[END_REF]; (e) Temporal distribution of the TLS surveys.

The site at Varengeville-sur-Mer is on the Scie syncline (at 6 km west of the faulted Bray anticline) and the sites of Dieppe are connected to the eastern part of Bray fault (Fig. 1). The cliff lithology at Varengeville is made up of Santonian chalk, covered by a bed of clay and sand of Paleogene period (due to the syncline prone strata preservation). Clay and sand are very prone to erosion (Fig. 2b). The cliff section of Dieppe is made up of Coniacian chalk covered with residual flint formation (Fig. 2c). Even if each facies within a stage has some subtle resistance contrasts, the Coniacian chalk is close to the physical and lithological characteristics of Santonian chalk [START_REF] Laignel | Caractérisation et dynamique érosive de systèmes géomorphologiques continentaux sur substrat crayeux. Exemple de l'Ouest du Bassin de Paris dans le[END_REF] (Table 1). However, Coniacian chalk is less fine, rougher to the touch than Santonian, with conglomeratic, yellowish and hardened chalk beds that are a little more numerous [START_REF] Auffret | Dieppe (est)[END_REF].

Table 1: Physical and lithological characteristics of the Coniacian and Santonian chalk stages of the western part of the Parisian Basin [START_REF] Laignel | Caractérisation et dynamique érosive de systèmes géomorphologiques continentaux sur substrat crayeux. Exemple de l'Ouest du Bassin de Paris dans le[END_REF] In the studied sites, the regional set (N110-N130E) is apparent in the stereograms, with master joints at Varengeville-sur-Mer and Dieppe 2 N (Fig. 2d). In Dieppe 1 W, the strike is about N025E, which is the secondary fracture set observed by [START_REF] Genter | Multiscale fracture analysis along the French chalk coastline for investigating erosion by cliff collapse[END_REF]. The bedding dips are very low (0.5-1%) [START_REF] Bignot | Dieppe (ouest)[END_REF].These fracture orientations and lithology result in falls with a dominant vertical failure type in the three sections. Infrequently, sliding failure type might occur in Varengeville-sur-Mer between Lower-Middle Santonian and Upper Santonian [START_REF] Duperret | Coastal chalk cliff instability in NW France: role of lithology, fracture pattern and rainfall[END_REF].

In contact with the Bray faulted anticline (Fig. 1) the fracturing is high. [START_REF] Genter | Multiscale fracture analysis along the French chalk coastline for investigating erosion by cliff collapse[END_REF] measured, for the total of fracture types, a mean spacing between fractures of 5-10 m for the eastern part of Dieppe, against 15-20 m in Varengeville-sur-Mer. Fracturing difference may impact on rates and modalities of retreat due to "pre-cutting" of the cliff but, according to [START_REF] Duperret | Coastal chalk cliff instability in NW France: role of lithology, fracture pattern and rainfall[END_REF], the empirical relationship between numerous falls and intense fracturing is not systematic probably because the natural fracture properties and the interaction with other parameters may be more complex. Indeed, the higher value of fractures in Dieppe (5-10 m) than in Varengeville (15-20 m) does not lead to higher historical annual retreat rates. Besides, subtle differences in resistance between both stages cannot explain such contrasting regressive dynamics. This is explained by the efficacy (or inefficacy) of the external erosion forcing.

The studied cliff faces are characterized by: ( 1

Material and methods

TLS surveys

A terrestrial laser scanner is an optical active remote-sensing technology. The two-way time of flight of a laser pulse reflected by a point is used to measure the point position (range and angles) relative to the device. The laser beam covers the environment with vertical scanning by an oscillating mirror and horizontal scanning by rotating the head. This instrument is widely used and numerous applications of the TLS technique on slope instabilities are given by the review of [START_REF] Abellán | Terrestrial laser scanning of rock slope instabilities[END_REF].

In Varengeville-sur-Mer, the two scanner stations are positioned on the foreshore at about 75 m from the cliff face (Fig. 3). In Dieppe, the two scanner stations are precisely positioned on the ground at a horizontal distance of 33 m (Dieppe 1 W) and 37 m (Dieppe 2 N) from the subvertical cliff face respectively. The site configuration and the survey design (with one target and a Trimble M3 total station) reduce occlusion and poor incident angles and enable large spatial coverage with a very high resolution (median spacing from 0.02 m in Dieppe to 0.05 m in Varengeville-sur-Mer). TLS survey is described in detail in Letortu et al. (2015b;[START_REF] Young | Decadal-scale coastal cliff retreat in southern and central California[END_REF].

Data processing and error margin

Data processing has 4 steps: (1) are retained to study the spatial distribution of material departures (dates in gray in Fig. 6). In February 2013, stabilization work was carried out in Dieppe. These consisted of clearing hazardous parts of the cliff (human erosion in Fig. 5) and setting up safety nets. To monitor the natural evolution of the cliff, these human activities are not considered in the calculations.

Data analysis

This 3D dataset allows the analysis of different characteristics of the cliff face evolution such as the eroded surface and volume, calculations of retreat rates, the modalities of retreat and the profile evolution. [START_REF] Emery | Sea cliffs: their processes, profiles, and classification[END_REF] present the profile of active cliffs as an indicator of the power balance between the resistance and homogeneity of the material outcropping on the cliff face and the combined aggressiveness of marine and subaerial agents (including ground water) associated with gravity. A steep slope at the foot of a cliff generally indicates dominance of marine erosion, while a gentle slope at the base means that subaerial erosion dominates. This classification has been included in numerous publications (e.g. [START_REF] Kuhn | Coastal cliff monitoring and analysis of mass wasting processes with the application of terrestrial laser scanning: A case study of Rügen, Germany[END_REF] and is used in this paper to identify, from a geomorphological point of view, the dominant forcing on erosion of the studied cliffs.

