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A longitudnal wave is represented by x=x...

A longitudnal wave is represented by `x=x_(0) sin 2pi(nt - x/lambda)`. The maximum particle velocity will be four times the wave velocity if

A

`lambda = (pi x_(0))/4`

B

`lambda - 2pi x_(0)`

C

`lambda = (pi x_(0))/2`

D

`lambda = 4pi x_(0)`

Text Solution

Verified by Experts

The correct Answer is:
C
The Gien wave equation, is,
`x =x_(0) sin2pi(nt -x/lambda)`
`x =x_(0)sin(2pint -(2pix)/lambda)`
Compare it with the standard wave equation,
`x =A sin(omega t-kx)`
we get, `omega =2pin, k =(2pi)/lambda`
Wave velocity, `v= omega/k =(2pi n)/(2pi//lambda) = nlambda`.......(i)
Particle velocity, `v_(p) =(dx)/(dt) = 2pin x_(0) cos(2pint -(2pix)/lambda)`
Maximum particle velocity, `(v_(p))_("max") = 2pinx_(0)`......(ii)
According to given problem, `(v_(p))_("max") = 4v`
Using (i) and (ii), we get `2pinx_(0) = 4nlambda`
`rArr lambda = (2pi nx_(0))/(4n) = (pi x_(0))/2`
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