The following equations represent transverse waves :
`z_(1) = A cos(kx - omegat)`,
`z_(2) = A cos (kx + omegat)`, `z_(3) = A cos (ky - omegat)`
Identify the combination (s) of the waves which will produce (i) standing wave(s), (ii) a wave travelling in the direction making an angle of `45^(@)` with the positive `x` and positive `y` axes. In each case, find the positions at which the resultant is always zero.

Answer

(i) `z_(1)` and `z_(2) : x = (2n + 1)(pi)/(2k) rArr (2n+1) lamda//4`
where `n = 0,pm 1,pm2…` etc
(ii) `z_(1)` and `z_(3) : x- y = (2n +1)(pi)/(k)`
where `n = 0,pm 1, pm 2,…` etc.

Explanation

(i) Combination of waves producing standing wave : `Z_(1) +Z_(2)`
(ii) Combination of waves producing a wave travelling along `x = y` line : `Z_(1)+Z_(3)`
(iii) Position of nodes in case `(i)x=(2n+1)(pi)/(2k)` ltbrltgt case (ii) `x-y=(2n+1)(pi)/(k)`