Two waves travelling in opposite directions produce a standing wave. The individual wave funcations are
`y_(1) = (4.0 cm) sin(3.0x - 2.0t)`
`y_(2) = (4.0 cm) sin(3.0x + 2.0t)`
where `x` and `y` are in centimeter.
(a) Find the maximum displacement of a particle of the medium at `x = 2.3 cm`,
(b) Find the position of the nodes and antinodes.

Explanation

(a) When the two waves are summed. The result is a standing wave whose mathematical representation is given by Equation, with `A = 4.0 cm` and `k = 3.0 rad//cm,`
`y = (2A sin kx)cos omegat = [(8.0 cm) sin 3.0 X] cos 2.0 t`
Thus, the maximum displacement of a particel at the position `x = 2.3 cm` is
`y_(max) = [(8.0 cm) sin 3.0x]_(x = 2.3 cm)`
`= (8.0 m) sin (6.9 rad) = 4.6 cm`
(b) Because `k = 2pi//lambda = 3.0 rad//cm`, we see that `lambda = 2pi//3cm`. Therefore, the antinodes are located at
`x = n((pi)/(6.0)) cm (n = 1, 3, 5, ........)`
and the nodes are located at
`x = n(lambda)/(2-)((pi)/(3.0)) cm (n = 1, 2, 3, .........)`