Three transverse progressive waves arez_(1) = A cos (kx - omega t) ,z_(2) = A cos (ks + omega t)andz_(3) = A cos (ky - omega t). How may these be superposed to generate(i) a stationary wave,(ii) a wave propagating in a direction inclined at an angle of 45^(@) withboth the positive x and y - axes ?In each case, find out the positions where the resultant intensity would always be zero .

Three transverse progressive waves arez_(1) = A cos (kx - omega t) ,z_

1.	Three transverse progressive waves arez_(1) = A cos (kx - omega t) ,z_(2) = A cos (ks + omega t)andz_(3) = A cos (ky - omega t). How may these be superposed to generate(i) a stationary wave,(ii) a wave propagating in a direction inclined at an angle of 45^(@) withboth the positive x and y - axes ?In each case, find out the positions where the resultant intensity would always be zero .
Answer» Solution :(i)The first and the SECOND waves are TWO identical but oppositely directed waves. So, they would generate a stationary wave. Theequation of the resultant wave would be ` Z = z_(1) + z_(2) = A [COS (kx - omega t) + cos (kx + omega t)]` `= 2 a cos kx . cos omega t ` The resultant intensity is zero , where 2 A cos kx = 0 ` :. cos kx = 0 or , x = ((2n + 1)pi)/(2k) [ n = 0 , 1 , 2 , 3, .....]` (ii)The first wave directed along positive x - axis and the third wave directed along positive y - axis are identical waves.So the resultantwave propagates in a direction which is inclined at `45^(@)`with both the x are the y-axes.Theequation of the resultant wave would be ` z = z_(1) + z_(3) = A [cos (kx - omega t) + cos (ky - omega t)]` `= 2 A"cos"(k(x+y)-2omegat)/(2)*"cos"(k(x-y))/(2)` The resultant intensity is zero, where 2 a cos ` (k(x - y))/(2) = 0 ` `:."cos"(k(x-y))/(2)=0` or, `(k(x - y))/(2) = ((2n + 1) pi) /(2)[ n = 0 , 1 , 2 , 3 , .....]` or,` x - y = ((2 n + 1)pi)/(k)`.