= Solution
In <SI> units, the <Maxwell equations> in <vacuum> with prescribed <charge density> $\rho$ and <current density> $\mathbf J$ are
$$
\nabla\cdot\mathbf E=\frac\rho{\epsilon_0},\qquad
\nabla\cdot\mathbf B=0,\qquad
\nabla\times\mathbf E=-\partial_t\mathbf B,\qquad
\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\epsilon_0\partial_t\mathbf E.
$$
When $\rho=0$ and $\mathbf J=\mathbf0$, take the <curl> of the third equation and substitute the fourth. Using the <curl of the curl identity> and $\nabla\cdot\mathbf E=0$ gives the <electromagnetic wave equation>:
$$
\nabla\times(\nabla\times\mathbf E)=-\nabla^2\mathbf E
=-\mu_0\epsilon_0\partial_t^2\mathbf E,\qquad
\boxed{\nabla^2\mathbf E-\frac1{c^2}\partial_t^2\mathbf E=0,
\quad c=(\mu_0\epsilon_0)^{-1/2}.}
$$
For example, $\mathbf E=E_0\widehat{\mathbf x}\cos(kz-\omega t)$, $\mathbf B=(E_0/c)\widehat{\mathbf y}\cos(kz-\omega t)$ is a <plane electromagnetic wave> provided $\omega=ck$. The <electric field>, <magnetic field> and propagation direction are mutually perpendicular.
The <Maxwell equations> are linear, so the <superposition principle> applies to fields. However, the <Poynting vector> is quadratic:
$$
\langle\mathbf S\rangle=\frac1{\mu_0}\langle
\mathbf E_1\times\mathbf B_1+\mathbf E_2\times\mathbf B_2
+\mathbf E_1\times\mathbf B_2+\mathbf E_2\times\mathbf B_1\rangle.
$$
One may add the average <wave intensities> only when the cross terms vanish under the actual detector average. For co-propagating <plane electromagnetic waves> of parallel <linear polarization>, averaging over optical cycles but potentially resolving the difference <frequency> gives
$$
I(t)=I_1+I_2+2\sqrt{I_1I_2}\cos[(\omega_1-\omega_2)t+\phi].
$$
If the detector integrates for $T$, the last term acquires $\operatorname{sinc}[(\omega_1-\omega_2)T/2]$, with <sinc function> convention $\operatorname{sinc}x=\sin x/x$, evaluated at the midpoint phase. Thus $T\gg1/|f_1-f_2|$ suppresses <beats>, and the infinite-time average adds the <wave intensities> exactly for distinct fixed <frequencies>. Random mutually uncorrelated <wave phases> also remove the cross terms in an ensemble average; orthogonal <linear polarizations> remove them for co-propagating waves in polarization-insensitive detection. Merely having different <frequencies> does not remove a resolved <beat> signal: two nearly equal <frequencies> with fixed relative <wave phase> are a concrete counterexample. For different propagation directions, the relevant addition is of average <Poynting vectors>, or their components through the chosen detector <surface>.
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