# Kinematic Stretching

Linear wave theory only provides information about the water kinematics at and below mean sea level (MSL). If the velocity or acceleration within points above MSL is of interest (i.e. in a wave crest), extrapolation or stretching methods become necessary Nestegård et al.1.

In the literature, several wave stretching methods have been introduced Nestegård et al.1, OrcaFlex2, FRyDoM3. Their general approach is to model the respective scaling factor $$E(z)$$ (introduced in Wave Kinematics) for points above MSL $$(z > 0)$$ by stretching or extrapolating its values. In the following, the three methods that have been implemented into QBlade are introduced briefly. For further information the reader is referred to Nestegård et al.1, OrcaFlex2, FRyDoM3.

## Vertical Stretching

This method assumes that all points above MSL equal the kinematic conditions at MSL $$(E(z) = 0).$$ $$E$$ below MSL is left unchanged

## Extrapolation Stretching

This method extrapolates $$E(z)$$ above MSL linearly by using its gradient along the z-axis. Again, $$E(z)$$ below MSL is left unchanged.

## Wheeler Stretching

This method modifies $$E(z)$$ so it always is stretched (or contracted) to the instantaneous wave elevation ($$z = \zeta$$). This is done by replacing z with a scaling factor $$z'$$ that modifies $$z$$ linearly so the following statement is always valid $$h < z' < 0$$, where $$h$$ Is the water depth.

1(1,2,3)

Arne Nestegård, Marit Ronæss, Øistein Hagen, Knut Ronold, and Elzbieta Maria Bitner-Gregersen. On environmental conditions and environmental loads. In DNV Recommended Practice DNV-RP-C205. 05 2006.

2(1,2)

OrcaFlex. Kinematic stretching. https://www.orcina.com/webhelp/OrcaFlex/Content/html/Waves,Kinematicstretching.htm, 2021.

3(1,2)

FRyDoM. Kinematic stretching. http://theory.frydom.org/src/environment/waves/linear_wave_theory/wave_stretching.html, 2021.