Two waves are passing through a region in the same direction at the same time . If the equation of these waves are
`y_(1) = a sin ( 2pi)/(lambda)( v t - x)`
and `y_(2) = b sin ( 2pi)/( lambda) [( vt - x) + x_(0) ]`
then the amplitude of the resulting wave for `x_(0) = (lambda//2)` is

Options

A. `|a-b|`
B. `a+b`
C. `sqrt(a^2+b^2)`
D. `sqrt(a^2+b^2+2ab cosx)`

Answer

A

Explanation

Let `phi_1` and `phi_2` represent angles of the first and second waves , then
`phi_2=(2pi)/(lambda)[(vt-x)+x_0]`
and `phi_1=(2pi)/(lambda)(vt-x)`
But `x_0=(lambda)/(2)`
`therefore phi_2-phi_1=pi`
Hence, phase difference, `phi =pi `: So, amplitude of the resultant wave
`R=sqrt(a^2+b^2+2abcosphiz)`
`=sqrt(a^2+b^2+2abcospi)`
`=sqrt((a-b)^2)`
`=a-b`
or `R=|a-b|`