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) + 2 ab cos x)`

Answer

A

Explanation

Let `phi_(1) and phi_(2)` represent angles of the first and second waves , then
`phi_(2) = ( 2 phi)/( lambda ) [( vt - x) + x_(0)]`
and `phi_(1) = ( 2pi)/(lambda) ( v t - x)`
But `x_(0) = (lambda)/(2)`,
` phi_(2) - phi_(1) = pi`
Hence , phase difference , `phi = pi `. So , amplitude of resultant wave
`R= sqrt(a^(2) + b^(2) + 2 ab cos phi)`
` = sqrt(a^(2) + b^(2) + 2 ab cos pi) = sqrt(( a - b)^(2)) = a - b`
or `R = | a - b |`