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 combineation (s) of the waves which will produce (i) standing wave(s), (ii) awave travelling in the direction making an angle of `45^(@)` degrees with the positive `x` and positive `y` axes. In each case, find the positions at which the resultant is always zero.

Answer

A, B, C, D

Explanation

(i) KEY CONCEPT : When two progressive waves having same amplitude and period, but travelling in opposite direction with same velocity superimpose, we get standing waves.
The following two equations qualify the above criteria and hence produce standing wave
`z_(1) = A cos(kx - omegat)`
`z_(2) = A cos(kx + omegat)`
The resultant wave is given by `z = z_(1) + z_(2)`
rArr `z = A cos (kx - omegat) + A cos (kx + omegat)`
`=2A cos kx cos omegat`
the resultant intensity will be zero when
`2A cos kx = 0`
rArr `cos k x = cos((2n + 1)/(2)pi`
rArr `k x = (2n +1)/(2)pi` rArr `x = ((2n +1))pi/(2k)`
where `n = 0,1 ,2,......`
(ii) The transverse waves
`z_(1) = A cos (kx - omegat)`
`z_(3) = A cos (ky - omegat)`
Combine to produce a wave travelling in the direction making an angle of `45^(@)` with the positive `x` and positive `y` axes.
The resultant wave is given by `z = z_(1) + z_(3)`
`z = A cos (kx - omegat) + A cos (ky - omegat)`
rArr `z = 2A cos (x - y)/(2)cos[(k(x + y) - 2omegat)/(2)]`
The resultant will be zero when
(2Acos) (k(x-y))/2 =0` rArr (cos)(k(x - y))/(2) = 0`
rArr `(k(x - y))/(2) = (2n + 1)/(2)pi` rArr `(x - y) = ((2n + 1))/(k)pi`