Explain the formatin of stationary wave by analytical method. Show that nodes and antinodes are equally spaced in a stationary wave.
The speed limit for a vehicle on road is 120 km/hr. A policeman detects a drop of 10 % in the pitch of horn of a car as it passes him. Is the policeman justified in punishing the car driver for crossing the speed limit ?
( Given : Velocity of sound = 340 m/s )

Explanation

conditions for antinodes : when ` A = pm 2 a ` , the particles are these points vibrate with maximum amplitude which are called antinodes.
We have ` A = 2a cos "" ( 2pi x ) / ( lamda) `
` therefore pm 2 a = 2a cos "" ( 2 pi x ) /( lamda ) `
` therefore cos "" ( 2pi x ) /( lamda) = pm 1 `
` therefore ( 2pi x ) /( lamda ) = 0, pi, 2pi,...... `
` therefore x = ( lamda p) /( 2 ) = p(( lamda ) /( 2 )) `
where p = 0, 1, 2 .............
For x = 0 , ` ( lamda) /( 2 ) , lamda , ( 3lamda) /( 2 ) `.... antinodes are prouduced.
The distance between any two successive antinodes is ` ( lamda ) / ( 2 ) `
Condition for nodes : When A = 0 the particles at these points are permanently at rest which are called nodes.
We have, ` A = 2a cos "" ( 2pi x ) / ( x ) `
` 2a cos (( 2pi x ) /( lamda ) ) = 0 `
` therefore cos (( 2pi x ) /( lamda ) ) = 0 `
`( 2 pi x ) /( lamda ) = (pi) / ( 2 ), ( 3pi ) /( 2 ), ( 5pi ) /(2) `....
` x = ( 2P - 1 ) ( lamda)/( 4 ) `
Where P = 1, 2 , ............
For ` x = ( lamda ) /( 4 ) , ( 3lamda ) / ( 4 ), ( 5lamda ) / ( 4 ) .... ` nodes are produced
The distance between any two successive nodes is ` ( lamda ) /( 2 ) `
The distance between any two adjacent antinode is ` ( lamda ) /( 4 ) `.
Hence, in stationary waves, the nodes and antinodes are equispaced.
Numerical :
Given : ` " " n_ 2 = n _ 1 - 10 % n _ 1 , v= 340 m//s `
`" " n _ 2 = n _ 1 - ( 1)/( 10) n _ 1 = ( 9 ) /( 10 ) n _ 1 `
When the car approaches the policeman, apparent fequency is
` n _ 1 = ( v + v _ 0 ) /( v - v _s ) xx n " " `... (i)
where ` v _ 0 = 0 ` and when the car passes the policeman apparent fequency is
` n _ 2 = ( v - v _ 0 )/( v + v _ s ) xx n " " ` ... (ii)
where, ` v _ 0 = 0 ` and `v _ s ` is the speed of car.
On dividing equation (i) by (ii)
` therefore " " ( n_ 1 ) /( n _ 2 ) = ( v + v _ s ) /( v - v _ s ) `
` therefore ( n _ 1 )/((9 ) / ( 10 ) n _ 1 ) = ( 340 + v _ s ) /( 340 - v _ s ) `
` therefore ( 10 )/( 9 ) = ( 340 + v _ s ) /( 340 - v _ s ) `
` v _ s = 17.89 ` m/s
` = 64.42 km//hr ` ( speed of car )
The speed limit is 120 km/hr whereas the speed of car is 64.42 km/hr.
Hence the policeman is not justified in his action.