Question #43172

Two coherent monochromatic plane electromagnetic waves, Wave 1 and Wave 2, propagate in a vacuum. Wave 1 propagates along the positive $z$-axis with its electric field polarized along the $x$-axis: $\vec{E}_1(z,t) = E_0 \cos(kz - \omega t) \hat{i}$. Wave 2 propagates along the positive $y$-axis with its electric field also polarized along the $x$-axis: $\vec{E}_2(y,t) = E_0 \cos(ky - \omega t) \hat{i}$. Here, $E_0$ is the electric field amplitude in $\text{V/m}$, $k$ is the wave number in $\text{rad/m}$, and $\omega$ is the angular frequency in $\text{rad/s}$. Let $\langle u \rangle$ denote the time-averaged total electromagnetic energy density at a spatial coordinate $(x, y, z)$. What is the ratio of the maximum to the minimum value of $\langle u \rangle$ in the region where the two waves superimpose?