Two travalling wavews of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a standing wave having the equation `y = A cos kx sin omega t` in which `A = 1.0 mm, k = 1.57 cm^(-1) and omega = 78.5 s^(-1)` (a) Find the velocity of the component travelling waves. (b) Find the node closet to the origin in the x gt 0. (c ) Find the antinode closet to the origin in the region x gt 0 (d) Find the amplitude of the particle at x = 2.33cm.

Explanation

`1.` The above standing wave is formed by the superposition of the waves as given below
`y_(1)=(A)/(2)sin(omegat-kx)` and `y_(2)=(A)/(2)sin(omegat+kx)`
The wave velocity of either of the wave is `v=(omega)/(k)`,
`v=(78.5)/(1.57)=50cm//s`
`2` For a node amplitude `=0impliescoskx=0`
The smallest value of positive `x` satisfying this relation is given by `kx=(pi)/(2)impliesx=(pi)/(2xx1.57)=1cm`
`3`. For antinode , amplitude becomes maximum and for that `|coskx|=1` smallest positive value of `x` satisfying this relation is given by `kx=pi`
`x=(pi)/(1.57)=2cm`
`4`. Amplitude of the particle
`a_(0)=|acoskx|=(1.0mm)(cos.(1.57)/(cm)xx2.33cm)`
`=1.0mmcos.(7pi)/(6)=0.86mm`