A string of length `'l'` is fixed at both ends. It is vibrating in its `3^(rd)` overtone with maximum ampltiude `'a'`. The amplitude at a distance `(l)/(3)` from one end is `= sqrt(p)(a)/(2)`. Find `p`.

Options

A. `a`
B. `0`
C. `((sqrt(3) a))/(2)`
D. `(a)/(2)`

Answer

C

Explanation

For a string vibrating in its ` nth ` overtone `( n + 1) th` harmonic
`( n + 1) (lambda)/(2) = L rArr lambda = ( 2L)/( n + 1)`
`kx = (2 pi x)/( lambda) = ( pi ( n + 1) x )/( L)`
`y = 2 A sin ((( n + 1) pi x)/( L)) cos omega t`
here ` 2 A = a and n = 3`.
for ` x = (L)/(3) , y = [ a sin (( 4 pi)/(L) xx (L)/(3))] cos omega t`
` = a sin ( 4 pi )/(3) cos omega t = -a (( sqrt(3))/(2)) cos omega t`
i.e., at ` x = L//3` , the amplitude is `sqrt(3) a//2`.