Saint-Venant's principle and its relatives - part 2

In part 1 of this discussion we had concluded that the Laplacian was a "zero-sum game", i.e. that a static modulation along one space dimension was exactly matched by a decay in the perpendicular dimension: \(q_x^2 + q_z^2 = 0\).

What happens for a time-dependent field distribution? For simplicity, let us assume a purely harmonic dependence: \(f(x,z,t) = F(x,z) \exp(i \omega t)\), with translation invariance along \(y\), where \(f\) stands for a component of the electric or magnetic field (in the scalar wave approximation).

The field now obeys:

\begin{equation}
\label{eq:dalemb}
\Box \, f = \frac{1}{c^2} \partial _t^2 f(x,z,t) - \underbrace {( \partial _x^2 + \partial _z^2)}_{\Delta} f(x,z,t)= 0
\end{equation}

where \(c\) is the speed of light and \(\Delta\) is the Laplacian operator discussed in part 1. The wave operator \(\Box \) is often called d'Alembertian.

Saint-Venant's principle and its relatives - part 1

This venerable principle (published in 1855) and a whole family of analogous results can be explained in a very simple, almost geometrical, manner based on the observation that "the Laplacian is a zero-sum game". Within this class of results (with applications ranging from optical microscopes to the metallic mesh on the door of microwave ovens). Saint-Venant's principle is probably the only one to have an official name, but not the easiest to understand, so we will begin by a simple example from electrostatics.

Power-law distribution of war magnitudes

While reading Steven Pinker's The better angels of our nature I stumbled upon the following argument for the magnitude of war (number of casualties) following a power-law distribution (page 220):

Recall the mathematical law that a variable will fall into a power-law distribution if it is an exponential function of a second variable that is distributed exponentially. My own guess is that the combination of escalation and attrition is the best explanation for the power-law distribution of war magnitudes.