= Solution
In a source-free vacuum, <Maxwell equations> are
$$
\nabla\cdot\mathbf E=0,\quad\nabla\cdot\mathbf B=0,\quad\nabla\times\mathbf E=-\partial_t\mathbf B,\quad\nabla\times\mathbf B=\mu_0\epsilon_0\partial_t\mathbf E.
$$
Taking a <curl> of the last two equations and using $\nabla\times(\nabla\times\mathbf A)=\nabla(\nabla\cdot\mathbf A)-\nabla^2\mathbf A$ gives the <electromagnetic wave equations>
$$
\boxed{\mathbf E_{tt}=c^2\nabla^2\mathbf E,\quad\mathbf B_{tt}=c^2\nabla^2\mathbf B,\quad c=\frac1{\sqrt{\mu_0\epsilon_0}}}.
$$
Substitution of the complex <plane wave> amplitudes into <Maxwell equations>, then taking real parts, gives
$$
\mathbf k\cdot\mathbf e=\mathbf k\cdot\mathbf b=0,\qquad\mathbf k\times\mathbf e=\omega\mathbf b,\qquad\mathbf k\times\mathbf b=-\frac\omega{c^2}\mathbf e.
$$
For a nonzero propagating wave these imply the <dispersion relation> $\boxed{\omega^2=c^2|\mathbf k|^2}$. Conversely choose any nonzero transverse complex $\mathbf e$ and set $\mathbf b=\mathbf k\times\mathbf e/\omega$ to obtain such a wave. The static uniform-field sector at $\omega=|\mathbf k|=0$ is separate from propagation.
At a <perfect conductor>, the oscillatory boundary conditions are $\mathbf n\times\mathbf E=0$ and $\mathbf n\cdot\mathbf B=0$. Surface charge and surface current allow nonzero normal electric and tangential magnetic fields. The magnetic condition here concerns the time-dependent wave field; a perfect conductor can retain a pre-existing static magnetic field.
Let $R=I-2\mathbf n\mathbf n^T$ be the orthogonal reflection across the boundary plane, so $\det R=-1$. The proposed reflected amplitudes are $\mathbf k'=R\mathbf k$, $\mathbf e'=-R\mathbf e$, and $\mathbf b'=R\mathbf b$. For an orthogonal reflection, $(R\mathbf a)\times(R\mathbf d)=-R(\mathbf a\times\mathbf d)$. Hence
$$
\mathbf k'\times\mathbf e'=\omega\mathbf b',\qquad\mathbf k'\times\mathbf b'=-\frac\omega{c^2}\mathbf e',\qquad\mathbf k'\cdot\mathbf e'=\mathbf k'\cdot\mathbf b'=0.
$$
Also $|\mathbf k'|=|\mathbf k|$, so the reflected wave obeys the same <dispersion relation>. At $\mathbf n\cdot\mathbf x=0$, the incident and reflected phases coincide because $\mathbf k'-\mathbf k$ is parallel to $\mathbf n$. Their summed amplitudes satisfy
$$
\mathbf e+\mathbf e'=2(\mathbf n\cdot\mathbf e)\mathbf n,\qquad\mathbf b+\mathbf b'=2[\mathbf b-(\mathbf n\cdot\mathbf b)\mathbf n].
$$
The first has zero tangential part and the second zero normal part, proving both boundary conditions. Finally $\mathbf k'\cdot\mathbf n=-\mathbf k\cdot\mathbf n>0$, as required for reflection into the vacuum.
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