${\displaystyle {\partial ^{2}u \over \partial t^{2}}=c^{2}\nabla ^{2}u\!}$

${\displaystyle {\partial ^{2}u \over \partial t^{2}}=c(u)^{2}\nabla ^{2}u}$

${\displaystyle \left[{\frac {\partial }{\partial t}}-c{\frac {\partial }{\partial x}}\right]\left[{\frac {\partial }{\partial t}}+c{\frac {\partial }{\partial x}}\right]u=0}$

${\displaystyle {\Big \Downarrow }}$

${\displaystyle {\frac {\partial u}{\partial t}}-c{\frac {\partial u}{\partial x}}=0\qquad {\mbox{and}}\qquad {\frac {\partial u}{\partial t}}+c{\frac {\partial u}{\partial x}}=0}$

${\displaystyle {\partial ^{2}u \over \partial t^{2}}-c^{2}{\partial ^{2}u \over \partial x^{2}}=0\!}$

${\displaystyle x+ct=\xi \!}$? ${\displaystyle x-ct=\eta \!}$

${\displaystyle u_{\xi \eta }={\partial ^{2}u \over \partial \xi \,\partial \eta }=0\!}$

${\displaystyle u=f(\xi )+g(\eta )\!}$

${\displaystyle \omega ={\frac {2\pi }{T}}=2\pi f\quad ,\quad k={\frac {2\pi }{\lambda }}\quad ,\quad c={\frac {\lambda }{T}}}$