Results

Annual retreat rates

Unsurprisingly, the wave-battered cliff currently retreats faster than the abandoned ones but with a significant difference: 36 times faster than Dieppe 2 N. The annual retreat rate is 0.36 m/year in Varengeville-sur-Mer (13366 ± 156 m 3 fell in 85 months over 5214 m², thus an average fall volume of 1887 m 3 /year). It is 0.00 m/year in Dieppe 1 W (2 m 3 ± 9 m 3 fell in 85 months over 434 m², thus an average fall volume of 0.3 m 3 /year) and 0.01 m/year in Dieppe 2 N (14 m 3 ± 30 m 3 fell in 85 months over 1018 m², thus an average fall volume of 2 m 3 /year) (Fig. 4). Despite the very high precision of the method presented in this study, this quantitative assessment may not be representative for the abandoned cliffs because erosion results are below the margin of error. However, these annual retreat rates on the cliff face observed by TLS over seven years are close to those observed by the cliff top photointerpretation between 1966 and 2008 [START_REF] Letortu | Vitesses et modalités de recul des falaises crayeuses de Haute-Normandie (France): méthodologie et variabilité du recul[END_REF] with 0.38 m/year in Varengeville and 0.04-0.06 m/year in Dieppe. Taking into account the error margin of ± 0.03 m/year of the photo-interpretation method, the 2010-2017 TLS results seem to be representative of the historical retreat rate values (42 years, 1966-2008).

Retreat modalities and the responsible forcing

Whereas erosion occurs almost everywhere on the cliff face in Varengeville-sur-Mer, erosion is rare and localized on the cliff face in Dieppe (Fig. 4). The patterns of erosion are different between sites. According to the classification of slope movements by [START_REF] Varnes | Slope movement types and processes[END_REF] The abandoned cliffs of Dieppe are only affected by debris falls (Fig. 5, dark blue) over the 7year period. However, research in local newspapers between 1950 and 2010 reported that rock falls had already occurred on the studied abandoned cliffs. In March 2001, a rock fall was observed near Dieppe 1 W. We must bear in mind that these events have return periods of several decades [START_REF] Letortu | Vitesses et modalités de recul des falaises crayeuses de Haute-Normandie (France): méthodologie et variabilité du recul[END_REF]. For the 7-year monitoring period, debris falls are mostly located at the cliff foot or at the cliff top (Fig. 5) and are responsible for the annual retreat rate of up to 0.01 m/year in Dieppe 2 N. The active cliff in Varengeville-sur-Mer is affected by debris falls (Fig. 6, dark blue) and rock falls (Fig. 6, from light blue to dark red). The location of debris falls are variable: they are located at the cliff foot (DoD 1, 4), at the cliff top (DoD 5, 10), and mainly on the whole cliff face (DoD 2,11,14,15,16,18,19,20). In order to compare both environments (abandoned and active cliffs), we observe debris falls which are present in both contexts. In the context of the active cliff, debris falls are responsible for less than 2 % of the total erosion (218 m 3 ) measured over the period with retreat rates of 0.00-0.01 m/year (over 0.36 m/year). These retreat rates are consistent with the erosion observed on the abandoned cliffs where only debris falls are observed (Fig. 7). This value of 2 % is lower than those evoked in the literature (from 10 % [START_REF] May | The nature and rates of change on chalk coastlines[END_REF][START_REF] Hénaff | Le recul des falaises crayeuses du Pays de Caux: détermination des processus d'érosion et quantification des rythmes d'évolution/Retreat of chalk cliffs in the Pays de Caux: processes and rates[END_REF] to 25 % (Letortu et al., 2015b)). This could be due to different monitoring time periods and the higher error margin of the previous methods of quantification (extrapolation from several debris falls observed in the field). Because debris falls occur mainly at the cliff top of the abandoned cliff of Dieppe 2 N, the retreat rate by debris falls observed on the active cliff top could be due to the same forcing.

The retreat rate appears negligible in Varengeville-sur-Mer (0.00 m/year), so subaerial forcing is probably low for the erosion by debris falls but they may participate in the erosion by rock falls.

Profile analysis

Applying the classification scheme of [START_REF] Emery | Sea cliffs: their processes, profiles, and classification[END_REF] to the investigated cliff sections, the abandoned cliffs present homogeneous profiles underlining the homogeneity of the rock (Coniacian chalk) (Fig. 8). The profiles are also steady and subvertical (80-85°) underlining the low aggressiveness of subaerial processes on these cliff faces. Indeed, the expected slope value for a continentalized profile is from 5 to 30° in temperate climates (Pinot, 1998) but the continentalization of between 30 to more than 120 years is short in terms of geomorphological time scales.

For the active cliff in Varengeville-sur-Mer (Santonian chalk), changes in the shape of the profiles are observed. These various spatial patterns may highlight differences in the spatial and temporal variability of marine and subaerial processes (Fig. 8). During the 7 years of monitoring, marine processes seem largely dominant in the active cliff context with regard to vertical profiles, because the latter are maintained by the undermining and removal role of marine agents. However, subaerial processes also contribute to the retreat.

Our erosion results have been linked to external factors to be more precise about the dominant forcing. Two "dead times" and two "high points" of erosion can be identified and have been linked to environmental conditions (Fig. 9):  October 2010 to September 2012: A low erosion period occurred with a maximum retreat of 2.31 m and an average annual retreat rate of 0.07 m/year. This period of 23. were mild (+72 % of mean temperature in November and 147 % in December, 177 % in January) and with storm wave conditions in October but the winter was rainy (+84 % in January, +108 % in February), the spring of 2016 was "classic", summer 2016 was warm (+15 % of mean temperature for the three summer months) and almost dry (-85 % in July 2016), followed by a "classic" October and November but a very dry December (-85 % rainfall).

