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

Explanation

When the wave hits the rigid boundary it bounces back to the original medium and can interfere with the original waves. A pattern is forme,d that are known as standing waves or waves staionary. Let us consider two harmonic progresive waves (formed by strings) that thave 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-omegat) " "....(1)`
(waves move toward right)
The dispacement of the second wave (refected wave) is
`y_(2)=A sin (kx+omegat) " "...(2)`
both will interfere each othe rby the principle of superpositon, the net displacement is
`y=y_(1)+y_(2) " "....(3)`
By substituting equation (1) and equaiton (2) in equantion (3), we get
`y={{:(A sin (kx-omegat)),(+A sin (kx+omegat)):}" ".....(4)`
Using trigonometric identity, we rewrite equation (4) as
`y(x,t)=2 A cos (omegat) sin (kx)" "...(5)`
This represents a stationary wave or standing wave, it is meant that this wave does not move either forward or backward, whereas progressive or travelling waves will move forward. In addition, the displacement of the paritcle in equation (5) can be written in more compact form,
`y(x,t) =A' cos(omegat)`
Where `A'=2A sin (kx)`. it is implied implying that the parituclar element of the string executes simplr harmonic motion with amplitudes equals to A. the maximum of this amplitude occurs at positins for which
`sin (kx)=1`
`rArr kx=(pi)/(2),(3pi)/(2),(5pi)/(2)....=m pi`
Where m takes half integer of half integral values. The position of maximum amplitude is know in terms of wavelength, let us represent the anti-nodal positional as
`x_(m)=((2m+1)/(2))(lambda)/(2) " "....(6)`
where `m=0,1,2,`
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)=(5lambda)/(4)`
and so on.
This distance between two successive antinodes can be computed by,
`x_(m)-x_(m-1)=((2m+1)/(2))(lambda)/(2)-(((2m+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`
`rArr kx=0, pi , 2pi, 3pi, .... =npi`
where n takes integer or integral values. It is noted that the elements at these points do not vibrate (not move) and the points are called node.s The `n^(th)` nodal positions is given by
`x_(n)=n(lambda)/(2) " "...(7)`
where `n=0,1,2,...`
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 minimum at
`x_(2)=lambda`
and so on.
The distance between any two successsive nodes can be calculate as
`x_(n)-x_(n-1)=n(lambda)/(2)-(n-1)(lambda)/(2)=(lambda)/(2)`