`y_1 = 8 sin (omegat - kx) and y_2 = 6 sin (omegat + kx)` are two waves travelling in a string of area of cross-section s and density rho. These two waves are superimposed to produce a standing wave.
(a) Find the energy of the standing wave between two consecutive nodes.
(b) Find the total amount of energy crossing through a node per second.

Answer

A, B, C

Explanation

(a) Distance between two nodes is `lambda/2` and `pi/k`. The
volume of string between two nodes is
`V = pi/k s` ......(i)
Energy density (energy per unit volume ) of
each wave will be
`u_1 = 1/2 rho omega^2 (8)^2 = 32 rho omega^2`
and ` u_2 = 1/2rho omega^2 (6)^2 = 18 rho omega^2 `
`:.` Total mechanical energy between two
consecutive nodes will be
`E = (u_1 + u_2)V`
`= 50 pi/k rho omega^2S`
(b) `y= y_1 + y_2 `
`= 8 sin (omegat - kx) + 6 sin (omegat + kx)`
` = 2 sin (omegat - kx) + { 6 sin (omegat + kx)`
` + 6sin (omegat + kx)}`
`= 2 sin (omegat - kx ) + 12 cos kxsin omegat `
Thus, the resultant wave will be a sum of
standing wave and a travelling wave.
Energy crossing through a node per second
= power or travelling wave.
`:. P = 1/2 rho omega^2 (2)^2 Sv `
`= 1/2 rho omega ^2 (4)(S)(omega/k)`
`= (2rho omega^3 S)/k` .