In the appendix B of "Functional Analysis" by Peter Lax, author claims that for any pair of distributions $l$ and $m$ in a single variable, $$ u = l(x+t) + m(x-t) $$ is a solution of 1D wave equation $$ u_{tt}-u_{xx}=0 $$ in the sense of distributions.
However I cannot understand the expressions $l(x+t)$ and $m(x-t)$. In the same book the composition of a distribution and an invertible $C^{\infty}$ mapping has been defined, but $(x,t)\mapsto x+t$ is not invertible.
How can I interpret $l(x+t)$? And with that proper interpretaion, how can I show that such distribution is a solution of a wave equation in a distribution sense?
A broad answer is to consider distributions as limit of sequences of smooth functions converging in the sense of distributions, from which many operators can be applied naturally and in a consistent way (derivative, Fourier transform, change of variable, composition with functions, convolution, tensor product, multiplication with a smooth function..) $$\langle l(x),\phi(x)\rangle =\lim_{n\to \infty} \int_\Bbb{R} l_n(x) \phi(x)dx$$ where $l_n(x)=\langle l(y),n\varphi(n(x-y))\rangle$ given some $\varphi\in C^\infty_c(\Bbb{R}),\int\varphi=1$ fixed a priori.
$$\langle l(x+t),\Phi(x,t)\rangle= \lim_{n\to \infty} \int_{\Bbb{R}^2} l_n(x+t) \Phi(x,t)dxdt$$