Xavier Fritel
email: xavier.fritel@univ-poitiers.fr

Professor Virginie Ringa

MD, PhD Emeline Quiboeuf

MSc Arnaud Fauconnier

Female urinary incontinence, from pregnancy to menopause, a review of epidemiologic and pathophysiologic findings Authors

Keywords: Female urinary incontinence, a review BMI, body mass index, LE, level of evidence, MUI, mixed urinary incontinence, SUI, stress urinary incontinence, UI, urinary incontinence, UUI, urge urinary incontinence Urinary incontinence, pregnancy, delivery, cesarean, epidemiology

Hypotheses that might explain urinary incontinence during pregnancy and after childbirth have been examined. Urinary incontinence prevalence reaches a maximum during pregnancy and decreases after childbirth. Cesarean delivery is associated with lower rates of stress incontinence than vaginal delivery. Women delivered by cesarean section differ from women who had a vaginal delivery through pre-existing characteristics associated with the incontinence risk, producing a selection bias. The only randomized trial (intention of vaginal delivery versus elective caesarean section) was not conclusive. The suspected etiological factors consist of congenital elements, obesity, aging, pregnancy and vaginal delivery. They are likely to have an effect at different times and portions of the urethral sphincter complex.

Unfortunately there exist no cohorts facilitating description of the natural history of female urinary incontinence. Given the small amount of usable data available, it is not possible to draw any conclusions concerning the possible long-term protective effects of cesarean section.

Introduction

Urinary incontinence (UI) is a symptom frequently encountered in women, and it can evolve into a disabling pathology necessitating costly treatment [START_REF] Wilson | Annual Direct Cost of Urinary Incontinence[END_REF][START_REF] Ho | Personal and treatment costs of childbirth related incontinence[END_REF][START_REF] Lecomte | Aides techniques aux personnes handicapées : Situation actuelle, données économiques, propositions de classification et de prise en charge[END_REF]. Despite its frequent occurrence, many aspects of its pathophysiology remain uncertain. Urinary continence is due to the fact that bladder pressure remains lower than urethral closure pressure, and incontinence may result from bladder or urethral impairment; when closure pressure is lower than bladder pressure, leakage occurs. Three types of incontinence are generally distinguished: stress urinary incontinence (SUI); urge urinary incontinence (UUI), and mixed urinary incontinence (MUI), which associates the first two.

The pathophysiology of UUI has yet to be clearly elucidated. It appears to be connected with poor transmission or processing of information between the bladder and the nervous system [START_REF] Groat | The urothelium in overactive bladder: passive bystander or active participant?[END_REF]. The main identified disorders with regard to SUI are impaired urethral support and sphincter deficiency [START_REF] Delancey | Stress urinary incontinence: Relative importance of urethral support and urethral closure pressure[END_REF]. Nobody really knows why, how, and at what precise moment these disorders arise [START_REF] Delancey | Why do women have stress urinary incontinence?[END_REF].

In a majority of cases urinary incontinence in women is stress-related before the age of 50 and mixed after the age of 50 [START_REF] Hannestad | A community-based epidemiological survey of female urinary incontinence: The Norvegian EPINCONT Study[END_REF]. The time lapse between average age at first childbirth and average age at the time of SUI surgery ranges from 20 to 30 years; this wide range underscores the difficulty of establishing a causal relation between childbirth and SUI. And yet, the obstetrical trauma hypothesis according to which vaginal childbirth results in irreversible traumatic lesions to the urinary continence system is generally accepted. Some practitioners even favor caesarean section as a way of preventing urinary incontinence [START_REF] Al-Mufti | Obstetricians' personal choice and mode of delivery[END_REF].

Our objective in this review of the literature is to examine the arguments supporting the traumatic obstetrical theory and, if need be, to put forward alternative hypotheses that might more adequately explain the occurrence of urinary incontinence in women.

Methods

The studies included in our manuscript originate from two sources. The first is a review of available epidemiologic or clinical studies in which urinary incontinence was the main outcome of interest and obstetrical factors (related to pregnancy or delivery) were analyzed as explanatory factors. We have included a review of all cohorts as regards urinary incontinence during first pregnancy and postpartum (22 studies). The second part consists of a panel of selected studies on the pathophysiology of urinary incontinence associated with anatomical, hormonal, or physiological changes during pregnancy and after childbirth. Selection of these studies was not based on a systematic search; they are rather cited to illustrate some underlying pathophysiological mechanisms that may explain the epidemiological evidence.

The level of evidence [LE] is indicated for each study through the standard model developed by the Oxford Centre for Evidence-Based Medicine (www.cebm.net): LE1, randomized comparative studies; LE2, cohort studies; LE3, case-control studies; LE4, case-series.

We wish to emphasize that most available information on the connection between childbirth and middle-aged UI came from cross-sectional studies that could not determine the onset of urinary symptoms. Only three teams have published results on longitudinal follow-up of women studied for more than two years subsequent to childbirth [START_REF] Dolan | Stress incontinence and pelvic floor neurophysiology 15 years after the first delivery[END_REF][START_REF] Macarthur | on behalf of the ProLong study group. Exclusive caesarean section delivery and subsequent urinary and faecal incontinence: a 12-year longitudinal study[END_REF][START_REF] Viktrup | Risk of stress urinary incontinence twelve years after the first pregnancy and delivery[END_REF].

Results

Urge urinary incontinence, pregnancy and delivery

UUI prevalence in young women is quite low (7% at 30 years of age in EPINCONT, see below); it is higher in older women (13% at 50 years of age and 19% at 70) [LE3b] [START_REF] Hannestad | A community-based epidemiological survey of female urinary incontinence: The Norvegian EPINCONT Study[END_REF].

During pregnancy, its prevalence increases as the term approaches (12% at end of pregnancy)

[LE2b] [START_REF] Viktrup | The risk of lower urinary tract symptoms five years after the first delivery[END_REF]. In the first stages of postpartum, UUI prevalence is lower in cases of cesarean section [LE2a] [START_REF] Press | Does cesarean section reduce postpartum urinary incontinence? A systematic review[END_REF]. However, in the long term after childbirth, cross-sectional and longitudinal observational studies have not shown any association between parity and UUI, or mode of delivery and UUI [LE2a] [START_REF] Press | Does cesarean section reduce postpartum urinary incontinence? A systematic review[END_REF][START_REF] Rortveit | Age-and type-dependent effects of parity on urinary incontinence : The Norwegian EPINCONT Study[END_REF][START_REF] Rortveit | Urinary incontinence after vaginal delivery or cesarean section[END_REF][START_REF] Hirsch | Parity is not associated with urgency with or without urinary incontinence[END_REF][START_REF] Foldspang | Parity as a correlate of adult female urinary incontinence prevalence[END_REF]. Given the lack of connection between UUI and pregnancy or delivery, the remaining review will be essentially devoted to SUI.

