Exponentially conjugated wave equation ($({\partial }_t+\gamma {)}^{2}u=c^2{u}_{xx},v=e^{\gamma t}u$)

For real constants $\gamma ,c$, the substitution $v=e^{\gamma t}u$ turns the displayed equation into the wave equation $v_{tt}=c^2{v}_{xx}$. If $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$, then $v(x,0)=f(x)$ and $v_t(x,0)=g(x)+\gamma f(x)$. For $v_0=|c|>0$, D'Alembert's formula consequently givesThe extra $\gamma f$ term is necessary even when the original initial velocity is zero. For $c=0$, take the limit to obtain $u=e^{-\gamma t}[f+t(g+\gamma f)]$. The formula gives classical solutions for suitably smooth data and weak solutions for less regular data. Positive $\gamma$ damps the transformed wave, whereas negative $\gamma$ amplifies it.