Derivation of the ray equations

Mathematically rays are characteristics of the dispersion equation. In an inhomogeneous medium the dispersion equation reads:
       $F(\omega ,k$_i,t,x_i)=0
Here $\omega$ and $k$_i are derivatives of the phase function $\psi (t,x$_i), i.e. $\omega \; =\; \partial \psi /\partial t$ and $k$_i = -∂ψ/∂x_i.

Thus equation (1) is an example of a nonlinear partial differential equation of first order.
In general such an equation is written as: $F(\psi ,\; \psi$_,i, x_i)=0 where $\psi$_,i are the partial derivatives of $\psi$ with respect to independent coordinates $x$_i. Such an equation has characteristics for which holds (see e.g. Webster, 1955, section 24, eq. 45):
       $dx$_i/du = -∂F/∂ψ_,i
       $d\psi /du\; =\; \Sigma \; \psi$_,i ∗ ∂F/∂ψ_,i,
       $d\psi$_,i/du = -∂F/∂x_i - ψ_,i ∗ ∂F/∂ψ

Equation (1) is written more specifically as:
       $\omega -k$_iU_i-[g k tanh(kd)]^1/2 = 0,
we obtain the following ray equations:
       $dt/du\; =\; \partial F/\partial \omega \; =\; 1$, so that the parameter $u$ in fact is time $t$.

Furthermore:
       $dx$_i/dt = -∂F/∂k_i = U_i + ∂σ/∂k ∗ k_i/k ,    where    $\sigma \; =\; [g\; k\; tanh(kd)]1/2$
       $d\psi /dt\; =\; \omega \; \ast \; \partial F/\partial \omega \; +\; \Sigma \; k$_i ∗ ∂F/∂k_i = ω - k_i U_i - k_i ∗ ∂σ/∂k_i,
       $d\omega /dt\; =\; -\partial F/\partial t\; =\; k$_i ∗ ∂U_i/∂t - ∂σ/∂d ∗ ∂d/∂t, showing that $\omega$ is constant in stationary cases.
       $dk$_i/dt = ∂F/∂x_i = -k_j ∗ ∂U_j/∂x_i - ∂σ/∂d ∗ ∂d/∂x_i

Refraction can be found from the last equation since $\theta$ is the direction of the vector $k$_i:
       $d\theta /dt\; =\; -k$_i/k ∗ ∂U_i/∂n - 1/k ∗ ∂σ/∂d ∗ ∂d/∂n,
where    $n$ is the spatial coordinate perpendicular to $\theta$.
This equation is used in SWAN together with the frequency shift equation:
$d\sigma /dt\; =\; \partial \sigma /\partial d\; \ast \; dd/dt\; +\; \partial \sigma /\partial k\; \ast \; dk/dt$

The picture below shows a ray pattern for the Haringvliet estuary in the Netherlands. At the same time it shows the weakness of the ray method: it is unsuitable to predict wave heights in areas with irregular bottom.

Action conservation

In presence of a current (also in the spacial case that the current velocity is 0) wave action is conserved. The action density is denoted as N.
If N is formulated in (x, k)-space it is constant along wave rays assuming that source terms (dissipation and generation) are 0.
       $dN/dt\; =\; \partial N/\partial t\; +\; c$_xi∂N/∂x_i + c_ki∂N/∂k_i = 0
This is equivalent to the well-known conservation equation:
       $\partial N/\partial t\; +\; \partial /\partial x$_i[c_xiN] + ∂/∂k_i[c_kiN] = 0
because it can be seen from the equations above that
       $\partial /\partial x$_i [dx_i/dt] = -∂²F/∂k_i∂x_i = -∂/∂k_i [dk_i/dt]
If N is formulated in other variables, e.g. $\sigma$ and $\theta$, the only valid form is the conservation equation:
       $\partial N\text{'}/\partial t\; +\; \partial /\partial x$_i [c_xiN'] + ∂/∂σ [c_σN'] + ∂/∂θ [c_θN'] = 0

In a grid model the conservation form is the efficient formulation; in a ray-based model it is more efficient to use the constancy of N(t,x,k). Thus also $(c$_g/k) N'(t,x,σ,θ) is constant along a wave ray.
In presence of source terms N is not constant; now one can use:
       $d/dt\; [N(t,$x,k)] = S_N(t,x,k)
or: $d/dt\; [(c$_g/k) N'(t,x,σ,θ)] = (c_g/k) S'(t,x,σ,θ).

References

Webster, A.G. (1955)

Partial Differential Equations of Mathematical Physics, Dover Publ. 1955

back