Two waves travelling in opposite directions produce a standing wave . The individual wave functions are given by `y_(1) = 4 sin ( 3x - 2 t)` and `y_(2) = 4 sin ( 3x + 2 t) cm` , where `x` and `y` are in cm

Options

A. The maximum displacement of the motion at `x = 2.3 cm is 4.63 cm`.
B. The maximum displacement of the motion at `t = 2.3 s is 4.63 cm`.
C. Nodes are formed at `x` values given by
` 0 , pi//3 , 2 pi//3 , 4 pi//3 , ….`
D. Antinodes are formed at `x` values given by
`pi//6 , pi//2 , 5 pi//6 , 7 pi//6,…`

Answer

A, C, D

Explanation

`y = y_(1) + y_(2)`
y = 4[ sin ( 3x - 2t) + sin( 3x + 2t)]`
`y = 4[ 2 sin ( 3x) cos(2 t)]`
`y = 8 sin ( 3 x) cos (2t)`
`y = R cos ( 2t)`
`R = Resultant Amplitude ` = 8 sin ( 3x)`
`R = 8 sin [ 3 (2.3)]`
`R = 8 sin (6.9)`
`R = 4.63 cm`
Nodes are formed at points of zero intensity , i.e.,
`I_(R) = R^(2) = 0`
`sin^(2) ( 3x) = 0`
`sin ( 3 x) = 0`
`3x = 0 , pi , 2 pi , 3 pi , 4 pi ,...`
`x = 0 , (pi)/(3) , ( 2pi)/(3) , pi , (4 pi)/(3) ,...`
Antinodes are formed in between.