 January to November 2017: A high erosion period (maximum retreat of 6.21 m and an average annual retreat rate of 0.28 m/year) occurred with a mild (except January with 14 freeze/thaw cycles), calm and dry winter (-15 % and -36 % rainfall in January and March ; +40 % of mean temperature in February and March), the spring and summer of 2017 were quite warm and dry (-51 % rainfall and +25 % of mean temperature in June, -34 % in July but +71 % in August) but the month of September had storm wave conditions and the fall had warmer temperatures than the climate normal (+37 % of mean temperature in October) and had variable rainfall (-47 % in October, +37 % in November). As per Letortu et al. (2015a), the whole cliff face erosion occurs particularly during high rainfall and high wave conditions over the 7-year monitoring period. However, the relationships are not systematic as in August and October 2014 when high rainfall and high wave conditions occurred respectively without significant erosion (Fig. 9). Due to the imprecise dating of falls (time frequency of 4-5 months), it is not possible to be more explicit about the main processes responsible for erosion. Moreover the determination has four main sources of difficulty:

 First, the non-linear evolution of the coastal environment limits direct cause and effect relationships. Thresholds, hysteresis, synergies/combinations, positive and negative feedbacks, and 'purge' effect (described in detail in [START_REF] Viles | Linking weathering and rock slope instability: non-linear perspectives[END_REF] all complicate the analysis of relationships. For example, the relative role of marine and subaerial processes in cliff erosion may change over time, because platforms gradually widen over long time periods, which affects the amount of wave energy delivered to the foot of the cliff (negative feedback);  Secondly, the boundary conditions (sea level, climate, geology, tide and wind wave) modulate the domination of agents and processes over the evolution of the coast at a given place and time, requiring spatial and temporal scales to be jointed together. For example, the erosive power of the ocean will depend on the relative balance between tidal and wave climates (Kennedy et al., 2014b);  Third, the cliff is also inherited past environmental changes [START_REF] Trenhaile | Rock coasts, with particular emphasis on shore platforms[END_REF]: in our understanding of current processes and their impacts, we must also integrate long-term rock degradation [START_REF] Griggs | Coastal cliffs and platforms[END_REF][START_REF] Swenson | Bluff recession rates and wave impact along the Wisconsin coast of Lake Superior[END_REF][START_REF] Joyal | Influence of lithostratigraphy on the retreat of an unconsolidated sedimentary coastal cliff (St. Lawrence estuary, eastern Canada)[END_REF];  Finally, studies often focus on the cliff face while the cliff belongs to a vast system including nearshore, platform, cliff and hinterland [START_REF] Trenhaile | The geomorphology of rock coasts[END_REF]. Studies that focus on a single component are likely to provide only a partial view of the behavior of the entire slope [START_REF] Lim | The rock coast of the British Isles: cliffs, in: Rock Coast Geomorphology: A Global Synthesis[END_REF].

Basal notch creating instability on the whole cliff face

Figs. 6 and8 show one of the retreat modalities previously described along the Seine-Maritime coastline (Costa et al., 2006a;[START_REF] Dewez | Probabilistic coastal cliff collapse hazard from repeated terrestrial laser surveys: Case study from Mesnil Val (Normandy, northern France)[END_REF]Letortu et al., 2015b): the creation of a basal notch (created by debris falls or rock falls) that, by overhanging, will gradually destabilize the whole cliff face and generate a rock fall from the whole cliff face. Three cases are observed during the seven years of monitoring on both sides in Varengeville-sur-Mer (Fig. 6, on the western part: a to d; on the eastern one: e to g, and h to l). For example, debris fall 'a' generates a maximal basal notch on the western side of 0.2 m, reactivated by debris fall 'b' (maximal notch of 0.25 m). A few months later, the created overhang is eroded by rock fall 'c' from the whole cliff face (1875 m 3 ) which is reactivated a month later by rock fall 'd' (4616 m 3 ).

Discussion

Profile analysis

The profile analysis as a proxy of the power balance between the resistance and homogeneity of the material outcropping on the cliff face and the combined aggressiveness of marine and/or subaerial forcing has several limitations. First, the profile evolution is often the result of multiple factors which interact in time and space which can make the definite interpretation of the responsible factor more complex. Second, the analysis can be complicated by local effects (as in profiles 1 and 2 in Varengeville). Third, "equifinality" can occur: the same profile can be obtained by different processes.

Roles of structural variations and of the gravel barrier

The Dieppe cliff profiles (1 W and 2 N) and profile 1 in Varengeville-sur-Mer are similar in their subverticality and their homogeneity. This highlights (1) the low efficacy of subaerial forcing (or, for the active cliff, the very high efficacy of marine forcing) in cliff erosion, and [START_REF] Trenhaile | Rock coasts, with particular emphasis on shore platforms[END_REF] a close resistance between Santonian and Coniacian chalk stages with a common mode of failure: the vertical failure mode.

But at the Varengeville site, profiles 1 and 2 are different (Fig. 8). Marine forcing is predominant but the role of subaerial processes is much more visible in profile 2 with a more convex cliff top (softer slope). A hypothesis could be the greater thickness of the overlying Tertiary strata on the western part, which, due to the presence of perched groundwater [START_REF] Bignot | Dieppe (ouest)[END_REF], would accelerate the retreat at the top of the cliff. However, cliff face retreat rates were calculated for each side of the dry valley: annual retreat rates are almost the same with 0.37 m/year for the western part and 0.35 m/year for the eastern part. Difference in thickness of Tertiary strata does not seem to influence retreat rates but it can influence the modality of retreat. Another hypothesis could be a difference in fracturing between the two sides of the dry valley, modifying drainage and infiltration. A more detailed analysis of the fracturing needs to be carried out. The last hypothesis could be the influence of the descending road to the sea which, acting as a groin, accumulates gravel on the front western part (up to 5 m thick) to the detriment of the immediate downdrift side (eastern part, up to 2.5 m thick). The gravel barrier could cause negative feedback with increased wave protection at the cliff foot, leading to a likely decrease in the efficacy of marine processes on the western part of the cliff foot. In addition, profile 2 has no basal notch although this is the case on profile 1, highlighting the role of marine forcing on the eastern part. However, the amount and configuration of beach material can also create positive feedback with higher wave erosive efficacy by providing abrasive elements [START_REF] Sunamura | Geomorphology of Rocky Coasts[END_REF], as evoked in our previous paper in the study area (Letortu et al., 2015b) (Fig. 6, DoDs 1 and 4).

