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if two progressive waves travelling in the same direction interfere, then obtain the instaneous displacement of the combined wave. |
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Answer» Solution :Let `y_(1) =A_(1) SIN ( omega t - kx_(1))` and `y_(2) = A_(2) sin ( omega t - kx_(2))` interfere with eacth other. `y = y_(1) + y_(2)= A_(1) sin ( omega t - kx_(1)) + A_(2) sin (omega t - kx_(2))` i.e., `y = R sin ( omega t - kx)` Where `R = sqrt(A_(1)^(2) + A_(2)^(2) + 2A_(1)A_(2) cos ( k ) ( x_(1) x_(2))) ` and `kx = phi = tan^(-1) [ (A_(1) sin kx_(1) + A2 sin kx_(2))/(A_(1) cos kx_(1) + A2 cos kx_(2))]` If `A_(1) =A_(2) = A` then `R = sqrt( A^(2) + A^(2) + 2A^(2) cos k(x_(1)-x_(2)))` i.e., `R= sqrt( 2)a sqrt(1+cosk(x_(1)-x_(2)))` For `R = R_("max"), cos k ( x_(1)-x_(2))=1rArr` `k(x_(1)-x_(2)) = 0,2pi, 4pi,"..."2n PI` i.e., Phase difference between the two waves `= 2n pi`. For `R =R_("MIN") cos k( x1-x2) = 1pi, 3pi,5pi,"......" ( 2n + 1) pi`. i.e., Phase differences between the twowaves `= ( 2n +1 ) pi `. For path difference PUT `pi = ( lambda )/( 2)`. |
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