1.

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

Answer»

`lambda = (pi x_(0))/4`
`lambda - 2PI x_(0)`
`lambda = (pi x_(0))/2`
`lambda = 4pi x_(0)`

Solution :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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