Without any assumption on which of these parameters is the most influential on wave runup, a characteristic length

parameter selleck compound L∗L∗ can be introduced for the dimensional analysis. As three dependent potential energies can be considered (i.e., EPEP, EP+, EP-), a characteristic energy E∗E∗ is also introduced. The functional relationship between the independent variables L∗L∗, E∗E∗, ββ, ρρ, and g  , can be expressed as: equation(13) R=f(L∗,E∗,ρ,g).R=f(L∗,E∗,ρ,g).The beach slope parameter is a dimensionless quantity (and an invariant in the present experiments), therefore not included in (13). The Buckingham Pi theorem (Hughes, 1993) was applied to (13) and out of this analysis (see Charvet, 2012) two dimensionless groups, Π1Π1 and Π2Π2, were formed: equation(14a,b) Π1=RL∗,Π2=(L∗)4ρgE∗.The characteristic length scale L∗L∗ may be the flume width (ww), wave amplitude (a   or a-a-), height (HH), wavelength (LL), or water depth (hh). As the present experiments were carried out in two dimensions, w   can be taken as a unit width so the following equation applies here

to a number of combinations of three possible variables for L∗L∗. The functional relationship between the two groups can be expressed as: equation(15) RL∗=ΨL∗3ρgE∗.By plotting Π1Π1 against Π2Π2 for a sample of simple combinations of L∗L∗, we can see that the data www.selleckchem.com/HSP-90.html is best described by a power law ( Fig. 9). All the data was used in these graphs. The cases where the correlation was poor were discarded. Therefore, we infer the functional relationship to be of the form: equation(16) RL∗=KL∗3ρgE∗k,where K and k are coefficients empirically determined from the dataset. Regression analysis is necessary to identify

the forms of (16) that can give a satisfactory fit to the data by optimizing values of K and k. Moreover, the scatter plots in Fig. 9 show that a significant proportion of the data tends to be clustered for large values of the predictor variable, which confirms the need for it to be partitioned into different wave categories. The uncertainty associated with (16) is quantified using a regression analysis. Linear regression can be performed using the variables in (16) by writing it as: equation(17) logRL∗=logK+klogL∗3ρgE+ε.It is necessary to find the best estimates (i.e., unbiaised) for the Gefitinib regression coefficients of the model, thus minimize the uncertainty associated with the prediction. To do so, the total error between the response data and the predicted response is reduced (as described in Appendix B) and the non-violation of the relevant statistical assumptions is checked. More details on regression analysis methods can be found in Chatterjee and Hadi (2006). To capture potential differences in runup regime between long waves, very long waves, elevated waves and N-waves, the wave data is divided into different populations.