Lois d'ouvrages: ajout du seuil triangulaire épais

Bos (1989) gives the following relation between submergence ratio (H2/H1) and the submergence flow reduction factor (S)

Extract from Bos, M.G., 1989. Discharge measurement structures., 3rd edition. ed, Publication. International Institute for Land Reclamation and Improvement, Wageningen, The Netherlands. p.142

Abaque_Bos_1989_p142_-_WebPlotDigitilizer.csv

df <- read.table("Abaque Bos 1989 p142 - WebPlotDigitilizer.csv", sep = ";", header = F, stringsAsFactors =  F)
x <- 1- df[,2]
S <- df[,1]

modele = nls(
    S ~ sin(c1 *x^c2),
    start = list(c1 = 7, c2 = 0.4)
)
summary(modele)

## 
## Formula: S ~ sin(c1 * x^c2)
## 
## Parameters:
##    Estimate Std. Error t value Pr(>|t|)    
## c1 4.434899   0.130123   34.08   <2e-16 ***
## c2 0.608720   0.009778   62.25   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01493 on 28 degrees of freedom
## 
## Number of iterations to convergence: 8 
## Achieved convergence tolerance: 2.378e-06

formula = paste("S = sin(", round(coef(modele)[1], 4), "(1 - H2/H1) ^", round(coef(modele)[2],4), ")")

plot(x,S, main = "Submergence reduction factor S = f(1 - H2/H1)", xlab = "1 - H2/H1")
xe = seq(min(x), max(x), (max(x) - min(x))/100)
regS <- function (x) {sin(coef(modele)[1] * x ^ coef(modele)[2])}
lines(xe, regS(xe), col="green")
legend(0.07,0.4, legend = c("Bos 1989", formula), col = c("black", "green"), lty = c(NA, 1), pch = c(1, NA) )

plot(x, (regS(x) - S) / S *100, main = "Regression relative error (%)")