Subaerial forcing inferior to marine forcing but non-negligible

The subvertical profiles of the active cliff and the current annual retreat rate, 36 times higher when the cliff is wave-battered, highlight the supremacy of marine over subaerial controls in the erosion of chalk cliffs around Varengeville-sur-Mer between 2010 and 2017. However, subaerial processes are present (the freeze/thaw action is visible in DoD 19 in Fig. 6 and is supported by Fig. 10, the runoff zones are visible at the cliff top in Figs. 5,6), but they seem much less active than marine processes in destabilizing cliffs. The slow continentalization of the abandoned cliffs in Dieppe may be explained by the low efficacy of subaerial processes in erosion but it is important to note that these morphological adjustments take time, and, local factors, notably hydrogeomorphological, could explain this slow evolution. The plateau here is characterized by a high fracturing of the chalky block (close to the Bray fault) where a karstic system may extend along pre-existing fracturing. Dieppe 1 W and 2 N have a well-developed karstic network that was widened during the Second World War, before being roughly filled. Moreover, the plateau is also locally characterized by a slope and a dip towards the hinterland, lowering the chalk water table level at the foot of the cliff, and directing a large part of the surface flows towards the dry valley a few hundred meters behind the cliff (stereogram of Dieppe 2 N; Fig. 2d). The monitored chalk cliffs could be less saturated than elsewhere due to these local factors.

The combined marine and subaerial processes alternate in time and space in the active cliff context. TLS monitoring has made it possible to visualize "erosion patterns" in Varengevillesur-Mer which appear to be seasonal: between September and February, the cliff foot is often eroded partly by marine processes while between March and September, it is rather the cliff top that is eroded due to subaerial activity (Fig. 6). Moreover, erosion increased in Varengeville-sur-Mer during the winter of 2013-2014, when conditions were particularly stormy on the Atlantic and Channel coasts [START_REF] Blaise | Bilan des tempêtes de l'hiver 2013-2014 sur la dynamique de recul du trait de côte en Bretagne[END_REF][START_REF] Earlie | Coastal cliff ground motions and response to extreme storm waves[END_REF] (Fig. 9).

Our results confirm that marine processes play the most important erosive role during the studied period, and also that of removing falls at the foot of the cliffs. Indeed, the roles of marine processes are visible following stormy periods with 1) the cliff foot bleached due to wave erosive action and 2) a milky sea due to the dissolution of chalk falls (Fig. 11). Thus, a high spatial variability of removal rates is observed. Beyond the elapsed time interval between the two surveys, the removal rate is explained by a multitude of factors which influence the modalities and the efficacy of the removal (size of blocks and chalk types, kinematics of rock falls, volume of rock falls depending on the height of the cliffs and the fracturing of the chalk, volume of gravel barrier etc.).

Conclusions

Erosion of three chalk cliff faces in Normandy was monitored every 4-5 months since October 2010 using a terrestrial laser scanner. These cliff faces with close structural characteristics (including lithology with Upper Cretaceous chalk with flint bands of Coniacian and Santonian ages) are exposed differently to marine processes: the active Varengevillesur-Mer cliff, affected by marine and subaerial processes, and the two abandoned Dieppe cliffs evolving under the action of subaerial processes only. The choice of sites and the high frequency, high resolution and centimeter precision monitoring (± 0.02 to 0.03 m) make it possible to improve the understanding of 1) the cliff face retreat rates, 2) the modalities of cliff evolution, and 3) the efficacy of marine and subaerial controls in erosion of the studied cliff faces.

Over the 85 months between October 2010 and November 2017, our results highlight the predominance of marine processes in the retreat of the coastal chalk cliffs because: 1) the current annual retreat rate is 36 times higher in the active cliff context than in the abandoned cliff context; 2) the cliff profiles are subvertical due to the primacy of marine processes in the active cliff context and the low effectiveness of subaerial processes in the abandoned cliff context; and 3) the location of erosion highlights the great effectiveness of marine processes in destabilizing cliffs (from the generation of a basal notch through the overhang generating instability to the triggering of a rock fall of the whole cliff face) and in the removal of falls.

These results indicate that TLS data can be very useful in monitoring coastal cliff erosion and in contributing to the debate on the primacy of agents and processes responsible for erosion and the triggering of falls. Research perspectives using our database are numerous, including the implementation of "rock fall" hazard warning systems. The various studies conducted in continental [START_REF] Abellán | Detection and spatial prediction of rockfalls by means of terrestrial laser scanner monitoring[END_REF][START_REF] Royán | Spatio-temporal analysis of rockfall pre-failure deformation using Terrestrial LiDAR[END_REF][START_REF] Kromer | Rockfall risk management using a pre-failure deformation database[END_REF] and coastal areas [START_REF] Rosser | Patterns of precursory rockfall prior to slope failure[END_REF] focus on two types of precursory signs: the activity of falls before failure and deformations before failure. One of the important issues in forecasting systems lies in the coupling of spatial and temporal predictions, which are still rare in the scientific literature (e.g. [START_REF] Rosser | Patterns of precursory rockfall prior to slope failure[END_REF][START_REF] Arosio | Towards rockfall forecasting through observing deformations and listening to microseismic emissions[END_REF][START_REF] Kromer | Rockfall risk management using a pre-failure deformation database[END_REF] because of their great complexity [START_REF] Abellán | Terrestrial laser scanning of rock slope instabilities[END_REF].

