Two harmonic travelling waves are described by the equations `y_(1)=a sin(kx-omega t)`  and `y_(2)=a sin(-kx+omega t+phi)`. The amplitude of the superimposed wave is 

Options

A. `2acos(phi)/(2)`
B. `2asinphi`
C. `2acos phi`
D. `2asin(phi)/(2)`

Answer

D

Explanation

Let `kx-omegat=alpha`
`y_(1)=a sin alpha, y_(2)=a sin(phi-alpha)`
Superimposed wave, `y=y_(1)+y_(2)`
`implies" "y=[a sin alpha+a sin phi cos alpha-a cos phi sin alpha]`
`implies" "y=[a(1-cos phi)sin alpha+(a cos phi)cos alpha]`
Let `R cos theta =a(1-cos phi)` and `R sin theta=a sin phi`.
Then, magnitude of R will be the magnitude of resultant wave,
`R_(2)=a^(2)[sin^(2)phi+(1-cosphi)^(2)]`
`=a^(2)[1+1-2cos phi]`
`implies" "R^(2)=4a^(2)sin^(2)(phi)/(2)impliesR=2a sin(phi)/(2)`