Hyperbolic conservation law

We observe that the gradient components of the eikonal equation satisfy a hyperbolic conservation law. First, use $\u = \t_x$ to rewrite equation (1) as  

$$\t_z\ =\ \sqrt{s^2 - \u^2}.$$ (2)

Second, take the derivative of this equation with respect to x:  

$$\u_z\ =\ F_x(\u),$$ (3)

where the function $F(\u)$ is defined as

$$F(\u) \ =\ \sqrt{s^2 - \u^2}.$$ (4)

$F(\u)$ is called the conserved flux; if $F(\u) = 0$, the rays do not ``flow'' downward anymore, but travel horizontally. Thus by choosing a positive sign in front of the square root in equation (2), and by using $\u = \t_x$ instead of $w=\t_z$ as the substitution variable, we limit ourselves to time-fields with downward-traveling rays.

By following analogous reasoning for the z-derivative w of the time field, and by making the appropriate choices of square roots, we can build equations for rays traveling in other directions. These other equations will have to be solved, for instance, when the field of a point source is being computed, because the rays then move in all directions.

An alternative approach, which we will follow here, is to write the eikonal equation in polar coordinates $(\r,\th)$, 

$$\Bigl( \t_\r\Bigr) ^2 \ +\ \Bigl( {\t_\th\over r} \Bigr) ^2 \ =\ s^2(\r,\th),$$ (5)

and solve it along expanding circular fronts. The conserved-flux function in polar coordinates becomes

$$F(\u) \ =\ \sqrt{s^2 - {{\u^2}\over{\r^2}}},$$ (6)

with $\u = \t_\th$ satisfying  

$$\u_\r\ =\ F_\th(\u).$$ (7)

The above equation (and equation (3)) now has the form of a hyperbolic conservation law, or flux-conserving equation. The meaning of this equation is illustrated with an example in the next section.