Question

Explain the formation of stationary waves by analytical method. Show that nodes and antinodes are equally spaced in stationary waves.

Answer 1

Analytical treatment of Stationary Waves : Consider two simple harmonic progressive waves having same amplitude (a), frequency (n) and period (T) travelling along same medium in opposite direction. The wave along positive direction of X-axis is given by :

`Y_1= alpha sin2pi(nt - x/λ )`    ............................(1)

The wave along negative direction of x-axis is given by:

`Y_2= alpha sin2pi(nt + x/λ )`    ............................(2)

By principle of superposition, the resultant displacement of stationary wave is given by:

`Y=Y_1+Y_2`

`therefore Y= alpha sin 2pi (nt - x/λ )+alpha sin 2pi (nt+ x/λ )`

`therefore Y= 2alpha sin\ (2pi)/2  (nt - x/λ +nt +x/λ).cos\ (2pi)/2(nt - x/λ - nt x/λ)`

Using sin C+sin D = 2 sin `((C+D)/2) cos ((C-D)/2)` and cos `(- θ )= cos θ)`

`therefore Y = 2a sin 2 pi n t . cos 2pi ((-x)/λ)`

`therefore Y = 2a [sin (2 pi nt).cos  ((2 pi x)/λ)]`

`therefore Y = [2 a cos ((2pix)/λ) ] sin 2 pi\ nt`              ....(3)

`therefore Y = Asin (2 pi\ n t )`

where A =  2a cos `(2pi x)/λ`  where A is the amplitude of the resutant stationary wave.

The above expression shows that resultant wave is a simple harmonic motion having same period but new amplitude. The absence of the term (x) in the sine function shows that the resutant waves do not move forward or backward. Such waves are called stationary waves.

Conditions for antinodes:
The points of medium which vibrate with maximum amplitude are called antinodes. Antinode is formed when A is maximum.

`A = +- 2a`

`therefore 2 a cos\ (2pi\ x)/λ = +- 2a`

`therefore  cos\ (2pi x )/λ = +-1`

`therefore (2pi x )/ λ = 0, pi,2pi`

`therefore (2pi x )/ λ = ppi`

`therefore x= (λP)/2 ` Where P = 0,1,2.......

When `(2pi x )/λ = 0,` then  x  = 0 .

When `(2pi x )/λ = pi` then  x = λ / 2. 

When `(2pi x )/λ = 2pi` then  x = λ.

`therefore x = 0, λ/2, λ.....`

The particles at these points vibrate with minimum amplitude. Such points are called antinodes. The distance between two successive antinodes is λ/2.
Conditions for nodes:
The points of medium which vibrate with minimum amplitudes are called nodes. Amplitude at nodes is zero.

A = 0

`therefore 2acos((2pix)/lambda)=0`

`therefore cos  (2pix)/lambda=0`

`therefore(2pix)/lambda=pi/2,(3pi)/2,(5pi)/2`

`x=((2P-1)/4)` where p = 1,2,.........

When`(2pix)/lambda=pi/2,` then x = λ/4.

When`(2pix)/lambda=(3pi)/2,` then x = 3λ/4.

When`(2pix)/lambda=(5pi)/2,` then x = 5λ/4.

`therefore x=lambda/4,(3lambda)/4,(5lambda)/4,...........`

The particles at these points vibrate with minimum or zero amplitude. The distance between any two successive nodes is λ/2.
Thus the distance between two successive antinodes and nodes is equal to λ/2. Therefore antinodes and nodes are equally spaced in a stationary wave.