 Solution

Write $y(x,t)=Y(\xi ,\eta )$ with $\xi =x-ct$, $\eta =x+ct$. The chain rule gives the differential operatorsTheir squared difference isConsequently the wave equation becomes $Y_{\xi \eta }=0$. Integrating first in $\eta$ gives $Y_{\xi }=F^{\prime }(\xi )$; integrating in $\xi$ gives the general solutionThe arbitrary twice differentiable functions describe right-moving and left-moving waves, respectively. Conversely the chain rule shows that each such sum satisfies the wave equation.