Scaling of waves between monotone slopes

Long crested waves above monotone bathymetries with different steepness are shown to be related by a time scaling. The scaling is explicitly present in the usual WKB approximation and more generally in the Hamiltonian potential theory for incompressible, irrotational inviscid fluid motion (Zakharov, 1968). The scaling uses the depth instead of the spatial distance as position marker, which is a canonical transformation in the action functional. This implies that waves above different slopes are related by a simple space-time scaling. At depths before near-coastal effects of run-up become relevant, the scaling property is valuable for understanding the wave propagation and may reduce laboratory experiments. Taking into account non-Hamiltonian coastal effects of breaking and coastal run-up, nonlinear simulations show correlations above 0.8 for waves above different slopes until a typical depth for many offshore activities of 15 m (Forrsitall, 2004). Numerical simulations with second- and third order nonlinearity are performed with HAWASSI software (Kurnia and Van Groesen, 2014), a variant of a higher order spectral method (Dommermuth and Yue, 1987; West et al., 1987). An example of the scaling is also shown to be present for an air-water CFD potential simulation (Aggarwal et al., 2020).