What are stationary waves ? Obtain its equation.

Explanation

Stationary waves : "When two waves of same amplitude, same time period, same frequency, same wavelength are travelling in opposite directions on the same line superpossed, then resultant wave formed is called stationary Waves.
OR
When two waves having the same amplitude and frequency (i.e. wavelength) and travelling in mutually opposite directions are superposed, the resultant wave formed loses the property of propagation and a stationary pattern is created in the medium. Such a wave is called a stationary wave.
These resultant waves cannot move in any direction. Hence, they cannot conduct energy.
Suppose, a resultant wave, moving in negative x-direction obtained by reflection of wave moving in positive x-direction of same wavelength and amplitude. Suppose, initial phase `phi =0` which is shown in figure,

`therefore y _(1) (x,t) = a sin (kx - omega t )` (Wave travelling in the positive direction of x-axis)
`y _(2) (x,t) = a sin (kx + omega t)` (Wave travelling in the negative direction of x-axis)
The principle of superpostion gives, for the combined wave,
`y (x,t) =y _(1) (x,t) +y _(2) (x,t)`
`= a sin (kx - omega t ) + a sin (kx + omegat )`
`=a [2 sin [ ((kx - omega t ) + (kx + omega t ))/( 2)]`
`cos [ ((kx - omegat ) - (kx + omega t ))/( 2) ]]`
`= 2 a sin kx cos (-omega t )`
`therefore y (x,t) =2 a sin kx cos omega t " "...(1)`
`[ because cos [-theta)=cos theta]`
`[because sin (A+B) + sin (A-B) =2 sin A cos B]`
Equation (1) is general equation of stationary wave.
Amplitude of stationary wave is 2a sinkx and different points are oscillating with different amplitude.
kx and ot are different terms but not combined. Hence, stationary wave is not function of `f(kx - omegat)` Hence, they are not progressive waves.
Wave pattern is not moving on left or right hand so, that it is called stationary wave.