Association between obstetrical trauma due to vaginal delivery and stress urinary incontinence SUI prevalence is higher in older women, and reaches a maximum level at approximately the age of 50 (27% in EPINCONT) [LE3b] [START_REF] Hannestad | A community-based epidemiological survey of female urinary incontinence: The Norvegian EPINCONT Study[END_REF][START_REF] Maclennan | The prevalence of pelvic floor disorders and their relationship to gender, age, parity and mode of delivery[END_REF]. In a majority of studies on the subject less SUI prevalence following cesarean section has been found in comparison with vaginal childbirth [LE2a] [START_REF] Viktrup | Risk of stress urinary incontinence twelve years after the first pregnancy and delivery[END_REF][START_REF] Press | Does cesarean section reduce postpartum urinary incontinence? A systematic review[END_REF][START_REF] Rortveit | Urinary incontinence after vaginal delivery or cesarean section[END_REF][START_REF] Fritel | Stress urinary incontinence four years after the first delivery: a retrospective cohort study[END_REF]. EPINCONT is a large-scale cross-sectional survey that compared SUI prevalence in 669 women aged from 20 to 64 years having delivered only by cesarean section and 11 299 women who had only vaginal deliveries [START_REF] Rortveit | Urinary incontinence after vaginal delivery or cesarean section[END_REF]. SUI prevalence was higher in the vaginal childbirth group (OR 2.4; IC95% 1.7-3. Hannah, who compared planned cesarean births with planned vaginal delivery for breech presentation at term [START_REF] Hannah | Term Breech Trial 3-Month Follow-up Collaborative Group. Outcomes at 3 months after planned cesarean vs planned vaginal delivery for breech presentation at term: the international randomized Term Breech Trial[END_REF], showed lower SUI prevalence in the planned cesarean group three months later (4.5 vs. 7.3%) [LE1b]. The risk of postnatal UI was aggravated when the second stage of labor was prolonged, and lessened in cesarean childbirth, especially when the section was carried out prior to labor or at the onset of labor [LE2b] [START_REF] Brown | Effects of prolonged second stage, method of birth, timing of cesarean section and other obstetric risk factors on postnatal urinary incontinence: an Australian nulliparous cohort study[END_REF]. SUI prevalence 12 years after childbirth was 55% in the event of only vaginal deliveries as opposed to 40% in the event of only cesarean deliveries [LE2b] [START_REF] Macarthur | on behalf of the ProLong study group. Exclusive caesarean section delivery and subsequent urinary and faecal incontinence: a 12-year longitudinal study[END_REF]. These observations suggest a causal link between the trauma of vaginal childbirth and SUI.

Obstetrical trauma

A childbirth-related pelvi-perineal trauma may be immediately observed by simple clinical examination or using complementary postnatal tests [START_REF] Delancey | On the biomechanics of vaginal birth and common sequelae[END_REF]. Perineal tears frequently occur, particularly on the occasion of a first childbirth, in which a 1.8% to 16% prevalence of thirddegree perineal tear has been reported [LE2b] [START_REF] Anthony | Episiotomies and the occurrence of severe perineal lacerations[END_REF][START_REF] Lowder | Risk factors for primary and subsequent anal sphincter lacerations: a comparison of cohorts by parity and prior mode of delivery[END_REF]. After vaginal childbirth, investigations have shown increased pudendal nerve terminal latency, widening of the levator hiatus, perineal descent, and increased urethral mobility at stress [LE2b] [START_REF] Sultan | Pudendal nerve damage during labour: prospective study before and after childbirth[END_REF][START_REF] Toozs-Hobson | The effect of mode of delivery on pelvic floor functional anatomy[END_REF]. Dietz found a reduction of reflex pelvic floor activation on coughing that is more marked after vaginal delivery [START_REF] Dietz | Does childbirth alter the reflex pelvic floor response to sudden increases in intra-abdominal pressure?[END_REF]. Twenty per cent of primiparas show levator ani lesions at MRI after vaginal childbirth, and the lesions are associated with an increased risk of SUI (72% with SUI in cases of MRI-evidenced lesion vs. 45%) in this cross-sectional study of 160 primiparas examined 9-12 months subsequent to vaginal delivery [LE3b] (29). Dietz and Lanzarone objectified similar levator ani lesions through ultrasonography in 36% of primiparas having undergone vaginal childbirth and examined 2-6 months subsequent to delivery [START_REF] Dietz | Levator trauma after vaginal delivery[END_REF]. In their work, no link between levator ani lesions and SUI was reported [LE3b].

Vaginal delivery may consequently entail the creation of irreversible lesions that could be aggravated by aging and subsequent childbirth, finally leading to urinary incontinence. Such is the theory put forward by DeLancey, who strongly suggested that an obstetrical trauma paved the way to urinary incontinence, which can occur immediately if the trauma is severe or later in cases where the trauma is partially repaired [START_REF] Delancey | Graphic integration of causal factors of pelvic floor disorders: an integrated life span model[END_REF].

Shortcomings of the trauma paradigm

Several observations call the predictions given by the trauma model into question: SUI prevalence is maximal not postpartum, but rather in the prenatal stage (Figure 1), during which 30% to 50% of pregnant women have reported SUI [LE2b] [START_REF] Brown | Effects of prolonged second stage, method of birth, timing of cesarean section and other obstetric risk factors on postnatal urinary incontinence: an Australian nulliparous cohort study[END_REF][START_REF] Viktrup | The symptom of stress incontinence caused by pregnancy or delivery in primiparas[END_REF][START_REF] Francis | The onset of stress incontinence[END_REF][START_REF] Stanton | The incidence of urological symptoms in normal pregnancy[END_REF][START_REF] Dimpfl | Incidence and cause of postpartum urinary stress incontinence[END_REF]. Such findings mean that the pathology (SUI) may actually precede exposure (delivery) and suggest that mechanisms other than obstetrical traumatism are involved prior to delivery in the genesis of SUI.

A follow-up of 305 primiparas carried out by Viktrup (32) shows a remission of incontinence over the first 12 months of postpartum, with SUI prevalence being divided by 6 as it decreased from 19% in the immediate postpartum period to 6% three months later and to 3% after one year [LE2b]. A comparable drop in SUI prevalence following delivery and over the initial months postpartum has been observed by others in the past [START_REF] Francis | The onset of stress incontinence[END_REF][START_REF] Stanton | The incidence of urological symptoms in normal pregnancy[END_REF][START_REF] Dimpfl | Incidence and cause of postpartum urinary stress incontinence[END_REF][START_REF] Buchan | Médecine domestique ou traité complet des moyens de se conserver en santé, de guérir & de prévenir les maladies, par le régime et les remèdes simples[END_REF] or by contemporary authors identified in our bibliographic review (Figure 1), which recapitulates all published longitudinal studies pertaining to primiparous women [START_REF] Brown | Effects of prolonged second stage, method of birth, timing of cesarean section and other obstetric risk factors on postnatal urinary incontinence: an Australian nulliparous cohort study[END_REF]. These observations tend to suggest a deleterious, yet reversible, effect of pregnancy on urinary continence.

