Ondas: conservacion

\section{Energy of a wave according to Borowitz} \subsection{Energy and flow of real wave equation in one dimension\footnote{ S. Borowitz, Fundamentals of Quantum Mechanics, W.A. Benjamin, NY (1967), pp.71-74}} \[ \frac{\partial ^2\psi }{\partial z^2}-\frac{1}{v^2}\frac{\partial ^2\psi }{\partial t^2}=0 \] multiply by $\frac{\partial \psi }{\partial t}=\dot {\psi }$, then $\dot {\psi }\left( {\frac{\partial ^2\psi }{\partial z^2}-\frac{1}{v^2}\frac{\partial \dot {\psi }}{\partial t}} \right)=0$. Notice that $\frac{\partial }{\partial t}\left( {\frac{\partial \psi }{\partial z}} \right)^2=2\frac{\partial \psi }{\partial z}\frac{\partial \dot {\psi }}{\partial z}$ and $\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)=\dot {\psi }\frac{\partial ^2\psi }{\partial z^2}+\frac{\partial \psi }{\partial z}\frac{\partial \dot {\psi }}{\partial z}$ so that their difference yields $\textstyle{1 \over 2}\frac{\partial }{\partial t}\left( {\frac{\partial \psi }{\partial z}} \right)^2-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)=-\dot {\psi }\frac{\partial ^2\psi }{\partial z^2}$; whereas the second time derivative may be written as $\dot {\psi }\frac{\partial \dot {\psi }}{\partial t}=\textstyle{1 \over 2}\frac{\partial \dot {\psi }^2}{\partial t}$, so that the equation is: \[ \textstyle{1 \over 2}\frac{\partial }{\partial t}\left( {\frac{\partial \psi }{\partial z}} \right)^2-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)+\frac{1}{v^2}\textstyle{1 \over 2}\frac{\partial \dot {\psi }^2}{\partial t}=0 \] grouping temporal and spatial derivatives, we obtain a continuity type equation: \begin{equation} \label{eq1} \frac{\partial }{\partial t}\textstyle{1 \over 2}\left[ {\left( {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right]-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)=0 \end{equation} and thus a density and flux may be defined as ${e}'=\textstyle{1 \over 2}\left[ {\left( {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right]$, ${S}'=\dot {\psi }\frac{\partial \psi }{\partial z}$. The dimensionless density or flow (with units of velocity) are translated into energy density and flow multiplying by $\sigma =\rho v^2$, where $\rho $ is the mass per unit length. \begin{equation} \label{eq2} e=\textstyle{1 \over 2}\rho v^2\left[ {\left( {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right] \end{equation} \begin{equation} \label{eq3} S=\rho v^2\dot {\psi }\frac{\partial \psi }{\partial z} \end{equation} the wave function may be real or complex. If it is complex, the energy and flow are also complex.

\section{Energy of a wave according to Borowitz} \subsection{Energy and flow of real wave equation in one dimension\footnote{ S. Borowitz, Fundamentals of Quantum Mechanics, W.A. Benjamin, NY (1967), pp.71-74}} \[ \frac{\partial ^2\psi }{\partial z^2}-\frac{1}{v^2}\frac{\partial ^2\psi }{\partial t^2}=0 \] multiply by $\frac{\partial \psi }{\partial t}=\dot {\psi }$, then $\dot {\psi }\left( {\frac{\partial ^2\psi }{\partial z^2}-\frac{1}{v^2}\frac{\partial \dot {\psi }}{\partial t}} \right)=0$. Notice that $\frac{\partial }{\partial t}\left( {\frac{\partial \psi }{\partial z}} \right)^2=2\frac{\partial \psi }{\partial z}\frac{\partial \dot {\psi }}{\partial z}$ and $\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)=\dot {\psi }\frac{\partial ^2\psi }{\partial z^2}+\frac{\partial \psi }{\partial z}\frac{\partial \dot {\psi }}{\partial z}$ so that their difference yields $\textstyle{1 \over 2}\frac{\partial }{\partial t}\left( {\frac{\partial \psi }{\partial z}} \right)^2-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)=-\dot {\psi }\frac{\partial ^2\psi }{\partial z^2}$; whereas the second time derivative may be written as $\dot {\psi }\frac{\partial \dot {\psi }}{\partial t}=\textstyle{1 \over 2}\frac{\partial \dot {\psi }^2}{\partial t}$, so that the equation is: \[ \textstyle{1 \over 2}\frac{\partial }{\partial t}\left( {\frac{\partial \psi }{\partial z}} \right)^2-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)+\frac{1}{v^2}\textstyle{1 \over 2}\frac{\partial \dot {\psi }^2}{\partial t}=0 \] grouping temporal and spatial derivatives, we obtain a continuity type equation: \begin{equation} \label{eq1} \frac{\partial }{\partial t}\textstyle{1 \over 2}\left[ {\left( {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right]-\frac{\partial }{\partial z}\left( {\dot {\psi }\frac{\partial \psi }{\partial z}} \right)=0 \end{equation} and thus a density and flux may be defined as ${e}'=\textstyle{1 \over 2}\left[ {\left( {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right]$, ${S}'=\dot {\psi }\frac{\partial \psi }{\partial z}$. The dimensionless density or flow (with units of velocity) are translated into energy density and flow multiplying by $\sigma =\rho v^2$, where $\rho $ is the mass per unit length. \begin{equation} \label{eq2} e=\textstyle{1 \over 2}\rho v^2\left[ {\left( {\frac{\partial \psi }{\partial z}} \right)^2+\frac{1}{v^2}\dot {\psi }^2} \right] \end{equation} \begin{equation} \label{eq3} S=\rho v^2\dot {\psi }\frac{\partial \psi }{\partial z} \end{equation} the wave function may be real or complex. If it is complex, the energy and flow are also complex.

categoría:ondas

Anónimo

Buscar

Ondas: conservacion

Espacios de nombres

Más

Acciones de página

Navegación

Navegación

Herramientas wiki

Herramientas wiki

Anónimo

Buscar

Ondas: conservacion

Navegación

Herramientas wiki

Herramientas de página