= Solution
The <space of test functions> is
$$
\mathcal D(\mathbb R^n)=C_c^\infty(\mathbb R^n).
$$
A sequence $\varphi_j$ converges to $\varphi$ in $\mathcal D$ when all supports lie eventually in one <compact set> $K$ and
$$
\sup_{x\in K}|D^\alpha(\varphi_j-\varphi)(x)|\longrightarrow0
$$
for every <multi-index> $\alpha$. A <distribution> is a <linear functional> $u:\mathcal D\to\mathbb C$ such that, for every compact $K$, there are $C_K>0$ and $m_K\in\mathbb N_0$ with
$$
|\langle u,\varphi\rangle|
\leq C_K\max_{|\alpha|\leq m_K}
\sup_{x\in K}|D^\alpha\varphi(x)|
$$
whenever $\operatorname{supp}\varphi\subseteq K$. Convergence in $\mathcal D'$ is <weak convergence of distributions>[pointwise convergence on test functions].
Continuity plainly implies sequential continuity. Conversely, suppose the linear form is sequentially continuous but the displayed estimate fails for some compact $K$. For every $j$, choose $\varphi_j$ supported in $K$ such that
$$
\max_{|\alpha|\leq j}\sup_K|D^\alpha\varphi_j|\leq\frac1j,
\qquad
|\langle u,\varphi_j\rangle|\geq1.
$$
Then $\varphi_j\to0$ in $\mathcal D$, while $u(\varphi_j)\not\to0$, a contradiction. Hence the seminorm estimate holds on every compact set and $u\in\mathcal D'$.
For a <vector>[translation vector] $h$ and a <multi-index> $\alpha$, define
$$
\langle\tau_hu,\varphi\rangle
=\langle u,\varphi(\mathord\cdot+h)\rangle,
\qquad
\langle D^\alpha u,\varphi\rangle
=(-1)^{|\alpha|}\langle u,D^\alpha\varphi\rangle.
$$
These definitions extend ordinary <translation of a distribution>[translation] and <derivative>[differentiation] to distributions.
If $\tau_{te_i}u=u$ for every $t$, differentiating its pairing at $t=0$ gives $\partial_i u=0$. Conversely, if $\partial_i u=0$, then for every test function
$$
\frac d{dt}\langle\tau_{te_i}u,\varphi\rangle
=\langle u,\partial_i\varphi(\mathord\cdot+te_i)\rangle
=-\langle\partial_i u,\varphi(\mathord\cdot+te_i)\rangle=0.
$$
The pairing is constant in $t$, hence $\tau_{te_i}u=u$. This proves both directions of <translation invariance and vanishing distributional derivative>.
Finally, the <distributional derivative>[distributional differentiation] obeys the linear <chain rule> under the linear coordinates $s=x-y$ and $t=x+y$. Thus
$$
\partial_x^2 f(x-y)=f''(x-y)=\partial_y^2 f(x-y)
$$
and likewise
$$
\partial_x^2 g(x+y)=g''(x+y)=\partial_y^2 g(x+y)
$$
as distributions. Adding the two identities gives
$$
\boxed{u_{xx}-u_{yy}=0},
$$
the <one-dimensional wave equation>; this is the low-regularity form of the travelling waves in the <D'Alembert formula>.
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