During the propagation of one progressive harmonic wave having amplitude 10 m, displacements and particles at 2 m and 16 m at respectively 2s and 8 s are 5 m and 5`sqrt3` m respectively. Find angular frequency and wave vector for this wave.

Explanation

`(i) y _(1) = A sin (omega t _(1) - kx _(1))`
`therefore 5 = 10 sin (2 omega -2k)`
`therefore sin (2 omega -2k) =1/2 = sin ""(pi)/(6)`
`therefore 2 omega - 2k = (pi)/(6)`
`therefore 12 omega - 12k =pi " "...(1)`
`(ii) y _(2) = A sin (omega t _(2) - kx _(2))`
`therefore 5 sqrt3 = 10 sin (8 omega -16k)`
`therefore sin (8 omega - 16k ) = (sqrt3)/( 2) = sin ""(pi)/(3)`
`therefore 8 omega - 16 k = (pi)/(3)`
`therefore 24 omega - 48 k =pi " "...(2)`
Multiplying equation (1) by `(-2)` we get,
`-24 omega + 24 k =- 2pi`
Adding equation (2) and (3),
`- 24 k =- pi`
`therefore k = (pi)/(24) (rad)/(m) " "...(4),`
From (1) and (4),
`12 omega -12 ((pi)/(24)) = pi`
`therefore 12 omega -(pi)/(2) =pi`
`therefore 12 omega = (3pi)/(2)`
`therefore omega = (pi)/(8) (rad)/(s) " "...(5)`