General solution of the one-dimensional wave equation

The affine change of variables $(x,t)\mapsto (\xi ,\eta )=(x-ct,x+ct)$ is a bijection of ${\mathbb{R}}^2$ with inverse $x=(\xi +\eta )\mathrm{/}2$, $t=(\eta -\xi )\mathrm{/}(2c)$, and the image $R^{\mathrm{\prime }}$ of the rectangle $R$ is a nonempty open convex affine image of $R$ (a parallelogram when $R$ is bounded); the projections $I$ of $R^{\mathrm{\prime }}$ onto the $\xi$-axis and $J$ onto the $\eta$-axis are nonempty open intervals, and every section $\{\eta :(\xi ,\eta )\in R^{\mathrm{\prime }}\}$ and $\{\xi :(\xi ,\eta )\in R^{\mathrm{\prime }}\}$ is a nonempty open interval. By [F2] the function $U(\xi ,\eta ):=u((\xi +\eta )\mathrm{/}2,(\eta -\xi )\mathrm{/}(2c))$ is $C^2$ on $R^{\mathrm{\prime }}$, and [F1] gives $u_{tt}-c^2{u}_{xx}=-4c^2{U}_{\xi \eta }$ on $R^{\mathrm{\prime }}$.

Suppose $u$ solves $u_{tt}=c^2{u}_{xx}$ on $R$; since $c>0$, $U_{\xi \eta }=0$ on $R^{\mathrm{\prime }}$. For fixed $\xi \in I$ the section $J_{\xi }:=\{\eta :(\xi ,\eta )\in R^{\mathrm{\prime }}\}$ is a nonempty open interval and ${\mathrm{\partial }}_{\eta }(U_{\xi }(\xi ,\cdot ))=U_{\xi \eta }(\xi ,\cdot )=0$ there, so by [F3] the value $U_{\xi }(\xi ,\eta )$ is independent of $\eta$ in that section; call it $p(\xi )$. To see $p$ is $C^1$, fix ${\xi }_0\in I$ and choose ${\eta }_0$ with $({\xi }_0,{\eta }_0)\in R^{\mathrm{\prime }}$; openness gives an interval $I_0$ about ${\xi }_0$ on which $(\xi ,{\eta }_0)\in R^{\mathrm{\prime }}$, so $p(\xi )=U_{\xi }(\xi ,{\eta }_0)$ on $I_0$. Since $U$ is $C^2$, this local representative is $C^1$, and hence $p\in C^1(I)$. Choose a primitive $P$ of $p$ on $I$ by [F4]; since $p\in C^1$, $P\in C^2(I)$. Then ${\mathrm{\partial }}_{\xi }(U-P)=0$ on $R^{\mathrm{\prime }}$, and the same section argument in the $\xi$-direction gives that $U-P$ is independent of $\xi$: there is $G:J\to \mathbb{R}$ with $U(\xi ,\eta )-P(\xi )=G(\eta )$ for all $(\xi ,\eta )\in R^{\mathrm{\prime }}$. To see $G$ is $C^2$, fix ${\eta }_0\in J$ and choose ${\xi }_0$ with $({\xi }_0,{\eta }_0)\in R^{\mathrm{\prime }}$; openness gives a neighbourhood $J_0$ on which $G(\eta )=U({\xi }_0,\eta )-P({\xi }_0)$, a $C^2$ function. Thus $G\in C^2(J)$ and $u(x,t)=P(x-ct)+G(x+ct)$ on $R$.

Conversely, if $F\in C^2(I)$ and $G\in C^2(J)$, then $(x,t)\mapsto F(x-ct)+G(x+ct)$ is $C^2$ on $R$ by [F2] and [F5], and two applications of the chain rule give ${\mathrm{\partial }}_t^2[F(x-ct)+G(x+ct)]=c^2{F}^{\mathrm{\prime }\mathrm{\prime }}(x-ct)+c^2{G}^{\mathrm{\prime }\mathrm{\prime }}(x+ct)=c^2{\mathrm{\partial }}_x^2[F(x-ct)+G(x+ct)]$; hence every such sum solves the wave equation.

Uniqueness of the pair. If $F_1(\xi )+G_1(\eta )=F_2(\xi )+G_2(\eta )$ for all $(\xi ,\eta )\in R^{\mathrm{\prime }}$, then $\mathrm{\Phi }(\xi ,\eta ):=F_1(\xi )-F_2(\xi )-G_2(\eta )+G_1(\eta )$ vanishes on $R^{\mathrm{\prime }}$. For fixed $\xi \in I$ the section in $\eta$ is a nonempty interval, so ${\mathrm{\partial }}_{\eta }\mathrm{\Phi }=G_1^{\mathrm{\prime }}-G_2^{\mathrm{\prime }}=0$ on $J$, and for fixed $\eta \in J$ similarly ${\mathrm{\partial }}_{\xi }\mathrm{\Phi }=F_1^{\mathrm{\prime }}-F_2^{\mathrm{\prime }}=0$ on $I$; hence $F_1-F_2$ and $G_2-G_1$ are the same constant $k$ by [F3], that is $F_1=F_2+k$ and $G_1=G_2-k$.

Therefore every $C^2$ solution of the one-dimensional homogeneous wave equation on a nonempty open rectangle has the form $F(x-ct)+G(x+ct)$ with $F,G$ of class $C^2$ on the projections, every such sum is a solution, and the decomposition is unique up to the additive shift $F\mapsto F+k$, $G\mapsto G-k$.