A simple harmonic plane wave propagatees along x-axis in a medium. The displacement of the particle as a function of time is shown in figure, for `x=0`(curve 1) and `x=7` (curve 2)
The two particle are with a span of onewavelength.

The wavelength of the wave is

Options

A. `6 cm`
B. `24 cm`
C. `12 cm`
D. `16 cm`

Answer

c

Explanation

Let general wave equation be `y=A sin(omegat-kx+phi)`
`v=(dy)/(dt)= Aomega cos(omegat-kx+phi)`
For curve `(1),x=0`
at `t=0,x=0,` we have `y=0`
`implies 0=A sin [phi]implies sin phi=0`
`implies phi=0` or `pi`
here `phi=pi` (because velocity is negative) for curve `(2),xx=7 cm`
`t=0,x=7 cm, y=-1`
`-1=sin(-kxx7+pi)`
`implies sin(-7k+pi)=-1//2`
`implies-7k+pi=2npi+(7pi)/(6)` or `2npi+(11pi)/(6)`
here `implies-7k+pi=2npi+(11pi)/(6)`
(because at `t=0`, velocity is positive)
`implies-7((2pi)/(lambda))=2npi+(5pi)/(6)`
implieslammbda=(-14pi)/((5pi)/(6)+2npi)`
`
`implieslambda=(-84)/(12n+5)`
for `n=-1,lambda=12 cm`
for `n=-2,lambda=(84)/(19)cm` (not possible)
because `lambdagt7 cm`
`v=flambda=100xx(12)/(100)=12 m//s`