A standing wave is formed by two harmonic waves, `y_1 = A sin (kx-omegat) and y_2 = A sin (kx + omegat)` travelling on a string in opposite directions. Mass density is 'ρ' and area of cross section is S. Find the total mechanical energy between two adjacent nodes on the string.

Explanation

The distance between two adjacent nodes is `lambda/2 or pi/k`.
`:.` Volume of string between two nodes will be
V = (area of cross-section) (distance between two nodes)
`=(s) = (pi/k)`. Energy density (energy per unit volume) of a travellIng wave is given `u = 1/2 rho A^2 omega^2`.
A standing wave is formed by two idential waves travelling in oposite directions. Therefore, the energy stored between two nodes in a standing wave E = 2[energy stored in a distance of
`pi/k` of travelling wave] = 2 (energy density) (volume)
`=2(1/2 rhoA^2 omega^2) ((pi s)/k) or E =(rho A^2 omega^2 pis )/k`