A travelling harmonic wave on a string is described by `y ( x,t ) = 7.5 sin ( 0.0050 x + 12 t + ( pi)/(4))`
(a) What are the displacement and velocity of oscillation of a point at x = 1 cm, t = 1s ? Is this velocity equal to the velocity of propagation ?
(b) Locate the points of the string which have the same transverse displacements and velocity as the x = 1cm point at t = 2s, 5s and 11s

Explanation

Given, ` y ( x,t ) = 7.5 sin ( 0.0050 x + 12 t + ( pi)/( 4))`
At `x = 1 cm, t = 1s`
`y = 7.5 sin ( 0.005 + 12 + ( pi )/( 4))`
`y = 7.5 sin ( 12.005 + ( pi)/( 4)) = 7.5 sin ( 12.790 ) = 7.5 sin ( 12.7^(@))` `|:(= 12.79xx(180)/(3.142)),(=732.7^(@)),(=12.7):}`
` =7.5 xx 0.2198 = 1.648 cm`
y = 1.65 cm
Velocity of oscillaiton of a particle `v = (dy)/( dt) = ( 7.5 ) ( 12) cos ( 0.005 x + 12 t + ( pi)/( 4))`
i.e., `v = 90 cos ( 0.005 x + 12 t + ( pi)/( 4))`
at `x = 1 cm, t = 1s`
`v = 90 cos ( 0.005 + 12 + ( pi)/(4)) = 90 cos ( 12.7^(@)) = 90 xx 0.9755`
`v = 87.8cms^(-1)`
Particle velocity will not be equal to the wave velocity
Wave velocity `v = ( omega )/( k ) = ( 12 )/( 0.005)`
i.e., `v = 2400 cms^(-1) = 24 ms^(-1)`
Wavelength `lambda = ( 2pi)/(k)`
i.e., ` k = ( 2 xx 3.142)/( 0.005) , lambda = 1256.8cm, lambda = 12.6 cm`,
All points at distance ` n lambda ( n = +- 1, +-2, ".......")` where `lambda = 12.6m` from the point x = 1cm, the displacement and velocity will be the same.