Motion of Waves
5.0 Energy associated with a wave
5.0 Energy associated with a wave
Every wave motion has energy associated with it. In wave motion, energy and momentum are transferred or propagated.
Energy density $u$
Energy density is defined as the total mechanical energy per unit volume of a plane progessive wave. Mathematically, $$u = {\Delta K} + {\Delta U}$$
where,
$\Delta K$: Kinetic energy per unit volume
$\Delta U$: Potential energy per unit volume
Kinetic energy per unit volume $\left( \Delta K \right)$
Kinetic energy $(K)$ of a particle of mass $\Delta m$ is given by, $$K = \frac{1}{2}\left( {\Delta m} \right){v^2}$$
Kinetic energy per unit volume is given by, $$\begin{equation} \begin{aligned} \Delta K = \frac{K}{V} = \frac{{\frac{1}{2}\left( {\Delta m} \right)v_p^2}}{V} \\ \Delta K = \frac{1}{2}\rho v_p^2\quad ...(i) \\\end{aligned} \end{equation} $$
Let the equation of wave be, $$y = A\sin (kx - \omega t)$$Speed of the particle $\left( {{v_p}} \right)$ is given by, $$\begin{equation} \begin{aligned} \frac{{dy}}{{dt}} = - A\omega \cos \left( {kx - \omega t} \right) \\ {v_p} = - A\omega \cos \left( {kx - \omega t} \right)\quad ...(ii) \\\end{aligned} \end{equation} $$ From equation $(i)$ & $(ii)$ is given by, $$\Delta K = \frac{1}{2}\rho {A^2}{\omega ^2}{\cos ^2}\left( {kx - \omega t} \right)\quad ...(iii)$$ The above equation is of the kinetic energy per unit volume.
Potential energy per unit volume $\left( \Delta U \right)$
Potential energy $(U)$ of a particle of mass $\Delta m$ is given by, $$U = \frac{1}{2}m{\omega ^2}{x^2}$$
Potential energy per unit volume is given by, $$\begin{equation} \begin{aligned} \Delta U = \frac{U}{V} = \frac{{\frac{1}{2}m{\omega ^2}{x^2}}}{V} \\ \Delta U = \frac{1}{2}\rho {\omega ^2}{x^2}\quad ...(iv) \\\end{aligned} \end{equation} $$
Let the equation of wave be, $$y = A\sin (kx - \omega t)$$ So, $$\Delta U = \frac{1}{2}\rho {\omega ^2}{A^2}{\sin ^2}\left( {kx - \omega t} \right)\quad ...(v)$$ The above equation is of the potential energy per unit volume.
Adding eqation $(iii)$ & $(v)$ we get, $$\begin{equation} \begin{aligned} \Delta E = \Delta K + \Delta U \\ \Delta E = \frac{1}{2}\rho {\omega ^2}{A^2}{\cos ^2}\left( {kx - \omega t} \right) + \frac{1}{2}\rho {\omega ^2}{A^2}{\sin ^2}\left( {kx - \omega t} \right) \\ \Delta E = \frac{1}{2}\rho {\omega ^2}{A^2} \\\end{aligned} \end{equation} $$
The above equation is of the energy density.
