
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:
$\omega k$_{i}U_{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 wellknown conservation equation:
$\partial N/\partial t\; +\; \partial /\partial x$_{i}[c_{xi}N]
+ ∂/∂k_{i}[c_{ki}N] = 0
because it can be seen from the equations above that
$\partial /\partial x$_{i} [dx_{i}/dt]
= ∂^{2}F/∂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_{xi}N']
+ ∂/∂σ [c_{σ}N']
+ ∂/∂θ [c_{θ}N'] = 0
In a grid model the conservation form is the efficient formulation;
in a raybased 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
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