Obtain an expression for a stationary wave formed by two sinusoidal waves travelling along the same path in opposite direction, also discuss the condition for the formation of Nodes and Anti-nodes?

Explanation

Let us 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)=Asin(kx-omegat)" "......(1)`
(waves move toward right)
and the displacement of the second wave (reflected wave ) is
`y_(2)=Asin(kx-omegat)" "......(2)`
(wave move toward left)
By the principle of superposition, both will interfere with each other the net displacement is
`y=y_(1)+y_(2)" "......(3)`
Substituting equation (1) and equation (2) in equation (3), we get
`{{:(Asin(kx-omegat)),(+Asin(kx+omegat)):}" "......(4)`
Using trigonometric identity, we rewrite equation (4) as
`y(x,t)=2Acos(omegat)sin(kx)" "......(5)`
It represents a stationary wave or standing wave, that means that this wave does not move either forward or backward, whereas progressive or travelling waves will move forward or backward. In addition, the displacement of the paricle in equation (5) can be written in more compact form,
`y(x,t)=A'cos(omegat)`
where, `A'=2Asin(kx)`, It is implied 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`
`implieskx=(pi)/(2),(3pi)/(2),(5pi)/(2).......`
`=mpi`
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))(lamda)/(2)" "......(6)`
where m=0,1,2....
For m=0 we have maximum at,
`x_(0)=(lamda)/(2)`
For m=1 we have maximum at,
`x_(1)=(3lamda)/(4)`
For m=2 we have maximum at,
`x_(2)=(5lamda)/(4)`
and so on.
The distance between two successive antinodes can be calculated by
`x_(m)-x_(m-1)=((2m+1)/(2))(lamda)/(2)-(((2m+1)+1)/(2))(lamda)/(2)`
`=(lamda)/(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`
`implieskx=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 nodes. The `n^(th)` nodal positions is given by,
`x_(n)=n(lamda)/(2)` where, n=0,1,2,.....