What are stationary waves? write down the characteristics of stationary waves.

Explanation

Explanation of stationary waves: When the wave hits the rigid boundary it bounces back to the original medium and can interfere with the original waves. A pattern is fonned, which are known as standing waves or stationary waves.
Explanation: Consider two harmonic progressive waves (formed by strings) that have the same amplitude and same velocity but move in opposite directions. Then the displacement of the first wave (incident wave) is
`y_(1) = A (sin kx - omega t)` (waves move toward right) `" " ........... (1)`
and the displacement of the second wave (reflected wave) is
`y_(2) = A sin (kx + omega t)` (waves move toward left) `" " ...... (2)`
both will interfere with each other by the principle of superposition, the net displacement is `y = y_(1) + y_(2) " " ......... (3)`
Substituting equation (1) and equation (2) in equation (3), we get
`y = A sin (kx - omega t) + A sin (kx + omega t) " " ... (4)`
Using trigonometric identity, we rewrite equation (4) as
`y (x , t) = 2 A cos (omega t) sin (kx) " " ..... (5)`
This represents a stationary wave or standing wave, which means that this wave does not move either forward or backward, whereas progressive or travelling waves will move forward or backward. Further, the displacement of the particle in equation (5) can be written in more compact form,
y(x , t) = A. cos `(omega t)`
where, A. = 2A sin (kx), implying that the particular element of the string executes simple harmonic motion with amplitude equals to A.. The maximum of this amplitude occurs at positions for which `sin (kx) = 1 implies kx = (pi)/(2) , (3pi)/(2) , (5pi)/(2) ....... = m pi`
where m takes half integer or half integral values. The position of maximum amplitude is _ known · as .antinode. Expressing wave number in terms of wavelength, we can represent the anti-nodal positions as
`x_(m) = ((2m + 1)/(2)) (lambda)/(2)` , where m = 0 ,1 , 2... `" " .... (6)`
For m =0 we have maximum at `x_(0) = (lambda)/(2)`
For m = 1 we have maximum at `x_(1) = (3lambda)/(4)`
For m = 2 we have maximum at `x_(2) = (5 lambda)/(4)` and so on .
The distance between two successive antinodes can be computed by
`x_(m) - x_(m- 1) = ((2m+1)/(2 )) (lambda)/(2) - (((2 m +1) + 1)/(2)) (lambda)/(2) = (lambda)/(2)`
Similarly, the minimum of the amplitude A. also occurs at some points in the space, and these points can be determined by setting
`sin (kx) = 0 implies kx = 0 , pi , 2pi ,3pi , ....= npi `
where n takes integer or integral values. Note that the elements at these points do not vibrate (not move), and the points are called nodes. The `n^(th)` nodal positions is given by,
`x_(n) = n""(lambda)/(2)` where n = 0 , 1 , 2 .... `" " ..... (7)`
For n = 0 we have minimum at `x_(0) = 0`
For n = 1 we have minimum at `x_(1) = (lambda)/(2)`
For n = 2 we have maximum at `x_(2) = lambda` and so on .
The distance between any two successive nodes can be calculated as
`x_(n) - x_(n-1) = n""(lambda)/(2) - (n - 1) (lambda)/(2) = (lambda)/(2)`
Characteristics of stationary waves:
1. Stationary waves are characterised by the confinement of a wave disturbance between two rigid boundaries. This means, the wave does not move forward or backward in a medium ( does not advance), it remains steady at its place. Therefore, they are called "stationary waves or standing waves".
2. Certain points in the region in which the wave exists have maximum amplitude, called as anti-nodes and at certain points the amplitude is minimum or zero, called as nodes.
The distance between two consecutive nodes (or) anti-nodes is `(lambda)/(2)`
The distance between a node and its neighbouring anti-node is `(lambda)/(4)`
5. The transfer of energy along the standing wave is zero.