The current TLS monitoring must continue to improve temporal representativeness. Only reliable, homogeneous and sustainable monitoring will allow a better understanding of cliff evolution. The annual retreat rates will be refined in Dieppe by extending the monitoring time, which will allow the margin of error to be expanded. An increase in the temporal sampling of measurements would be desirable to refine knowledge on the contribution of marine and subaerial agents and processes responsible for triggering falls in Varengeville-sur-Mer. A high temporal sampling would also make it possible to develop the research axis on the precursory signs of failure. The great diversity of current monitoring instruments makes it possible to be more reactive [START_REF] Letortu | Examining high-resolution survey methods for monitoring cliff erosion at an operational scale[END_REF].

One of the limitations of our results is that they are difficult to transpose, as they are very dependent on the three studied sites. Other methods such as the mobile laser scanning [START_REF] Michoud | Landslide detection and monitoring capability of boatbased mobile laser scanning along Dieppe coastal cliffs, Normandy[END_REF], make it possible to cover larger areas but their implementation costs and the necessary technical know-how are obstacles to their deployment. The spatial representativeness of the TLS monitoring we conducted has been widened in order to extend the scope of our results in various contexts. Since November 2017, TLS monitoring of the Sainte-Marguerite-sur-Mer cliff (6 km west from the TLS site in Varengeville-sur-Mer) has begun and it will be linked to a continuous cliff face multiparameter monitoring (temperatures, water table level, seismicity, rainfall, sea water level etc.) during the winter 2018-2019.

Fig. 1 :

 1 Fig. 1: Geological and structural framework in Seine-Maritime (Normandy) and the location of

Fig. 2 :

 2 Fig. 2: (a) Location of the sites; (b) Main characteristics of Varengeville-sur-Mer site (Chalk

  ) Coniacian and Santanian chalk stages prone to erosion; (2) two contexts -Varengeville is an active cliff (marine and subaerial processes act together) and Dieppe is an abandoned cliff (action of subaerial processes only); (3) their proximity (distance of 6 km), assuming weather conditions are similar; (4) their verticality (from 75° to overhang), their height (35-40 m) and the linearity of the coastline; and (5) their spatial extent (from 45 m long in Dieppe to 250 m long in Varengeville-sur-Mer).

Fig. 3 :

 3 Fig. 3: Description of the instruments used for the cliff face erosion monitoring (survey of

  georeferencing and point cloud alignment; (2) manual point cloud filtering (vegetation, people, foreshore); (3) meshing using Delaunay triangulation and generation of a 3D Digital Elevation Model (DEM); and (4) creating a DEM of Difference (DoD). Precision is the most important parameter in our monitoring, thus all the point clouds are fitted to the 6 July 2011 reference point cloud using best fit alignment algorithms (Cloudcompare® or 3DReshaper®). This adjustment reduces the error margin because it includes the TLS instrumental error and the cloud adjustment error (fitting) only. To assess precision, fixed parts of the point cloud are compared using the usual data processing. The precision in planimetry is 0.03 m for Varengeville-sur-Mer and Dieppe 2 N and 0.02 m for Dieppe 1 W. The volume precision is ± 156 m 3 in Varengeville-sur-Mer (surface of 5214 m²), ± 9 m 3 in Dieppe 1 W (434 m²) and ± 30 m 3 in Dieppe 2 N (1018 m²). Some disturbances occurred during data acquisition such as temporary ground vibrations due to the docking of a cross channel ferry (2011/07/06 and 2013/02/12 in Dieppe 1 W) or scanner sinking in wet sand (2011/02/24 and 2015/02/26 in Varengeville-sur-Mer). Thus, these four point clouds have artifacts and they are not used to quantify the rate of retreat, but

Fig. 4 :

 4 Fig. 4: Erosion results of DoD over 7 years of monitoring at each of the sites. For interpretation

  (which was updated by[START_REF] Dikau | Landslide Recognition: Identification, Movement and Causes[END_REF] and[START_REF] Hungr | The Varnes classification of landslide types, an update[END_REF]), two terms can be used to describe movements of coherent rock in relation to fallen volume. Debris fall is used where tiny blocks or flakes (up to a decimeter) fall from across the cliff face, and rock fall is used to describe large-scale movements that fall from all or part of the cliff face.

Fig. 5 :

 5 Fig. 5: DoDs of the cliff face in Dieppe 2 N between 2010/10/07 and 2017/11/02. For

Fig. 6 :

 6 Fig. 6: DoDs of the cliff face in Varengeville-sur-Mer between 2010/10/08 and 2017/11/02 (gray

Fig. 7 :

 7 Fig. 7: Annual retreat rates in function of location (site and departure area) and type of

Fig. 8 :

 8 Fig. 8: Main evolutions of the cliff profile in Varengeville-sur-Mer and Dieppe during 7 years

  5 months is characterized by calm wave conditions, a cold and humid fall in 2010 (17 daily freeze/thaw cycles in December), a mild and dry winter and spring in 2011, a wet summer in 2011 (+141 % rainfall in August in comparison with the 1971-2000 climate normal from Météo-France data), a mild and dry fall in 2011 and the winter of 2012, and a humid spring-summer in 2012.  September 2012 to March 2014: A high erosion period (maximum retreat of 12.93 m and an average annual retreat rate of 1.51 m/year) occurred for 17.5 months and was characterized by high rainfall in October and December 2012, and with a cold winter (24 daily freeze/thaw cycles); the spring of 2013 had a dry April and June (-62 % and -67 % rainfall in comparison with the climate normal, respectively) but summer months of July and August were warm and rainy (+23 % rainfall in July with thunderstorms, +110 % in August), the fall and winter of 2013-2014 were particularly mild (an average of +70 % of mean temperature for the three winter months) but characterized by violent storms and rainfall (+76 % in November) and with frequent storm wave conditions.  March 2014 to January 2017: A low erosion period (maximum retreat of 2.73 m and an average annual retreat rate of 0.02 m/year) occurred for 31.5 months. This period is characterized by a drier and warmer spring in 2014 than the climate normal, a very rainy summer (+73 % and +251 % in July and August, respectively), a dry fall (-83 % in September, -45 % in October and -42 % in November) but with storm wave conditions in October, a mild and rainy winter in 2014-2015 (+55 % rainfall in January 2015), a dry spring in 2015 (-55 % rainfall in April), and a warm summer in 2015 characterized by a very rainy July and August (+126 % rainfall in August); the fall and winter of 2015-2016