Partially reversible obstetrical trauma

In a majority of cases, perineal exploration undertaken at some time after childbirth show that perineal measurements are quite similar to those reported before childbirth or improved in comparison with measurements undertaken just after delivery. In the study published by Sultan (26), the terminal latency of the pudendal nerve had reverted to normal at six months postpartum in eight of the 12 women for whom it had increased at six months postpartum However, in this study, two women (of 12) recovered from levator avulsion, and bladder neck mobility showed improvement for 53 women delivered vaginally with no further births [START_REF] Shek | Does Levator Trauma heal?[END_REF] [LE2b]. These observations tend to suggest that several obstetrical lesions of the pelvic floor may spontaneously heal during the months following delivery. In that case, vaginal childbirth would account for largely reversible lesions of the urinary continence system. Such findings may help to elucidate observations showing that the apparent protective effect of a cesarean is attenuated in time. In the randomized trial performed by Hannah [START_REF] Hannah | Term Breech Trial 3-Month Follow-up Collaborative Group. Outcomes at 3 months after planned cesarean vs planned vaginal delivery for breech presentation at term: the international randomized Term Breech Trial[END_REF][START_REF] Hannah | Maternal outcomes at 2 years after planned cesarean section versus planned vaginal birth for breech presentation at term: the international randomized Term Breech Trial[END_REF], the protective effect with regard to SUI observed at three months postpartum in the planned cesarean group (RR = 0.6; IC95%: 0.4-0.9) diminished and was no longer significant 24 months subsequent to childbirth (RR = 0.8; 0.6-1.1) [LE1b]. Over the same time interval, the fraction of SUI risk attributable to vaginal delivery was halved, decreasing from 38% at three months to 18% at 24 months [LE1b]. Over the long term, after the age of 50, according to the EPINCONT [START_REF] Rortveit | Urinary incontinence after vaginal delivery or cesarean section[END_REF] as well as GAZEL studies [START_REF] Fritel | Mode of delivery and severe stress incontinence. A cross-sectional study among 2625 perimenopausal women[END_REF], there exists no significant difference in SUI prevalence between women having undergone vaginal childbirth and women having undergone cesarean childbirth, even though association with parity remains [LE3b]. In the cohort followed by MacArthur and his colleagues for up to 12 years following childbirth [START_REF] Macarthur | on behalf of the ProLong study group. Exclusive caesarean section delivery and subsequent urinary and faecal incontinence: a 12-year longitudinal study[END_REF], 40% of the women having delivered only by cesarean section reported urinary incontinence [LE2b]. These observations suggest that parity, or pregnancy itself, may contribute to the occurrence of UI independently of the mode of delivery [START_REF] Fritel | Stress urinary incontinence four years after the first delivery: a retrospective cohort study[END_REF][START_REF] Viktrup | The symptom of stress incontinence caused by pregnancy or delivery in primiparas[END_REF][START_REF] Iosif | Stress incontinence during pregnancy and in puerperium[END_REF].

Possible selection bias

Moreover, the apparent protective effect of cesarean reported in observational studies, including studies based on registers, may be explained in part by a selection bias. Prospective studies showed that prior to childbirth, women who are going to have a vaginal delivery present higher urethral mobility [START_REF] Toozs-Hobson | The effect of mode of delivery on pelvic floor functional anatomy[END_REF][START_REF] Dietz | Antenatal pelvic organ mobility is associated with delivery mode[END_REF] and levator hiatus [START_REF] Toozs-Hobson | The effect of mode of delivery on pelvic floor functional anatomy[END_REF], than those who are going to have cesarean delivery [LE2b]. It may be the case that some particular tissue or muscle characteristics entail both a higher risk of cesarean birth and a lower risk of SUI. In other words, the mode of delivery could be a consequence of pelvic tissue quality rather than a cause, and it is likewise possible that cesarean delivery selects women at little risk of developing SUI. This type of preliminary difference between women delivering by cesarean as opposed to vaginal childbirth does not substantiate observational studies attributing a protective effect to caesarean section. The existence of this kind of selection bias in the EPINCONT cross-sectional study [START_REF] Rortveit | Urinary incontinence after vaginal delivery or cesarean section[END_REF] is suggested by the fact that on average, the women delivering by cesarean were four years younger than those delivering vaginally (36 vs. 40 years), had higher BMI (26.2 versus 25.4 kg/m²) as well as lower parity (1.7 versus 2.2), and had more recently given birth (8 vs. 12 years) to children of lower weight (3396 vs. 3542g).

Since higher age is a cesarean section risk factor in low-risk nulliparas [LE3b] (65), the opposite difference would have been expected.

As a result, the obstetrical trauma theory does not suffice as a stand-alone explanation of stress urinary incontinence occurring either at the end of pregnancy or with aging. The 241woman cohort followed by Viktrup over 12 years [START_REF] Viktrup | Risk of stress urinary incontinence twelve years after the first pregnancy and delivery[END_REF][START_REF] Viktrup | The risk of lower urinary tract symptoms five years after the first delivery[END_REF][START_REF] Viktrup | Lower urinary tract symptoms 5 years after the first delivery[END_REF] underscores the highly diversified histories of urinary incontinence in women and consequently suggests comparably diversified etiologies. In the women presenting with SUI 12 years after initial childbirth, SUI appeared for the first time during pregnancy in 39% of cases, just after delivery in 15%, and after the postpartum period in 46% of cases [LE2b]. Subsequent to an initial remission during the first year of postpartum, (3% one year after delivery), SUI prevalence increased over the years following the initial childbirth (30% at 5 years and 42% at 12 years) [LE2b].

It is equally necessary to explain why SUI occurring transitorily during pregnancy constitutes a risk factor for later appearance of SUI [LE2b] [START_REF] Dolan | Stress incontinence and pelvic floor neurophysiology 15 years after the first delivery[END_REF][START_REF] Viktrup | Risk of stress urinary incontinence twelve years after the first pregnancy and delivery[END_REF][START_REF] Fritel | Stress urinary incontinence four years after the first delivery: a retrospective cohort study[END_REF].

Individual susceptibility

The hypothesis of a congenital factor may help to elucidate SUI observed in young or nulliparous women and the similarity frequently observed among members of the same family [START_REF] Wolin | Stress incontinence in young, healthy nulliparous female subjects[END_REF][START_REF] Buchsbaum | Prevalence of urinary incontinence and associated risk factors in a cohort of nuns[END_REF][START_REF] Harris | Urinary incontinence and pelvic organ prolapse in nulliparous women[END_REF][START_REF] Buchsbaum | Urinary incontinence in nulliparous women and their parous sisters[END_REF]. Buchsbaum [START_REF] Buchsbaum | Urinary incontinence in nulliparous women and their parous sisters[END_REF]

