\[\begin{array}{l}
part(a):\\
\\
\frac{{\partial e}}{{\partial t}} = {u_t}{u_{tt}} + {c^2}{u_x}{u_{xt}}\\
\frac{{\partial \rho }}{{\partial x}} = c{u_{tx}}ux + c{u_t}{u_{xx}},since\quad {u_{tt}} = {c^2}{u_{xx}}\\
\Rightarrow \frac{{\partial e}}{{\partial t}} = {c^2}{u_t}{u_{xx}} + {c^2}{u_x}{u_{xt}}\quad (1)\\
c\frac{{\partial \rho }}{{\partial x}} = {c^2}{u_{tx}}ux + {c^2}{u_t}{u_{xx}} = \frac{{\partial e}}{{\partial t}} = (1)\\
\frac{{\partial \rho }}{{\partial t}} = c{u_{tt}}{u_x} + c{u_t}{u_{xt}}\quad (2)\\
\frac{{\partial e}}{{\partial x}} = {u_t}{u_{xt}} + {c^2}{u_x}{u_{xx}}\quad \\
\Rightarrow \frac{{\partial e}}{{\partial x}} = {u_t}{u_{xt}} + {u_x}{u_{tt}}\\
c\frac{{\partial e}}{{\partial x}} = c{u_t}{u_{xt}} + c{u_x}{u_{tt}} = \frac{{\partial \rho }}{{\partial t}} = (2)\\
\\
part(b):\\
\\
from\quad (a)\quad known\quad that\quad c\frac{{\partial \rho }}{{\partial x}} = \frac{{\partial e}}{{\partial t}},\quad c\frac{{\partial e}}{{\partial x}} = \frac{{\partial \rho }}{{\partial t}}\\
thus,\quad c\frac{{{\partial ^2}\rho }}{{\partial xt}} = \frac{{{\partial ^2}e}}{{\partial {t^2}}}\quad (1),\quad \frac{{{\partial ^2}\rho }}{{\partial {t^2}}} = c\frac{{{\partial ^2}e}}{{\partial xt}}\quad (2)\\
likewise,\quad \frac{{{\partial ^2}e}}{{\partial xt}} = c\frac{{{\partial ^2}\rho }}{{\partial {x^2}}}\quad (3),\quad \frac{{{\partial ^2}\rho }}{{\partial xt}} = c\frac{{{\partial ^2}e}}{{\partial {x^2}}}(4)\\
from\quad (1),(4):\quad {e_{tt}} = {c^2}{e_{xx}}\\
from\quad (2)(3):\quad {\rho _{tt}} = {c^2}{\rho _{xx}}\\
As\quad shown,\quad both\quad e(x,t)\quad and\quad \rho (x,t)\quad from\quad the\quad same\quad wave\quad equation.
\end{array}\]
