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

Here, the equation of standing wave is
`y=Acoskx sinomegat`
where `A=1.0mm=0.1cm, k=1.57cm^(-1),`
`omega=78.5s^(-1)`
As travelling waves are of equal amplitudes equal frequencies and moving in opposite directions, their equations can be written as
`y_(1)=(A)/(2)sin(omegat-kx)`
`y_(2)=(A)/(2)sin(omegat+kx)`
`y=y_(1)+y_(2)`
`=(A)/(2)[sin(omegat-kx)+sin(omegat+kx)]`
`=Acoskx sinomegat.`
(a) Wave velocity of either wave
`upsilon=(omega)/(k)=(78.5s^(-1))/(1.57cm^(-1))=50cm//s`
(b) For a node, `y=0, `
` :. coskx=0,kx=pi//2`
`x=(pi//2)/(k)=(3.14//2)/(1.57)=1cm`
(c) For an antinode, `y=max, ` for which
`|cos kx|=1`
`kx=npi=pi` for smallest value of x
`x=(pi)/(k)=(3.14)/(1.57)=2cm`
(d) Teh amplitude of vibration `=|Acos kx|`
`r=1.0cos(1.57xx2.33radian)`
`=1.0cos(3.658rad)`
`=1.0cos(3.658xx57degree)`
`=1.0cos(209^(@))=-cos29^(@)`
`r=-0.875mm`