Aging

In women as in men, aging is accompanied by a gradual rarefaction of the striated muscle cells in the urethral sphincter [LE3b] [START_REF] Strasser | Urinary incontinence in the elderly and age-dependent apoptosis of rhabdosphincter cells[END_REF]. Pelvic floor strength is reduced and levator hiatus is increased in older women [LE3b] [START_REF] Weemhoff | Effects of age on levator function and morphometry of the levator hiatus in women with pelvic floor disorders[END_REF]. Urethral closure pressure likewise declines in older women [LE3b] [START_REF] Delancey | Why do women have stress urinary incontinence?[END_REF][START_REF] Trowbridge | Effects of aging on lower urinary tract and pelvic floor function in nulliparous women[END_REF]. Pelvic floor denervation is greater in older women [LE3b] whether or not they once bore children [START_REF] Smith | The role of partial denervation of the pelvic floor in the aetiology of genitourinary prolapse and stress incontinence of urine, A neurophysiological study[END_REF]. SUI prevalence regularly increases with age [LE3b] in parous and nulliparous women [START_REF] Rortveit | Urinary incontinence after vaginal delivery or cesarean section[END_REF]. With time, the aging effect is likely to dilute the risk attributable to obstetrical factors; little by little, any causal factor connected with the obstetrical trauma is prone to dwindle. DeLancey [START_REF] Delancey | The appearance of levator ani muscle abnormalities in magnetic resonance images after vaginal delivery[END_REF] showed the existence of an association between SUI and MRI levator ani lesion in women of an average age of 30 who were examined 9-12 months postpartum [LE3b]; no comparable link was found in a case-control study conducted by the same team in women of an average age of 48 [LE3b] [START_REF] Delancey | Stress urinary incontinence: Relative importance of urethral support and urethral closure pressure[END_REF]. In a population of women consulting an urogynecological unit, mean age 55, levator ani lesion was negatively associated with stress UI [LE3b] [START_REF] Dietz | Does avulsion of the puborectalis muscle affect bladder function?[END_REF].

Individual susceptibility and aging do not suffice to explain why the SUI risk is greater in women who have given birth, even by cesarean section, than in nulliparas. In EPINCONT [START_REF] Rortveit | Urinary incontinence after vaginal delivery or cesarean section[END_REF], the age-adjusted risk with regard to all types of UI, of SUI, and of mixed UI was in fact greater in women having delivered by cesarean section than in childless women: OR = 1.5

(1.2-1.9), 1.4 (1.0-2.0) and 1.7 (1.2-2.5), respectively [LE3b]. A heightened risk for women having given birth by cesarean section in comparison with nulliparas has also been found in other studies [LE3b] [START_REF] Fritel | Mode of delivery and severe stress incontinence. A cross-sectional study among 2625 perimenopausal women[END_REF][START_REF] Parazzini | Risk factors for stress, urge or mixed urinary incontinence in Italy[END_REF][START_REF] Lan | The epidemiological study of women with urinary incontinence and risk factors for stress urinary incontinence in China[END_REF]. Regardless of the mode of delivery, it is consequently highly likely that pregnancy has a specific impact on continence mechanisms, with long-term clinical effects that may be connected with mechanical or metabolic phenomena. Between the prenatal and postpartum stages, the observed volume of the urethral sphincter decreased by 8.2% in the women delivering vaginally and by 10.6% in those giving birth by cesarean section [LE2b] [START_REF] Toozs-Hobson | The effect of mode of delivery on pelvic floor functional anatomy[END_REF]. A 10 cm H 2 0 drop in urethral closure pressure is likewise observed following childbirth, and it remains similar, whether delivery be vaginal or by cesarean [LE2b] (81). Weight gain may partially explain the increased SUI prevalence during pregnancy, postpartum remission and SUI relapse with age. In EPINCONT (85), a high BMI was associated with each type of UI [LE3b]. A high BMI is also associated with the UI specific to pregnancy [LE3b] [START_REF] Solans-Domènech | Grup de Recerca del Sòl Pelvià; GRESP). Urinary and anal incontinence during pregnancy and postpartum: incidence, severity, and risk factors[END_REF]. Postpartum weight loss is associated with remission of UI [LE2b] [START_REF] Wesnes | Urinary incontinence and weight change during pregnancy and postpartum: a cohort study[END_REF]. Apart from this kind of mechanical effect, a metabolic effect of pregnancy cannot be ruled out. Indeed, pregnancy entails numerous hormonal or metabolic modifications that may be involved in UI occurrence. Estrogens are known to modify collagen synthesis in the tendons [LE2c] (86), and collagen metabolism is modified in the event of SUI [LE3b] (87). The metabolic and hormonal modifications connected with pregnancy may both explain some of the urinary symptoms observed during pregnancy and entail long-term consequences.

Indeed, the long-term metabolic impact of pregnancy is known in the context of other chronic pathologies, such as diabetes: the number of previous pregnancies increases the risk of abnormal glycemia in non-pregnant women [LE3b] [START_REF] Mcdonald | Diabetes Reduction Assessment with Ramipril and Rosiglitazone Medication Trial Investigators. Dysglycemia and a history of reproductive risk factors[END_REF]. In a manner analogous to UI, the prevalence of diabetes is raised during pregnancy, and there is an association between the diabetes of pregnancy and the diabetes of maturity.

Discussion

SUI prevalence is maximal at the end of pregnancy, it diminishes postpartum and is lower in cases of cesarean delivery. SUI prevalence increases with age and reaches a maximal level around the age of 50. After 50, SUI prevalence is similar in women delivered only by cesarean and women delivered vaginally. Several pathophysiological mechanisms are likely to be involved to explain these epidemiological observations. Urethral mobility increases during pregnancy [START_REF] Dietz | Does pregnancy affect pelvic organ mobility?[END_REF]. It would be interesting to learn if a higher mobility is associated with a higher risk of SUI during pregnancy. There exists a link between heightened prenatal urethral mobility and postnatal SUI [START_REF] King | Is antenatal bladder neck mobility a risk factor for postpartum stress incontinence?[END_REF], and pregnant women with a high prenatal urethral mobility are more likely to deliver vaginally [START_REF] Toozs-Hobson | The effect of mode of delivery on pelvic floor functional anatomy[END_REF][START_REF] Dietz | Antenatal pelvic organ mobility is associated with delivery mode[END_REF]. Higher urethral mobility may be provoked by a number of physiological, metabolic or hormonal modifications induced by pregnancy. Hormones such as estrogens or relaxin are involved in collagen metabolism [START_REF] Hansen | Ethinyl oestradiol administration in women suppresses synthesis of collagen in tendon in response to exercise[END_REF]. In postmenopausal women, estrogen therapy increases the risk of SUI [START_REF] Hendrix | Effects of Estrogen With and Without Progestin on Urinary Incontinence[END_REF], and a higher level of serum relaxin has been associated with SUI during pregnancy [START_REF] Kristansson | Reproductive hormones and stress urinary incontinence in preg-nancy[END_REF]. Pregnancy also modifies carbohydrate metabolism and diabetes is associated with a heightened UI risk [START_REF] Brown | Prevalence and risk factors for urinary incontinence in women with type 2 diabetes and impaired fasting glucose: findings from the National Health and Nutrition Examination Survey (NHANES) 2001-2002[END_REF]. Unfortunately, studies on this subject are lacking. Van Geelen did not find a correlation between closure pressure changes and female hormones changes during pregnancy [START_REF] Van Geelen | The urethral pressure profile in pregnancy and after delivery in healthy nulliparous women[END_REF].

Urethral closure pressure and sphincter volume diminish after childbirth [START_REF] Toozs-Hobson | The effect of mode of delivery on pelvic floor functional anatomy[END_REF][START_REF] Van Geelen | The urethral pressure profile in pregnancy and after delivery in healthy nulliparous women[END_REF]. Urethral closure pressure is lower in primiparous stress incontinent women than in primiparous continent women [START_REF] Delancey | Why do women have stress urinary incontinence?[END_REF]. Urethral electromyography is altered in stress incontinent women [START_REF] Kenton | Continent women have better urethral neuromuscular function than those with stress incontinence[END_REF].

