A travelling wave represented by
`y=Asin (omegat-kx)`
is superimposed on another wave represented by
`y=Asin(omegat+kx).` The resultant is

Options

A. standing wave having nodes at x=`(n+1/2)lamda/2`, n=0,1,2,….
B. satnding wave having nodes at `x=(nlamda)/2`,n=0,1,2..
C. wave travelling along +x direction
D. wave travelling along-x direction

Answer

B

Explanation

According to the principle of superposition, the resultant wave is
`y=asin(kx-omegat)+asin(kx+omegat)`
`=2asinkxcosomegat`….(i)
It represents a standing wave.
In the standing wave, there will be nodes ( where amplitude is zero) and antinodes (where amplitude is largest).
From Eq. (i), the positions of nodes are given by
`sinkx=0rArrkx=npi, n=0,1,2,....`
`or(2pi)/lamdax=npi,n=0,1,2,.....`
`or x=(nlamda)/2,n=0,1,2,...`
In the same way,
From Eq. (i), the positions of antinodes are given by
`abs(sinkx)=1`
`rArrkx=(n+1/2)pi,n=0,1,2,...`
`or (2pix)/lamda=(n+1/2)pi,n=0,1,2,..`
`orx=(n+1/2)lamda/2,n=0,1,2,...`