1.

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

Answer» Correct Answer - A
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|`


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