The pathophysiological mechanism of this alteration of the urethral sphincter is not wellknown, and it may be related to pregnancy itself. The Childbirth and Pelvic Symptoms study has shown an alteration of urethral electromyographic parameters during pregnancy that persists six months after delivery [START_REF] Weidner | Change in urethral sphincter neuromuscular function during pregnancy persists after delivery[END_REF]. The diminution of closure pressure and sphincter volume observed following childbirth is similar in women delivered by cesarean section and in women delivered vaginally, and it persist six months after delivery [START_REF] Toozs-Hobson | The effect of mode of delivery on pelvic floor functional anatomy[END_REF][START_REF] Van Geelen | The urethral pressure profile in pregnancy and after delivery in healthy nulliparous women[END_REF].

The pelvic floor function may be altered during vaginal delivery by muscle injuries as avulsion, widening levator hiatus, diminution of reflex pelvic floor activation on coughing, or increased urethral mobility at stress [START_REF] Toozs-Hobson | The effect of mode of delivery on pelvic floor functional anatomy[END_REF][START_REF] Dietz | Does childbirth alter the reflex pelvic floor response to sudden increases in intra-abdominal pressure?[END_REF][START_REF] Delancey | The appearance of levator ani muscle abnormalities in magnetic resonance images after vaginal delivery[END_REF][START_REF] Dietz | Levator trauma after vaginal delivery[END_REF]. The relation between these lesions and de novo postnatal SUI remains uncertain. DeLancey observed an association between postnatal SUI and urethral mobility [START_REF] Delancey | Why do women have stress urinary incontinence?[END_REF]. On the other hand, the same team did not observe any association between major levator ani injury and postnatal UI or SUI [START_REF] Heilbrun | Correlation between levator ani muscle injuries on magnetic resonance imaging and fecal incontinence, pelvic organ prolapse, and urinary incontinence in primiparous women[END_REF]. Similarly, Shek did not find any association between de novo postnatal SUI and levator avulsion occurrence, or increased urethral mobility [START_REF] Shek | The effect of childbirth on urethral mobility: a prospective observationnal study[END_REF]. Some obstetrical alterations of the pelvic floor may be reversible, while others are not. As regards women having vaginally delivered, measurement of the levator hiatus and bladder neck mobility carried out 6 months postpartum yields figures similar to those of prenatal measurements [START_REF] Toozs-Hobson | The effect of mode of delivery on pelvic floor functional anatomy[END_REF]. For postpartum levator ani avulsions, Shek and Dietz observed a recovery in two cases (out of 12) (60).

Postpartum remission of SUI (Figure 1) may be explained by partial healing [START_REF] Delancey | Graphic integration of causal factors of pelvic floor disorders: an integrated life span model[END_REF], or as a result of compensation that might restore continence despite persistence of the lesions. The compensation may be due to the pelvic floor muscles, which assume a urethral support role, and it may be favored by pelvic floor training. Since muscle force diminishes as one grows older, SUI may recur as the years pass (Figure 2). If the obstetrical trauma hypothesis were the only explanation, then the protective effect of a planned cesarean delivery would in all likelihood be observed at any age, the incidence of SUI (appearing after childbirth) would be closely associated with the mode of delivery, and SUI prevalence would necessarily be similar in nulliparas and in women having given birth by cesarean section.

According to the reversible trauma hypothesis with a proper effect of pregnancy, postnatal remission of SUI (Figure 1) would find its explanation in the resolution of hormonal and metabolic changes associated with pregnancy and the spontaneous healing of the traumatic lesions due to vaginal childbirth. Concerning the relapse associated with aging (Figure 2), this would be provoked by tissue aging and be aggravated in case of multiple exposures to pregnancy. If the reversibility hypothesis were altogether accurate, then the cesarean section protective effect would disappear as time passes, and SUI at a mature age would be more closely connected with the number of pregnancies than with the mode of delivery.

Conclusion

At different stages of women's lives, at least five etiological factors can be considered to help explain SUI occurrence: a congenital factor, obesity, aging, pregnancy and vaginal delivery.

At different moments, these factors are likely to affect different parts of the urethral sphincter complex. Some, such as aging, are continuous; others, like pregnancy, are occasional; still others, such as weight gain, are reversible. This is why risk factors are significant during certain phases of women's lives, and not during others. At present we do not have a sufficient number of cohorts to allow us to describe the complete natural history of SUI in women or to distinguish age-associated risk from the risk linked with pregnancy and the risk associated with delivery. Childbirth is followed by remission of SUI, and the main indicators measuring pelvic floor quality show improvement postpartum. Cesarean section is associated with a lower rate of postpartum SUI and less frequent recourse to remote SUI surgery. However, it is difficult to postulate this association as causal. In fact, women undergoing cesarean section differ from women undergoing vaginal childbirth with regard to several prenatal characteristics connected with the UI risk. Moreover the one available randomized trial (planned vaginal delivery vs. planned cesarean for breech delivery at full term) is not conclusive.

Further studies still need to be elaborated so as to better understand the respective impacts of pregnancy and obstetrical trauma on urinary continence.

Legends of figures

Urinary incontinence (continuous line), and stress urinary incontinence (broken line) prevalence (with 95% confidence interval) during first pregnancy and postpartum in longitudinal studies [START_REF] Brown | Effects of prolonged second stage, method of birth, timing of cesarean section and other obstetric risk factors on postnatal urinary incontinence: an Australian nulliparous cohort study[END_REF].

Figure 2 Theoretical prevalence of stress urinary incontinence (SUI) in women in accordance with the number of pregnancies, types of deliveries, and age. The large black line represents SUI prevalence of stress urinary incontinence in nulliparas. The double black line its course during the first pregnancy and afterwards, and the double gray line its course during the second pregnancy and afterwards in women delivered vaginally. The continuous line depicts SUI in accordance with the hypothesis of a reversible obstetrical trauma, and the broken line in accordance with the hypothesis of an irreversible traumatism. Women with cesarean deliveries, thin black line, are located somewhere between nulliparas (large black line) and primiparas (continuous double line) according to the theoretical model of pregnancy-specific effect.

