The heat-flow equation

The diffraction or migration equation could be called the ``wavefront healing'' equation. It smooths back together any lateral breaks in the wavefront that may have been caused by initial conditions or by the lens term. The 15$^\circ$ migration equation has the same mathematical form as the heat-flow equation. But the heat-flow equation has all real numbers, and its physical behavior is more comprehensible. This makes it a worthwhile detour. A two-sentence derivation of it follows.  (1) The heat flow H_x in the x-direction equals the negative of the gradient $- \partial / \partial x$ of temperature T times the heat conductivity $\sigma$.(2) The decrease of temperature $- \partial T / \partial t$ is proportional to the divergence of the heat flow $\partial H_x / \partial x$ divided by the heat storage capacity C of the material. Combining these, extending from one dimension to two, taking $\sigma$ constant and $C \,=\, 1$,gives the equation  

$${ \partial T \over \partial t} \eq \sigma\ \left( {\partial... ... \partial x^2} \ +\ {\partial^2 \over \partial y^2} \right)\ T$$ (1)