A string vibrates according to the equation `y=5"sin" (pix)/(3) cos 40 pi t`
where, x and y are in centimeters and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position ` x = 1.5 ` cm when `t = 9/8 s?`

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`