Fig. 9 :

 9 Fig. 9: Evolution of climate (rainfall, temperature, number of freeze/thaw cycles from Météo-

Fig. 10 :

 10 Fig. 10: Successive zooms of debris falls at the cliff foot of Varengeville-sur-Mer after

Fig. 11 :

 11 Fig. 11: Marine action visible after a stormy period (December 2017-January 2018) in

Fig. 12 :

 12 Fig. 12: Evolution of the cliff face in Varengeville-sur-Mer (2 km from the monitored site) between 2006/03/09 and 2006/12/19 with a rock fall of 5700 m 3 in late May. In August, the

Acknowledgements

This work is part of the Service National d'Observation DYNALIT, via the research infrastructure ILICO. This work was supported by the French "Agence Nationale de la Recherche" through the "Laboratoire d'Excellence" LabexMER [ANR-10-LABX-19-01] program, and co-funded by a grant from the French government through the "Investissements d'Avenir" and the Brittany Region. This work was also supported by the ANR project "RICOCHET: multi-risk assessment on coastal territory in a global change context" funded by the French Research National Agency [ANR-16-CE03-0008].

The authors are grateful for the constructive criticisms and comments of referees which have improved the manuscript.


Isabelle Bonnet

Julien Gabelli
email: julien.gabelli@u-psud.fr

Hyperbolic cosine function in physics

Keywords: catenary, chainette, physics, hyperbolic cosine

A catenary is the shape that an electric cable takes under its own weight if suspended only at its ends between two pylons. A rope hung between two masts is also described with a catenary equation which is based on the hyperbolic cosine function. We propose here to approach the hyperbolic cosine function with two simple experiments using a beaded chain. We first study the static form of a chain whose ends are suspended between two poles, when the distance between the two poles varies. Then, we study the dynamics of the same chain falling from the edge of a table. The simplicity of these two experiments, which can be done at home, makes them appropriate for undergraduate students interested in practicals.

Introduction

A catenary is a two-dimensional curve that describes the shape of an homogeneous flexible chain suspended by its ends so that it is sagging only subject to gravity. In the early 17th century, Galileo was the first to address the mathematical question of the shape of a chain hanging under gravity; he wrongly claimed that the shape would fit a parabola [START_REF] Galilei | Dialogues Concerning Two New Sciences trans. H Crew and A de Salvio[END_REF][START_REF] Galilei | Dialogue Concerning the Two Chief World Systems-Ptolemaic & Copernican[END_REF]. A few years later, in 1690, Jacob Bernoulli launched a challenge to find the mathematical description of a suspended chain shape. One year after, the curve equation was obtained by Leibniz et al: they called it 'the catenary equation' [START_REF] Leibniz | De Linea in Quam Flexile se Pondere Propio Curvat, Ejusque Usu Insigni ad Inveniendas Quotcunque Medias Proportionales et Logarithmos[END_REF][START_REF]Huygens 1691 Dynastae in züchelem Solutio Ejusdemproblematis[END_REF][START_REF] Bernoulli | Solutio Problematis Funicularis[END_REF]. It corresponds to the hyperbolic cosine function cosh, and is also known as a 'chainette' in textbooks. This mathematical function can be expressed in terms of exponential functions as cosh(x) = e x +e -x 2 .

In this paper, we propose to study the hyperbolic cosine function with two simple experiments that can be performed at home with everyday life objects: a chain, a smartphone camera and a computer. Both experiments deal with classical physical problems: the first experiment is static (the analysis of the form taken by a hanging chain [START_REF] Nedev | The catenary-an ancient problem on the computer screen[END_REF][START_REF] Behroozi | In praise of the catenary[END_REF]) while the second one is dynamics (the motion of a chain sliding from a horizontal surface [START_REF] Behroozi | The sliding chain problem with and without friction: a universal solution[END_REF]). They are thus appropriate for undergraduate students.

The shape of the chain

Theoretical description

We consider a chain of length L and mass M-which is uniform-and we denote μ = M L its mass per unit length. We use Cartesian coordinates (x, z), with x the horizontal axis and z the vertical axis (see figure 1 for notations). The two extremities of the chain are placed at x = 0 and x = h, so that z(0) = z(h). Moreover, we impose the origin of the z-axis such as z(0) = 0. Applying the equilibrium condition on a chain's element of length dl in the x and z directions leads to

d (T. cos α) = 0, d (T. sin α) -μgdl = 0, (1)
where dl = √ dx 2 + dz 2 = dx 1 + dz dx 2 and α the angle the segment dl makes with the x-axis. We then deduce

T(x). cos α(x) = A d (T. cos α. tan α) = μg 1 + dz dx 2 dx (2)
with A a constant, which depends on the distance h between the two chain's ends: we will denote it A(h), and dz dx = tan α. Knowing that d du argsh(u) = 1 √ 1+u 2 , we obtain, after integration, the following differential equation:

arg sh dz dx -arg sh dz dx x=h/2 = μg A x - h 2 . ( 3
)
Because of the symmetry, arg sh dz dx x=h/2 = 0, and we get dz dx = sinh μg A x -h 2 . Know-

ing that cos α = dx dl = 1 √ 1+ dz dx , we finally deduce z(x) = ξ cosh x -h/2 ξ -ξ cosh h 2ξ , T(x) = A cos α = A cosh x -h/2 ξ , (4)
with ξ(h) = A(h) μg . Picture of the chain laid on a table to distinguish the individual beads that make it up.