  2) [LE3b]. Leijonhufvud cross-checked the Swedish birth registry from 1973 to 1982 with the records of surgical interventions from 1973 to 2005 (20), and his study shows a heightened likelihood of SUI surgery in cases of vaginal childbirth (compared with cases of cesarean childbirth) [LE3b]. The randomized trial by

[

  LE2b]. Snooks showed that the terminal latency of the pudendal nerve had returned to normal at eight weeks postpartum, as had perineal descent after five years [LE2b] (58,59). In the work carried out by Toozs-Hobson (27) no difference was observed as regards bladder neck mobility or levator hiatus between prenatal dimensions and the dimensions reported six months after vaginal childbirth [LE2b]. Later postpartum, between 3-6 months and 2-3 years, Shek and Dietz did not observe any improvement in levator hiatus dimension (60) [LE2b].

  indeed showed striking SUI concordance among sisters, regardless of parity [LE3b]. In EPINCONT (70), SUI prevalence is higher in the daughters of mothers with declared UI (OR 1.3; IC95% 1.2-1.4) [LE3b]. A case-control survey revealed three times more cases of SUI in a first-degree relative when the woman in question suffered from SUI [LE3b] (71). Dietz et al found a correlation for bladder neck mobility (r= 0.61) in nulliparous identical female human twins [LE3b] (72) The authors estimate that 59% of the variance in bladder neck mobility in nulliparas may be explained by recessive genes, while the rest of the variance would be due to individual environmental factors. Pregnancy may act as a presenting symptom of underlying individual SUI-related fragility of partially congenital origin. King and Freeman (73) have shown that prior to delivery, primiparas who later suffer from postnatal SUI already present higher urethral mobility than those who will remain continent (7.1 vs. 4.8 mm) [LE2b]. Postpartum, resolution of the physiological, hormonal and anatomical modifications specific to pregnancy may explain the SUI remission observed over the months following delivery. At a later time, the effects of aging could provoke an additional impairment of continence mechanisms, that is to say, a new alteration leading to a SUI relapse in women with individual susceptibility.

Tetzschner ( 82 )

 82 highlighted the lengthening of pudendal nerve terminal latency 12 weeks after childbirth, and the figure is similar in women having benefited from spontaneous vaginal delivery and in those having undergone a cesarean section [LE3b]. Weidner (83) discovered an alteration of urethral sphincter electromyography (EMG) in pregnant nulliparas as compared with non-pregnant nulliparas of the same age, and this alteration was still present six months after delivery [LE3b]. Dietz (84) observed increased urethral mobility in pregnant as opposed to non-pregnant nulliparas [LE3b].

  Estrogen administration after menopause favours the occurrence of SUI [LE1b] (88). Several metabolic disorders observed in the metabolic syndrome (enlarged waist, high triglyceride level, low HDL cholesterol, arterial hypertension, hyperglycemia) are encountered during pregnancy and are likewise associated with urinary disorders in the mature adult. In a population of 984 women aged over 40 years, Kim et al (89) discovered an association between the presence of at least three elements of the metabolic syndrome and SUI [LE3b]. Brown et al (90) underlined an association between SUI and diabetes or raised fasting glycemia levels in 1461 women [LE3b].

Figure 2

 2 Figure 2

Acknowledgments

We wish to thank Prof. Peter Dietz for his critique and intensive review of this work, both of which have greatly contributed to its improvement.

Conflict of Interest:

The authors have no conflict of interest to declare.


Laura O'brien

Paul Christodoulides
email: paul.christodoulides@cut.ac.cy

Emiliano Renzi
email: emiliano.renzi@ucd.ie

Denys Dutykh
email: denys.dutykh@univ-savoie.fr

Frédéric Dias
email: frederic.dias@ucd.ie

F R Éd

Éric Dias

The Force of a Tsunami on a Wave Energy Converter

Keywords: Tsunami, wave energy converter, load

come   L'archive ouverte pluridisciplinaire

Introduction

Wave energy devices are slowly becoming a reality. Various prototypes are now being tested in harsh sea conditions (storms). What about tsunamis? Even if offshore wind turbines seem to have survived the 2011 tsunami in Japan, it is legitimate to ask whether WECs will resist tsunamis. For deep sea WECs, such as Pelamis (Figure 1), tsunamis are not anticipated to be a threat since they are located far from the shore (the present Pelamis prototype operating at EMEC, Orkney, is located 2 km from the shore). On the other hand, for nearshore WECs, such as Oyster (Figure 2) it is important to take a closer look at the effect of tsunamis (the present Oyster prototype operating at EMEC, Orkney, is located 500 m from the shore). Unfortunately there is very few tsunami wave data away from the shoreline. One exception is the Mercator yacht, anchored 1.6 km away from the shore in Thailand during the 2004 Indian Ocean tsunami. The water depth was about 12 -13 m and the yacht experienced four major waves, one "depression" wave (2.8 m) and three "elevation" waves (3.8 m, 1.7 m and 4.2 m) [START_REF] Rossetto | Physical modelling of tsunami using a new pneumatic wave generator[END_REF]. And the problem is quite different from the problem of wave forces acting on flap-type storm surge barriers [START_REF] Tomita | Wave Forces Acting on Flap-type Storm Surge Barrier and Waves Transmitted on It[END_REF] because the periods involved are different.

ǫ i = A i h i δ i = h i λ i γ i = A i λ i U ri = ǫ i δ 2 i
Table 1. Dimensionless numbers

h 1 = 3 km h 2 = 31 m ǫ i 3.3x10 -4 10 -1 δ i 0.03 3x10 -3 γ i 10 -5 3x10 -4 U ri 0.367 1.08x10 4 Table 2.
Values of dimensionless numbers for tsunami at two positions i = 1, 2 according to Equations 1.2. (1977) [8] considers the amplification of tsunamis which advance toward shore over a gentle slope using Green's law for tsunamis,

Kajiura

A 1 A 2 = λ 2 λ 1 1 2 = h 2 h 1 1 4 . (1.1)
where A i and λ i are the amplitude and wavelength of a tsunami at a depth of h i , at two different positions i = 1, 2. Four dimensionless parameters are defined in Table 1. These parameters are used to compare the importance of linear, non linear and dispersive effects. Green's Law (1.1) implies that for a tsunami at 2 different positions 1 and 2,

ǫ 2 = ǫ 1 h 1 h 2 5/4 δ 2 = δ 1 h 1 h 2 -1/2 γ 2 = γ 1 h 1 h 2 3/4 U r2 = U r1 h 1 h 2 9/4 . (1.2)
If one takes a typical tsunami wave with A 1 = 1 m, h 1 = 3 km, λ 1 = 100 km then the corresponding dimensionless parameters arising from Table 1 are shown in column 1 of Table 2.

These values indicate that linear theory can be used to describe the behaviour of the wave up to a certain depth and a slight dispersive effect should be included for the wave travelling over very large distances. However, when considering small travel distances over a few kilometers the linear shallow water equations are sufficient. As the wave approaches the shore, finite amplitude (non linear) effects come into play when ǫ 2 ≈ 10 -1 . According to Equations (1.2) this occurs at a depth of h 2 = 31 m. Assuming a sea bed slope of 0.02 this occurs at a distance of approximately 1.5 km from the shore which is about one seventh of the wavelength of a tsunami with a period of 10 minutes. The dimensionless parameters corresponding to this depth are shown in column 2 of Table 2. The wave steepness is γ 2 ≈ 0.0003 and the Ursell number is U r2 ≈ 10 4 ≫ 1, indicating that dispersion is relatively minor compared with the non-linearity except for the front part of the wave. From these considerations, it is reasonable to conclude that at this distance from the shore there is a shift in importance from linear to non linear effects. Therefore linear shallowwater equations used offshore should be matched to the inner solution of the nonlinear shallow-water equations at a distance from shore of about a seventh of a wavelength of the tsunami. As a first approximation, linear theory is used here to predict the force exerted on a WSC.

Model description

We consider here the following idealized problem: a flap-type structure mounted at the sea bottom pierces the surface of the ocean. The structure is assumed to be fixed. What is the load on the flap due to a tsunami wave?

