Waves `y_(1) = Acos(0.5pix - 100pit)` and `y_(2)Acos(0.46pix - 92pit)` are travelling along x-axis. (Here `x` is in `m` and `t` is in second)
(3) The number of times `y_(1) + y_(2) = 0` at `x = 0` in `1sec` is

Options

A. (a) `100`
B. ( b) `46`
C. ( c ) `192`
D. (d) `96`

Answer

D

Explanation

(d) The equations are `y_(1) = Acos (0.5 pix - 100 pit)` and `y_(2) = Acos (0.46 pix - 92 pit)` represent two prograssive wave travelling in the same direction with slight difference in the frequency. This will give the hpenomenon of beats.
Comparing it with equation
`y = A cos (kx - omegat)`, we get
`omega_(1) = 100pu` rArr `2pif_(1) = 50 Hz` and
`K_(1) = 0.5pi` rArr `(2pi)/(lambda_(1) = 0.5pi` rArr `lambda_(1) = 4m`
Wave velocity `= lambda_(1)f_(1) = 200m//s` [Alternatively use `v = (omega)/(K)`]
`omega^(2) = 2pif_(2) = 92pi` rArr `f_(2) = 46 Hz`
Therefore beat frequency `= f_(1) - f_(2) = ` Hz` and
`K_(2) = 0.46pi` rArr `(2pi)/(lambda_(2) = 0.46pi` rArr `lambda_(2) = (200)/(46)`
Wave velocity `= (200)/(46) xx 46 = 200m//s`