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.

Given below are some functions of x and to represent the displacement

1.	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.
Answer» Solution :(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.