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Given below are some functions of x and ...

Given below are some functions of x and to represent the displacement of an elastic wave.
(i) `y = 5 cos (4x) sin (20t)`
(ii) `y = 4 sin (5x - (t)/(2)) + 3 cos (5x - (t)/(2))`
(iii) ` y = 10 cos {(252 -250)pi t] cos {(252 +250)pi t]`
(iv) `y = 100 cos (100 pi t + 0.5 x)`
State which of these represent
(a) a travelling wave along-x direction
(b) a stationary wave
(c) beats
(d) a travelling wave along + x direction. Give reasons for your answers.

Text Solution

Verified by Experts

(a) Wave equation `y =100 cos (100 pi t + 0.5 x)` represents a wave propagating along -X direction `(because ` coefficient of x is positive)
(b) Stationary wave is `y = 5 cos (4x ) sin ( 20t)`
(`because ` it has the form `y = 2 a cos (kx ) sin (omega t ) )`
(c) Equation of resultant wave in the phenomenon called "Beats" is
`y = 10 [ (252 -250)pi t ] cos [(252 +250)pit ]`
(d) Equation of wave propagationg along +X direction is,
`y = 4 sin (5x - (t)/(2)) + 3 cos( 5x - (t)/(2)) `
`therefore y = 4 [ sin { - ((t)/(2) - 5x ) }] + 3 [ cos { - ((t)/(2) - 5x )}]`
`therefore y =- 4sin ((t)/(2) - 5x )+ 3 cos ((t)/(2) - 5x )`
`(because sin (-theta ) =- sin theta and cos (-theta) = cos theta )`
In above equation coefficient of x is negative and so it indicates wave propagating along +X direction.
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