Authorities tend to classify the different forces acting on a structure due to a tsunami in the following way. In the document entitled "Development of design guidelines for structures that serve as tsunami vertical evacuation sites" [16] several forces are described by a number of design codes [7, 5, 1] which include loading requirements based on equations given by [4]:

• Hydrostatic Forces: Occur when standing or slowly moving water encounters a structure. They are caused by an imbalance of pressure due to a differential water depth on opposite sides of structure and act perpendicular to the surface. • Buoyancy Force: Concerns structures with little resistance to lift eg. light wood frame buildings, basements, or swimming pools. These act vertically through the center of mass of the displaced volume. • Hydrodynamic Force: Caused by water flowing at a moderate to high velocity around a structure. These are a combination of the lateral forces caused by the pressure forces from the moving mass of water and the friction forces generated as the water flows around the structure. They include frontal impact, drag along the sides, and suction on the downstream side. This force is a function of flow velocity, fluid density and structural geometry. • Breaking Wave Force: This force is taken as the hydrodynamic force if the wave breaks on the structure. When considering a breaking wave, generally the two structures of interest are piles/columns and walls. Waves that break obliquely incident to the wall (not perpendicular) result in a lower force. The net force is assumed to act at the still water elevation. It is also assumed that a breaking wave against a wall causes a reflected or standing wave and the crest of the wave is some height above the still water elevation. • Surge Force: Another variety of hydrodynamic force caused by the leading edge of a surge of a tsunami impinging on a structure. • Impact Force: Results from debris or any object transported by floodwaters, striking against a structure.

Assuming that the load is mainly hydrodynamic, even within this idealized framework it is not clear what the main force is going to be. The suggested loading for a solid wall facing the shoreline by [16] (ignoring impact forces) is given by one of three forces: a breaking wave force

F brkw = (1.1C p + ς)ρgd 2 s w, (2.1)
a surge force

F s = 4.5ρgh 2 , (2.2)
or a hydrodynamic force

F d = 1 2 ρC d A r u 2 p , (2.3)
where ρ is the water density, g is the acceleration due to gravity, C p ∈ [1.6, 3.5] is the dynamic pressure coefficient, ς = 1.9 or 2.4 is a hydrostatic coefficient, w is the width of the wall, A r is the area of the wall, h is the surge height, C d ≈ 1.5 is the drag coefficient, u p = 2 √ gd s is the design flood velocity and d s is the surge depth. Furthermore, Chen & Scawthorn (2003) [START_REF] Chen | Earthquake Engineering Handbook[END_REF] give the form proposed by [START_REF] Cross | Tsunami surge forces[END_REF] [3] for the force on a seawall, of width w,

F wall = 1 2 ρgwη 2 (x wall , t) + C f (t)ρwη(x wall , t)v 2 ,
where η(x wall , t) is the water surface elevation on the wall, v is the surge or bore velocity, w is the width of the wall and

C f = (1 + tan θ 1.
2 ) where tan θ is the slope of the front face of the bore as it impacts the wall which is estimated using experimental and theoretical data.

These formulas produce values that are highly dependent on coefficient estimates and on the design flood velocity, which is highly conservative u p = 2 √ gd s , twice that of the wave speed of a tsunami given by shallow water theory. Another way (a more precise way) to look at forces is through the integral of the stress tensor. Since viscous effects are neglected, the only contribution comes from the pressure term. In turn the pressure term can be evaluated through Bernoulli's equation. In the linear model p = -ρ(gz + Φ t ), where -ρgz is the hydrostatic pressure and -ρΦ t is simply the dynamic pressure. In the fully nonlinear model p = -ρ(gz + Φ t + 1 2 |∇Φ| 2 ), so the dynamic pressure has an additional term -ρ 1 2 |∇Φ| 2 . We focus here on the linear model and calculate the dynamic pressure difference across a fixed plate.

Linear model

Consider an infinite array of equally spaced thin plates in the open ocean used for the purpose of wave energy conversion (e. g. WECs). The analysis of the scattering problem, in which the flaps are held fixed in incoming waves, is used here to calculate the velocity potential and so, the pressure exerted on the system. This is important in order to investigate whether an array of nearshore WECs would survive the impact from a tsunami. Periodicity of the problem allows the geometry to be reduced to that of a single plate within two waveguides at a mutual distance b, as shown in Figure 3.

With reference to Figure 3, the plate is represented by a rectangular box of width w and thickness 2a, hinged along a straight foundation at a distance c from the bottom of the ocean of depth h. The plate is in the middle of a channel of total width b. A plane reference system of coordinates x = (x, y, z) is also set, with x on the centre line of the channel, y along the axis of the plate and z positive upwards. Monochromatic waves of frequency ω are incoming from the left with wave crests parallel to the plate.

3.1. Model equations. The theoretical basis of the mathematical model is provided by [START_REF] Renzi | Resonant behaviour of an oscillating wave energy converter in a channel[END_REF] and summarised here. Within the framework of a linear potential-flow theory, the velocity potential Φ(x, y, z, t) must satisfy the Laplace equation

∇ 2 Φ(x, y, z, t) = 0 (3.1)
in the fluid domain, with ∇ the nabla operator. On the free-surface, the kinematic-dynamic boundary condition

∂ 2 Φ ∂t 2 + g ∂Φ ∂z = 0, z = 0
is applied, with g the acceleration due to gravity. Absence of normal flux at the bottom and through the lateral walls of the channel requires

∂Φ ∂z = 0, z = -h; ∂Φ ∂y = 0, y = ±b/2 (3.2)
respectively.

A no-flux boundary condition must be applied on the lateral surfaces of the plate, yielding

∂Φ ∂x = 0, x = ±a, |y| < w/2. (3.3)
Since the total thickness of the plate 2a ≪ b, the thin-plate approximation can be made [9] by which the boundary condition on the plate (3.3) is restated at x = ±0. Finally, the reflected and transmitted wave field respectively on the weather side and the lee side of the plate must be both outgoing at large distances from the origin. where

φ I (x, y, z) = - iA 0 g ω cosh kh cosh k(z + h)e -ikx (3.6)
is the potential of the incident wave and φ D the potential of the diffracted waves (see [9,[START_REF] Mei | Theory and applications of ocean surface waves: Linear aspects[END_REF]).

In Equation (3.6) A 0 and k are respectively the wave amplitude and wavenumber, the latter depending on the wave frequency according to the dispersion relation ω 2 = gk tanh kh. Application of the Green integral theorem to the governing system of equations (3.1)-(3.6) yields an integral equation for φ D with a strong kernel singularity. The latter is solved with a new analytical method based on the careful treatment of the singularity (for details see [START_REF] Renzi | Resonant behaviour of an oscillating wave energy converter in a channel[END_REF]).

The solution is expressed in terms of a series of Chebyshev polynomials of the second kind U p :

φ(x, y, z) = φ I (x, y, z) - 1 4 √ 2 Re igA 0 w kx cosh k(z + h) gh + (g/ω) 2 sinh 2 kh 1/2 × N p=0 β p +∞ m=-∞ 1 -1 1 -u 2 1/2 U p (u) H (1) 1 k x 2 + (y -1 2 wu -m) 2 x 2 + (y -1 2 wu -m) 2 du . (3.7)
In the latter expression H

(1) 1

the Hankel function of first kind and first order. Finally, the β p , p = 0 . . . N ∈ N, are the complex solutions of a system of linear equations which ensures that the boundary condition on the plate (3.3) is respected. This system is solved numerically with a collocation scheme, therefore the solution (3.7) is partly numerical.