Chain's characteristic

We used a beaded chain of length L = 952 mm and mass M = 21 g: its mass per unit length is thus

μ = M L = 22 g • m -1 .
The chain is composed of N beads = 221 beads and is suspended between its two ends separated by a distance h varying from 30 to 80 cm (figure 1). For each distance h, we took a picture of the chain with a smartphone fixed to a camera tripod equipped with a bubble level to ensure horizontality: we checked that the camera's optical axis was perpendicular to the plane of the chain. We measured the (x(n), z(n)) coordinates of points of the chain by clicking manually on the n ∈ [[0; 221]] beads with the multi-points tool in ImageJ [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF]. We then reconstructed the shape of the chain point by point, as shown on figure 2, which displays the measured profiles z(x) for the different values of h.

We first measured the distance a(n) between two successive beads a 2 and found a = a(n) = 4.4 ± 0.1 mm (inset of the figure 2), in very good agreement with the expected value a = L N beads -1 = 4.3 mm.

(n) = (x(n + 1) -x(n)) 2 + (z(n + 1) -z(n))

Experimental description of the chain curve

To compare the data with the catenary equation (equation ( 4)), we fitted our experimental data (figure 2) with the theoretical expectation z(x) = ξ cosh x-x 0 ξ ξ cosh x 0 ξ leaving x 0 (h) and ξ(h) as free parameters. The graph of x 0 as a function of h displays a linear behavior and we found x 0 h = 0.511 ± 0.002, as expected (figure 3(a)). The parameter ξ(h) can be related to the total chain's length

L since L = h 0 1 + dz dx dx = 2ξ sinh h 2ξ(h) .
We numerically solved this nonlinear equation, which allows us to deduce a theoretical expectation of ξ(h), as shown in figures 3(b) and 4. Moreover, the inset of figure 3(b) displays the graph sinh h 2ξ(h) as a function of L 2ξ(h) ; a linear fit gives a slope of 0.989 ± 0.004, in very good agreement with the theory.

(x) = ξ cosh x-h/2 ξ - ξ cosh h 2ξ
(black dashed lines, ξ and x 0 being free parameters). Inset: Experimental determination of the distance a between two successive beads; n is the bead index of the chain (n = 0 . . . 221); we find a 4.4 ± 0.1 mm. The pointing error is estimated to be 0.5 mm, except at the chain's extremities where the pointing is delicate because of the fixing.

Tension at the chain's extremities

Let us denote T(0) = T(h) ≡ T 0 and |α(0)| = |α(h)| ≡ α 0 the tension and the angle at the chain's extremities which both depend on h. The equilibrium condition for the whole chain is written as 2T 0 sin α 0 = 2T 0 cos α 0 tan α 0 = Mg. The angle α 0 is deduced from the ended points of the chain:

α 0 = 1 2 arctan z(N beads )-z(N beads -1) x(N beads )-x(N beads -1) -arctan z 0 -z 1 x 0 -x 1 . Since T 0 = Mg 2 sin α 0 and sin α = dz dl = dz/dx √ 1+(dz/dx) 2 = sinh x-h/2 ξ cosh x-h/2 ξ
, we obtain

T 0 = Mg 2 tanh h 2 ξ(h)
. Figure 4 shows the graph of T 0 (h) where the gravity at the Earth's surface was assumed to be 9.8 m • s -2 . Note that lim h→L T 0 (h) = +∞: a suspended horizontal cable would imply an infinite tension on the posts.

It is therefore impossible to have a perfectly horizontal catenary: that is why droppers are used to connect the catenary with a contact wire which is horizontal in the electrical overhead lines. Schematic of the forces acting on a chain falling from the edge of a table at a time t > 0. We denote z(t) the length of the chain hanging out of the table, the surface of the table being defined by z = 0. The reference time t = 0 corresponds to the beginning of the fall. We split the chain into two subsystems. The first one, S 1 (in blue), corresponds to the part of the chain in contact with the table at time t, and has a length of Lz(t). Four forces act on S 1 : its weight W 1 , the two components of the reaction (N 1 , F 1 ) and the tension T 1 exerted by the other part of the chain. The second subsystem is S 2 (in red) and is the part of the chain of length z(t) that is hanging over. S 2 is subject to two forces: the weight W 2 and the tension T 2 . The chain is supposed to be inextensible so that T 1 = T 2 .

Fall of the chain from the edge of a table

In this second part, we approach the mathematical function cosh differently by studying another classical physics problem: the dynamics of a chain falling from the edge of a table.

Theoretical description

We consider a similar beaded chain, but with different characteristics: it is now composed of N beads = 229 beads so that its length is now L = 982 mm and its mass is M = 22 g. The chain is fully stretched and lying on a table so that its direction is perpendicular to the edge of the table.

One part of the chain is horizontal and lying on the table while the other part, of length z(t), is hanging vertically off of the table (figure 5). We split the chain into two subsystems: S 1 corresponds to the part of the chain in contact with the table and has a length of Lz(t). The second subsystem is S 2 and is the part of the chain (of length z(t)) that is hanging off.

The chain will start to move as soon as its hanging part z > z 0, where z 0 is a threshold length, and corresponds to the minimal length of the chain required to trigger the fall of the entire chain by gravity. The start of the motion sets the reference time t = 0. We measured a threshold length corresponding to 35 beads, or z 0 = 154 mm.

