A standing wave is formed by two harmonic waves $y_1 = A \sin (kx - \omega t )$ and $y_2 = A (kx + \omega t…
A standing wave is formed by two harmonic waves $y_1 = A \sin (kx - \omega t )$ and $y_2 = A (kx + \omega t )$ travelling on a string in opposite directions. Mass density of the string is $\rho$ and area of cross-section is $S$. Find the total mechanical energy between two adjacent nodes on the string.
Solution
Sol. The distance between two adjacent nodes is $\lambda/2$ or $\pi/k$. ($\therefore\; k = 2\pi/\lambda$)
$\therefore$ Volume of string between two nodes,
$V =$ (Area of cross-section) (Distance between two nodes)
$= (S)\left(\frac{\pi}{k}\right)$
Energy density (energy per unit volume) of a travelling wave,
$u = \frac{1}{2}\rho A^2 \omega^2$
A standing wave is formed by two identical waves travelling in opposite directions. Therefore, the energy stored between two nodes in a standing wave,
$E = 2$ (energy stored in a distance of $\pi/k$ of a travelling wave)
$= 2$ (energy density) (volume) = $2\left(\frac{1}{2}\rho A^2 \omega^2\right)\left(\frac{\pi S}{k}\right)$
or
$E = \frac{\rho A^2 \omega^2 \pi S}{k}$
Answer: $E = \frac{\rho A^2 \omega^2 \pi S}{k}$