A standing wave pattern is formed on a string.One of the waves is given by equation `y_(1)=a cos(omega t-kx+(pi)/(3))` ,then the equation of the other wave such that `x=0` a node is formed is-

Options

A. `y_(2) = a sin(omegat + kx + (pi)/(3))`
B. `y_(2) = a cos(omegat + kx + (pi)/(3))`
C. `y_(2) = a cos(omegat + kx + (2pi)/(3))`
D. `y_(2) = a cos(omegat + kx + (4pi)/(3))`

Answer

D

Explanation

At x = 0 the phase difference should be `pi` for destructive interference
Alternate solution
We use `y_(2) = a cos(omegat + kx + phi_(0))`
Resultant after superposition is
`y = y_(1) + t_(2) = a cos(omegat - kx + (pi)/(3)) + a cos (omegat + kx + phi_(0))`
= `2a cos [omegat + ((pi)/(3) + phi_(0))/2] xx cos[kx + (phi_(0) -(pi)/3)/2]`
y = 0 at x = 0 for any t
`implies kx + (phi_(0) - (pi)/(3))/(2) = (pi)/(2) ` at x = 0
`phi_(0) = (4pi)/(3)`. `y_(2) = a cos(omegat + kx + (4pi)/(3))`