A string vibrate according to the equation `y = 5 sin ((pi x)/(3)) cos (40 pi t)` where x and y in cm's and t is in second.
(i) What is the equation of incident and reflected wave ?
(ii) What is the distance between the adjacent nodes ?
(iii) What is the velocity of the particle of the string at the position x = 1.5 cm when `t = (9)/(8) sec` ?

Answer

(i) `y_("incident")=2.5 sin(40pit+(pi)/(3)x)and y_("reflected") = -2.5sin(40t-(pi)/(3)x)`
(ii) 3 cm
(iii) 0

Explanation

`y=2sin((pix)/(3))cos(40pi)`
`=2.5sin(40pi+(pix)/(3))-2.5sin(40pit-(pix)/(3))`
(i) Equation of incident wave
`y_(1) = 2.5 sin (40 pi t + (pix)/(3))`
Equation of reflected wave
`y_(2)=-2.5 sin(40pit-(pix)/(3))`
(ii) `(2pi)/(lamda)=(pi)/(3)rArr lamda=6cm`
`:.` Distance between adjacent nodes `=3cm`
(iii) `v=-5(40pi)sin((pix)/(3))sin(40pit)`
`=-200pisin((pixx1.5)/(3))sin(40 pixx(9)/(8))=0`