Wave kinematics

Using the linear wave theory and the conversion of wave crests, the wave propagation velocities in spatial space within Cartesian framework and spectral space can be obtained from the kinematics of a wave train (Whitham, 1974; Mei, 1983)

$$\frac{d\vec{x}}{dt} = (c_x,c_y) = \vec{c_g} + \vec{u} = \frac{1}{... ...\vec{k}\vert d)}\right ) \frac{\sigma \vec{k}}{{\vert\vec{k}\vert}^2} + \vec{u}$$

$$\frac{d\sigma}{dt} = c_\sigma = \frac{\partial \sigma}{\partial d... ...abla_{\vec{x}} d\right ) -c_g \vec{k} \cdot \frac{\partial \vec{u}}{\partial s}$$ (2.13)

$$\frac{d\theta}{dt} = c_\theta = -\frac{1}{k} \left ( \frac{\parti... ...ial d}{\partial m} + \vec{k} \cdot \frac{\partial \vec{u}}{\partial m} \right )$$

where $c_x$, $c_y$ are the propagation velocities of wave energy in spatial $x-$, $y-$space, $c_\sigma$ and $c_\theta$ are the propagation velocities in spectral space $\sigma-$, $\theta-$space, $\vec{u} = (u_x,u_y)$ is the ambient current, $d$ is the water depth, $s$ is the space co-ordinate in the wave propagation direction of $\theta$ and $m$ is a co-ordinate perpendicular to $s$. The expression for $c_\theta$ is presented here without diffraction effects. These are treated separately in Section 2.5.4.


Furthermore,

  $$\vec{k} = (k_x,k_y) = (\vert\vec{k}\vert\cos \theta, \vert\vec{k}\vert\sin \theta)$$ (2.14)

and the ambient current $\vec{u}$ is assumed to be uniform with respect to the vertical co-ordinate. In addition, the operator $d/dt$ denotes the total derivative along a spatial path of energy propagation, and is defined as

  $$\frac{d}{dt} = \frac{\partial}{\partial t} + (\vec{c_g} + \vec{u}) \cdot \nabla_{\vec{x}}$$ (2.15)