The amplitudes of the forces acting on S 1 are (see figure 5 for notations):

W 1 = -Mg 1 - z L its weight, N 1 = -W 1 the normal reaction of the table, F 1 = -kN 1
the friction exerted by the table, T 1 the tension exerted by S 2 [START_REF] Bernoulli | Solutio Problematis Funicularis[END_REF] with k the coefficient of friction [START_REF] Feynman | The Feynman Lectures on Physics Volume 1: Mainly mechanics, Radiation, and Heat[END_REF]. For t 0, the chain is moving so that k = μ k < μ s where μ k and μ s = F 1 N 1 are the coefficients of kinetic and static friction, respectively. The latter can be Figure 6. Chain falling from a table. Comparison between experimental points (blue circles) of z(t) obtained by manually tracking the lowest bead of the chain, and theoretical expectation (dashed black line) during the fall of the chain (equation ( 8)).

estimated with z 0 . The equilibrium before motion (t < 0) gives N 1 + W 1 = 0 and F 1 + T 1 = 0 for S 1 and W 2 + T 2 = 0 for S 2 . Moreover, assuming the chain is inelastic, we have

|T 1 | = |T 2 |. Finally, μ s = F 1 N 1 = W 2 W 1 = z 0
L-z 0 0.186. Even if the system is not conservative, it is convenient to use the work energy theorem on the whole system:

dE dt = P, ( 6
)
where E is the total energy including kinetic energy K = M 2 dz dt 2 and the potential energy U = -μgz dz = -Mg 2L z 2 with the assumption U(z = 0) = 0. The dissipated power due to friction on the table is

P = F 1 dz dt = -Mgμ k 1 -z L dz dt .
Finally, we get a second-order differential equation:

d 2 z dt 2 - z τ 2 = -gμ k , ( 7 )
where τ = L g(1+μ k ) is the characteristic time of the fall. Since the chain falls without initial velocity and having z(0) = z 0 , we obtain

z(t) = z 0 - μ k 1 + μ k L cosh t τ + μ k 1 + μ k L, ( 8
)
where we have assumed that z(0) = z 0 + ε, with ε an infinitely small positive quantity, so that the motion is already initiated at time t = 0.

Experimental data

We manually tracked the lowest bead of the chain (which was painted in black). The theoretical fitting of the graph z = f(t) gave μ k = 0.1672 ± 0.0001 (figure 6) and we then deduced τ = 0.293 ± 0.002 s, where we have used a value of 9.8 m • s -2 for g. As expected, note that μ k < μ s . On figure 6, we see a slight discrepancy between the experimental data and the theoretical expectation. This is due to experimental uncertainty but also to the theoretical modeling which assumed a constant coefficient of kinetic friction. The coordinates measurement is actually complicated as soon as the chain's velocity exceeds the frame rate acquisition (Δt = 1/60 s): this is the case when dz > a Δt = 0.3 m • s -1 (frames 30-60 in figure We also noticed that the chain snakes horizontally starting from frame 31, maybe due to the chamfer of the table edge.

Conclusions

Through this study, we have illustrated the importance of the hyperbolic cosine function in simple mechanics. The experiments carried out deal with static and dynamic studies of the same system: a chain characterized by its mass M and its length L. They have allowed us to characterize a shape by a characteristic length ξ = T 0 L cos α 0 Mg and a movement by a characteristic time τ = L g(1+μ k ) . In the first experiment, the chain is maintained by posts at an angle α 0 and a tension T 0 while in the second it falls by rubbing on a table with a coefficient of kinetic friction μ k . In both cases the chain is subjected to gravity g so that the acceleration ξ τ 2 depends on the characteristics of the chain (its mass M), on the boundary conditions (T 0 and α 0 ) and on the support's characteristics (μ k ): ξ τ 2 = (1+μ k )T 0 cos α 0

M

. This relates the universal character of gravity in both experiments: static or dynamical.

The two experiments presented above can be easily conducted at home since they do not require any specialized equipment: they just require a camera, a chain, a measuring tape, a kitchen weighting scale a personal computer. Analyses can be achieved thanks to free software: ImageJ for the tracking, and Python for numerical solving of equations. This study is thus appropriate for undergraduate students.

Figure 1 .

 1 Figure 1. Picture of the beaded chain used in this paper, suspended only by its ends, with the notations used for the derivation of the catenary equation. The two extremities of the chain (of length L and mass M) are separated by a distance h. Vector quantities are represented by bold letters and we use the Cartesian coordinate plane (x, z): g is the acceleration produced by gravity at the Earth's surface. A segment of chain, of length dl, at coordinates (x, z) is subject to the tension of the chain T(x) at point x, to the tension of the chain T(x + dx) exerted at the point x + dx and to its weight μdlg with μ the linear mass density of the chain. This segment makes an angle α(x) with the x-axis. Inset: Picture of the chain laid on a table to distinguish the individual beads that make it up.

Figure 2 .

 2 Figure 2. Shape of the suspended chain when the distance h between its two ends is varying: h = 30, 35, 40, 45, 50, 55, 60, 65, 70, 75 and 80 cm. The blue dots are obtained by manually clicking on the beads on the picture of the chain. These data points are fitted with the theoretical curve of the catenary equation z(x) = ξ cosh x-h/2

Figure 3 .

 3 Figure 3. (a) Graph of the free parameter x 0 as function of the distance h between the fixed point displays a linear behavior with a slope of 1/2, as expected with equation (4). (b) The fitting parameter ξ as a function of h is in agreement with the theoretical expectation we numerically obtained.

Figure 4 .

 4 Figure 4. Graph of the chain tension at its extremities. Experimental points (blue circle) estimated with the formula

Figure 5 .

 5 Figure5. Schematic of the forces acting on a chain falling from the edge of a table at a time t > 0. We denote z(t) the length of the chain hanging out of the table, the surface of the table being defined by z = 0. The reference time t = 0 corresponds to the beginning of the fall. We split the chain into two subsystems. The first one, S 1 (in blue), corresponds to the part of the chain in contact with the table at time t, and has a length of Lz(t). Four forces act on S 1 : its weight W 1 , the two components of the reaction (N 1 , F 1 ) and the tension T 1 exerted by the other part of the chain. The second subsystem is S 2 (in red) and is the part of the chain of length z(t) that is hanging over. S 2 is subject to two forces: the weight W 2 and the tension T 2 . The chain is supposed to be inextensible so that T 1 = T 2 .

Figure 7 .

 7 Figure 7. Time-lapse of the chain falling. The time is indicated in frame; the total fall lasts one second. For sake of visibility, we painted two beads in black. Starting from frame 31, we have indicated the vertical axis with dashed red lines to emphasize the fact that the chain is snaking while falling.