Once the potential is known, the dynamic pressure on the plate can be found

p = -ρ ∂Φ ∂t = ρ Re iωφe -iωt . (3.8)
This is directly related to the hydrodynamic force F , acting on the plate by integrating over the wet surface of the body S B , i. e.

F = S B p(x, y, z, t) dA, (3.9)
where dA = dxdy the infinitesimal area on the plate. Substituting Equation (3.7) into (3.8) at x = 0 ± , the pressure jump across the plate is given by ∆p = p(0 -, y, z, t) -p(0

+ , y, z, t) = ρRe{iω[φ D (0 -, y, z) -φ D (0 + , y, z)]e -iωt } (3.10) since φ I (0 -, y, z) -φ I (0 + , y, z) = 0. (3.11)
Therefore, the maximum pressure jump can be found

|∆p| = ρgωA 0 (1 -u 2 ) N p=0 β p U p (u) √ 2 cosh k(z + h) h + ω -2 sinh 2 kh 1/2 , (3.12)
where u = 2y/w. Note that Equation (3.12) is truncated at p = N making it an approximate expression, which converges as N → ∞.

Results

The force of a tsunami on an array of nearshore WECs at a depth of 10.9 m is analysed. If the tsunami has an amplitude of 1 m offshore at a depth of 3 km, then according to Green's law for tsunamis (1.2) the amplitude of the wave will be approximately 4 m when it hits the devices. Applying the linear model from [START_REF] Renzi | Resonant behaviour of an oscillating wave energy converter in a channel[END_REF] we approximate the WECs as an array of a fixed plates with a spatial period b = 91.6 m and width w = 18 m and determine the pressure exerted on one plate from a tsunami with amplitude 4 m and period 10 minutes. The pressure jump across the plate is shown in Figure 4 (a). It is plotted against y which runs along the axis of the plate and calculated at 6 equally spaced depths from the still water level to the sea floor. The greatest overall pressure difference is felt at the center of the plate (y = 0) and is zero at the edges of the plate (y = ±9 m) but is invariant with depth. The maximum value is ∆p ≈ 3x10 3 N/m 2 = 0.03 bar.

In order to compare these results to a standard sea state, the pressure jump exerted by a typical swell with amplitude 3 m and period 5 s impacting on the plate is shown in Figure 4 (b). This clearly shows how the pressure changes with depth, the maximum effect felt at the free surface and decreasing towards the sea floor. Also the magnitude is much greater than that from the tsunami, with a maximum ∆p ≈ 3x10 5 N/m 2 = 3 bar. From these results we can conclude that the tsunami load exerted on the plate does not vary with depth since it is such a long wave relative to the depth. Moreover, the magnitude of load exerted by the tsunami is approximately 100 times less than that of a normal swell. We can therefore assume that an array of nearshore WECs would easily withstand the force from a tsunami according to linear theory. However, as previously noted, non linear effects will start to become important at approximately 1.5 km from the shore so further research into the non linear effects on the plate needs to be done.

Conclusions

The hydrodynamic load of a tsunami on an array of nearshore WECs was investigated here. First, the difficulty in knowing whether to use linear or non linear theory was demonstrated, showing that WECs of this type are usually located close to the boundary of dominance between linear and non linear effects. Different forces suggested by standard tsunami design codes were reviewed displaying the variety of formulas and their reliance on estimated coefficients and a conservative velocity estimate. Applying the linear model from [START_REF] Renzi | Resonant behaviour of an oscillating wave energy converter in a channel[END_REF] to an array of fixed plates, a first approximation for the hydrodynamic loading on a WEC was calculated through determining the jump in the -ρΦ t term. Results showed that the loading for a typical tsunami was invariant with depth and maximum loading is felt at the center of the plate. By comparison with the loading from a typical swell, it was shown that the maximum force of a tsunami on a nearshore WEC will be approximately one hundreth of the magnitude of a regular sea state. We therefore conclude that an array of WECs will withstand a tsunami. However, further research needs to be done on the non linear effects on nearshore WECs, in particular the effects of a sloping sea bed and multiple waves. [START_REF] Stefanakis | Local Runup Amplification by Resonant Wave Interactions[END_REF] [13] demonstrated resonant phenomena between the incident wavelength and the beach slope within the framework of the nonlinear shallow water equations in one dimension for multiple tsunami waves. A comparison between the velocities of resonant and non resonant states from [START_REF] Stefanakis | Resonant Long-Wave Run-Up On A Plane Beach[END_REF] are shown in Figure 5. Furthermore, if after the first wave recedes the device is left on dry land, a second wave may act as a shock on the plate and do more damage than it would to a partially submerged device. This effect is demonstrated using a two dimensional non linear shallow water solver, VOLNA [START_REF] Dutykh | The VOLNA code for the numerical modeling of tsunami waves: Generation, propagation and inundation[END_REF] in Figure 6 (note the wave is not exactly symmetric due to an unstructured triangular mesh being employed). We believe that dangerous configurations could be found with more detailed investigations.

Figure 1 .Figure 2 .

 12 Figure 1. Pelamis Wave Power Device, picture from http://www.pelamiswave.com

Figure 3 .

 3 Figure 3. Geometry of the array of WECs (plan view).

3. 2 .

 2 Solution. The system of governing equations (3.1)-(3.3) is solved via the introduction of a complex spatial potential φ(x, y, z) such that Φ = Re φ(x, y, z)e -iωt . (3.4) Due to the linearity of the problem, the spatial potential φ is analysed by the classical decomposition φ = φ I + φ D , (3.5)

Figure 4 .

 4 Figure 4. Various jumps in pressure across an 18 m plate for a typical tsunami (a) and a typical swell (b), in a depth of h = 10.9 m at 6 depths from the free surface to the ocean floor.

Figure 5 .

 5 Figure 5. (a) Non resonant and (b) resonant non dimensional velocities from a monochromatic wave at x = L = 12.5 m on a sloping beach with slope tan θ = 0.13 and initial shoreline at x = 0 [14].

  (a) Bathymetry of a fixed plate on a sloping sea bed. (b) Free surface η after a tsunami has inundated the shore and it begins to recede. (c) 45 seconds later, the plate is left on dry land. (d) Another 45 seconds later, a subsequent wave impacts the plate.

Figure 6 .

 6 Figure 6. VOLNA simulation from multiple tsunami waves hitting a 13 m high plate in an initial depth of 10m on a sea bed with slope 0.03 (plane view).

Acknowledgements

The authors would like to acknowledge the support provided by the Science Foundation Ireland (SFI) under the project High-end computational modelling for wave energy systems, by the Framework Program for Research, Technological Development, and Innovation of the Cyprus Research Promotion Foundation under the Project AΣTI/0308(BE)/05, by the Irish Research Council for Science Engineering and Technology (IRCSET) and by Aquamarine Power.