Derivation of the ray equations

Mathematically rays are characteristics of the dispersion equation. In an inhomogeneous medium the dispersion equation reads:
Here ω and ki are derivatives of the phase function ψ(t,xi), i.e. ω = ∂ψ/∂t and ki = -∂ψ/∂xi.

Thus equation (1) is an example of a nonlinear partial differential equation of first order.
In general such an equation is written as: F(ψ, ψ,i, xi)=0 where ψ,i are the partial derivatives of ψ with respect to independent coordinates xi. Such an equation has characteristics for which holds (see e.g. Webster, 1955, section 24, eq. 45):
       dxi/du = -∂F/∂ψ,i
       dψ/du = Σ ψ,i ∗ ∂F/∂ψ,i,
       ,i/du = -∂F/∂xi - ψ,i ∗ ∂F/∂ψ

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

       dxi/dt = -∂F/∂ki = Ui + ∂σ/∂k ∗ ki/k ,    where    σ = [g k tanh(kd)]1/2
       dψ/dt = ω ∗ ∂F/∂ω + Σ ki ∗ ∂F/∂ki = ω - ki Ui - ki ∗ ∂σ/∂ki,
       dω/dt = -∂F/∂t = ki ∗ ∂Ui/∂t - ∂σ/∂d ∗ ∂d/∂t, showing that ω is constant in stationary cases.
       dki/dt = ∂F/∂xi = -kj ∗ ∂Uj/∂xi - ∂σ/∂d ∗ ∂d/∂xi

Refraction can be found from the last equation since θ is the direction of the vector ki:
       dθ/dt = -ki/k ∗ ∂Ui/∂n - 1/k ∗ ∂σ/∂d ∗ ∂d/∂n,
where    n is the spatial coordinate perpendicular to θ.
This equation is used in SWAN together with the frequency shift equation:
dσ/dt = ∂σ/∂d ∗ dd/dt + ∂σ/∂k ∗ 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 = ∂N/∂t + cxi∂N/∂xi + cki∂N/∂ki = 0
This is equivalent to the well-known conservation equation:
       ∂N/∂t + ∂/∂xi[cxiN] + ∂/∂ki[ckiN] = 0
because it can be seen from the equations above that
       ∂/∂xi [dxi/dt] = -∂2F/∂ki∂xi = -∂/∂ki [dki/dt]
If N is formulated in other variables, e.g. σ and θ, the only valid form is the conservation equation:
       ∂N'/∂t + ∂/∂xi [cxiN'] + ∂/∂σ [cσN'] + ∂/∂θ [cθN'] = 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 (cg/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)] = SN(t,x,k)
or: d/dt [(cg/k) N'(t,x,σ,θ)] = (cg/k) S'(t,x,σ,θ).


Webster, A.G. (1955)
Partial Differential Equations of Mathematical Physics, Dover Publ. 1955


© 2012: Nico Booij