Toronto Math Forum
APM3462015S => APM346Home Assignments => HA2 => Topic started by: Victor Ivrii on January 27, 2015, 09:36:46 PM

For a solution $u(x, t)$ of the wave equation $u_{tt}=c^2u_{xx}$, the energy density is defined as $e=\frac{1}{2}\bigl(u_t^2+c^2 u_x^2\bigr)$ and the momentum density as $p =c u_t u_x$.
a. Show that
\begin{equation}
\frac{\partial e}{\partial t} = c\frac{\partial p}{\partial x} \qquad \text{and} \qquad \frac{\partial p}{\partial t} = c\frac{\partial e}{\partial x}. \label{eqHA2.11}
\end{equation}
b. Show that both $e(x, t)$ and $p(x,t)$ also satisfy the same wave equation.

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

It looks not as bad as your two previous solutions looked but again one can see the problem: you try to put everything in mathmode which is absolutely wrong. Only formulae should go there. Also array should not be used there.