Two travelling waves, `y _(1) = A sin [(x + ct)] and y_(2)= A sin [kt - ct)]` are superposed on a string. The distance between adjacent antinodes is

Options

A. `(ct)/(pi)`
B. `(ct)/(2pi)`
C. `(pi)/(2k)`
D. `(k)/(pi)`

Answer

A

Explanation

Given,
`y _(1) = A sin [ k (x + ct )]" "...(i) `
`and y _(2) = A sin [ k (x - ct) ] " "...(ii)`
By the principel of superposition, the resultant displacement of the particle is given by
`y = y _(1) + y _(2)`
`y = A [ sin { k (x + ct) } sin {k (x - ct ) }]`
By the formula
`sin C + sin D = 2 sin "" (C +D)/(2) cos ""(C-D)/(2)`
we have
`y = 2 A sin "" ( kx + kxt + kx - kct )/(3) cos ""(kx + kct- kx + kct)/(2)`
`y =2 A sin kx. cos kct `
For first antinode
`sin k x _(1) =1 `
`sin kx _(1) = sin "" (pi)/(2)`
`kx _(1) = (pi)/(2) " "...(iii)`
For second antinode
`sin kx_(2) =- 1 `
`sin k x _(2) = sin "" ( 3pi)/(2)`
`therefore` distance between adjacent antinodes
`kx _(2) - k x _(1) = (3pi)/(2) - (pi)/(2)`
`k ( x _(2) - x _(1)) = pi,`
`Delta x = ( pi)/(k)`