# Kinematic stretching¶

The linear wave theory is based on the linearization of the free surface condition on the mean water level $$z=0$$. In case of irregular wave profile, the superposition of regular wave components can lead to unrealistic velocity estimation particularly in the crest. Stretching method are applied on the scaling factor $$E(z)$$ to overcome this issue.

$E(z) = \frac{\cosh(k(z+H))}{\sinh(kH)}$

The different methods are presented in the following.

## Vertical stretching:¶

The vertical stretching method assumes that the scaling factor above the mean water level is constant and equals to its value at the mean water level $$E(0)$$.

$E(z) = \Biggl \lbrace { E(0) \text{ if } z \geq 0 \atop E(z) \text{ otherwise} }$

This method results in an abrupt, non-physical, $$E(z)$$ evolution when crossing the mean water level.

## Extrapolation stretching:¶

To correct this non-physical evolution, the extrapolation stretching method introduces a correction to the vertical method.

$E(z) = \Biggl \lbrace { E(0) + z \frac{\partial E}{\partial z}(0) \text{ if } z \geq 0 \atop E(z) \text{ otherwise} }$

## Wheeler stretching:¶

This method stretches the vertical scale factor, in order to obtain at $$z=\eta$$, the kinematic given by the linear theory at $$z=0$$. The corresponding transformation changes $$z$$ into $$z'$$ :

$z' = H \frac{z-\eta}{H+\eta}$

where $$H$$ is the water depth and $$\eta$$ is the instantaneous wave elevation given by the linear wave model.

The wheeler stretching is defined for : $$H < z < \eta$$

## Chakrabarti stretching:¶

In this method, proposed by Chakrabarti [Chakrabarti1971], the water depth $$h$$ in the denominator is corrected by the wave elevation $$\eta$$ :

$E(z) = \frac{\cosh(k(z+h))}{\sinh(k(h+\eta))}$

## Delta-stretching:¶

This method is a coupling between the Wheeler and the extrapolation stretching, proposed by Rodenbush and Forristall [Rodenbush1986]. While $$E(z)$$ is still replaced by $$E(z')$$, the transformation $$z \longrightarrow z'$$ becomes

• for $$z > -H_{\Delta}$$ : $$z' = (z + H_{\Delta}) \frac{H_{\Delta} + \Delta \eta}{H_{\Delta} + \eta} - H_{\Delta}$$
• for $$z < -H_{\Delta}$$ : $$z' = z$$

$$\Delta$$ is a parameter taken between 0 and 1, and $$H_{\Delta}$$ is the water height on which the stretching is applied.

When $$z'$$ is positive, the wave kinematic is linearly extrapolated from the one at $$z'=0$$, the same way it is done in the extrapolation stretching.

The Delta-stretching method can be downgraded to one of the two methods, in using specific values for $$H_{\Delta}$$ and $$\Delta$$:

• for $$H_{\Delta} = H$$ and $$\Delta = 0$$, the method is equivalent to the Wheeler stretching, as long as $$z'<0$$.
• for $$H_{\Delta} = H$$ and $$\Delta = 1$$, it is equivalent to the extrapolation stretching.

Rodenbush and Forristall recommend using $$\Delta = 0.3$$ and $$H_{\Delta} = H_S/2$$. However Molin [Molin2002] suggests that $$H_{\Delta} = H$$, the water depth, is usually chosen.

## References:¶

 [Chakrabarti1971] Chakrabarti, SK, Discussion on “dynamics of single point mooring in deep water”. J Waterwayss, Harbour and Coastal Eng Div ASCE, Vol 97, 588-590, 1971
 [Rodenbush1986] Rodenbusch, G, Forristall, GZ, An empirical model for random wave kinematics near the free surface Proc Offshore Technology Conf, Paper 5098, 1986
 [Molin2002] Molin, B., Hydrodynamique des Structures Offshore, Editions Technip, 2002